http://apcocoa.uni-passau.de/wiki/api.php?action=feedcontributions&user=132.231.10.53&feedformat=atomApCoCoAWiki - User contributions [en]2024-03-29T01:41:37ZUser contributionsMediaWiki 1.35.0http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.L01PSolve&diff=12245ApCoCoA-1:GLPK.L01PSolve2011-08-23T12:16:26Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>GLPK.L01PSolve</title><br />
<short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.L01PSolve(F:LIST, CuttingNumber:INT, QStrategy:INT, CStrategy:INT, MinMax:STRING):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It operates in two stages. Firstly, it models the problem of finding one solution of given polynomial system into a mixed integer linear programming problem. For this the system is first converted into an equivalent CNF form and then the CNF form is converted into a system of equalities and inequalities. Secondly, the mixed integer linear programming model is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>CuttingNumber</em>: Maximal support-length of the linear polynomials for conversion to CNF. The possible value could be from 3 to 6. </item><br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
<item>@return A list containing a zero of the system F.</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
CuttingNumber:=6;<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Converting to CNF with CuttingLength: 6, QStrategy: Standard, CStrategy: Standard.<br />
Converting CNF to system of equalities and inequalities...<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 2<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
CuttingNumber:=6;<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Converting to CNF with CuttingLength: 6, QStrategy: LinearPartner, CStrategy: Standard.<br />
Converting CNF to system of equalities and inequalities...<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 4<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
CuttingNumber:=5;<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Converting to CNF with CuttingLength: 5, QStrategy: Standard, CStrategy: CubicParnterDegree2.<br />
Converting CNF to system of equalities and inequalities...<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 1<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>l01psolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.l01pSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.RPCSolve&diff=12244ApCoCoA-1:GLPK.RPCSolve2011-08-23T12:15:36Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>GLPK.RPCSolve</title><br />
<short_description>Solves a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.RPCSolve(F:LIST, QStrategy:INT, CStrategy:INT, MinMax:STRING):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It uses Real Polynomial Conversion (RPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modelled mixed integer linear programming problem is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
<item>@return A list containing a zero of the system F.</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 2<br />
<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: LinearPartner, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 4<br />
<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: CubicParnterDegree2.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 1<br />
<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>rpcsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.rpcsolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.IPCSolve&diff=12243ApCoCoA-1:GLPK.IPCSolve2011-08-23T12:14:31Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>GLPK.IPCSolve</title><br />
<short_description>Solves a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.IPCSolve(F:LIST, QStrategy:INT, CStrategy:INT, MinMax:STRING):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It uses Integer Polynomial Conversion (IPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modelled mixed integer linear programming problem is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
<item>@return A list containing a zero of the system F.</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 2<br />
<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: LinearPartner, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 4<br />
<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: CubicParnterDegree2.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 1<br />
<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>ipcsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.ipcsolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Hom.LRSolve&diff=12236ApCoCoA-1:Hom.LRSolve2011-08-18T08:46:03Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>Hom.LRSolve</title><br />
<short_description>Solves a non-square zero dimensional homogeneous or non-homogeneous polynomial system of equations.</short_description><br />
<syntax><br />
Hom.LRSolve(P:LIST,HomTyp:INT)<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function computes isolated solutions of a non-square polynomial system by using the idea of randomization. Consider the non-square polynomial system <tt>F:=[ x^2-1, xy-1, x^2-x ]</tt>. After randomization this system will be converted into a system <tt>G := [ x^2-1+a(x^2-x), xy-1+b(x^2-x) ]</tt>, where a and b are complex numbers having absolute value near one. The system G is a randomization of system F. Sometimes the system G may have more solutions than F. This function first randomizes the given system to make it square and then call HOM4PS to solve it. This function and the function <ref>Hom.SRSolve</ref> do the same job but with a different technique of randomization. <br />
<br />
This functions provides two different kinds of computation depending on the input that you provide in ApCoCoAServer during execution. After passing the command <tt>Hom.LRSolve(P,HomTyp)</tt> in ApCoCoA you need to interact with ApCoCoAServer. At this stage ApCoCoAServer asks you to enter 1 for the polyhedral homotopy and enter 2 for the classical linear homotopy. As a result this function provides all isolated solutions of a zero dimensional system of polynomial equations. The system of polynomials may be homogeneous or non-homogeneous.<br />
<itemize><br />
<item>@param <em>P</em>: List of polynomials of the given system.</item><br />
<item>@param <em>HomTyp</em>: set it to 1 for polyhedral homotopy and to 2 for classical linear homotopy.</item><br />
<item>@return A list of lists containing the finite solutions of the system P.</item><br />
<br />
</itemize><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the classical linear homotopy.<br />
-- We want to find isolated solutions of the following system. <br />
<br />
Use QQ[x[1..3]]; <br />
P := [<br />
x[1]x[2]x[3] - x[1]x[2]-15, <br />
3x[1]x[2]-x[1]+5, <br />
7x[1]x[3] - x[1],<br />
24x[1]x[2]+x[3] - 3x[1]x[3] - 1, <br />
x[1]^2 - x[1] <br />
];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.LRSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- Since we want to use the classical linear homotopy therefore we enter 2.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[9455327382203569/5000000000000000, -25208009777282481/10000000000000000],<br />
[13172347071045859/1000000000000000000, 780259255441451/10000000000000000],<br />
[50662103933981573/100000000000000000, 24894084616179979/50000000000000000]],<br />
[[94045825811783779/10000000000000000000, -18561325122258089/500000000000000000],<br />
[-11252856171103929/500000000000000, 53756347909614881/10000000000000000],<br />
[43866568184785617/10000000000000000, -970984718484509/40000000000000]],<br />
[[23564339009933287/1000000000000000000, 37422202697036111/1000000000000000000],<br />
[-20929334925895049/1000000000000000, -24991129623196171/2500000000000000],<br />
[26847721395327557/10000000000000000, 20456859352398073/1000000000000000]],<br />
[[-5340666810400797/10000000000000000, 7138058708108771/2500000000000000],<br />
[157412137424673/4000000000000000, -15131835631465503/250000000000000000],<br />
[45533206002984217/1000000000000000000, -67237130550938307/100000000000000000]],<br />
[[2223557602823067/10000000000000, -19326230622413977/250000000000000],<br />
[-15392736087963673/2000000000000000, 18511778667155307/200000000000000000],<br />
[-25906948948013323/1000000000000000, 3338667600178357/50000000000000]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number.<br />
<br />
</example><br />
<br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to find isolated solutions of non-homogeneous polynomial system x[1]^2-1=0, x[1]x[2]-1=0, x[1]^2-x[1]=0. <br />
<br />
Use QQ[x[1..2]]; <br />
P := [x[1]^2-1, x[1]x[2]-1,x[1]^2-x[1]];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.LRSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- Since we want to use polyhedral homotopy therefore we enter 1.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[-9143436298249491/20000000000000000, 9937657539108147/50000000000000000],<br />
[-24282046571107613/50000000000000000, 18641461485865229/100000000000000000]],<br />
[[1, 0], [1, 0]]<br />
]<br />
<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 2 solutions the second solution is [[1, 0], [1, 0]] <br />
-- and we read it as x=2+0i, y=1+0i. Since imaginary part is zero therefore its a real solution. <br />
<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Hom.HSolve</see><br />
<see>Hom.SRSolve</see><br />
</seealso><br />
<br />
<key>lrsolve</key><br />
<key>hom.lrsolve</key><br />
<key>hom4ps.lrsolve</key><br />
<key>solve zero dimensional polynomial system</key><br />
<wiki-category>Package_hom4ps</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Hom.HSolve&diff=12235ApCoCoA-1:Hom.HSolve2011-08-18T08:42:49Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>Hom.HSolve</title><br />
<short_description>Solves a zero dimensional square homogeneous or non-homogeneous polynomial system of equations.</short_description><br />
<syntax><br />
Hom.HSolve(P:LIST,HomTyp:INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function can do two kinds of different computations depending on the input that you provide in ApCoCoAServer during execution. After passing the command <tt>Hom.HSolve(P)</tt> in ApCoCoA you need to interact with ApCoCoAServer. At this stage ApCoCoAServer asks you to enter 1 for the polyhedral homotopy and to enter 2 for the classical linear homotopy. As a result this function provides all isolated solutions of a zero dimensional system of polynomial equations. The system of polynomials may be homogeneous or non-homogeneous.<br />
<itemize><br />
<item>@param <em>P</em>: List of polynomials of the given system.</item><br />
<item>@param <em>HomTyp</em>: set it to 1 for polyhedral homotopy and to 2 for classical linear homotopy.</item><br />
<item>@return A list of lists containing the finite solutions of the system P.</item><br />
<br />
</itemize><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0. <br />
<br />
Use S ::= QQ[x,y]; <br />
P := [x^2+y^2-5, xy-2];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.HSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- If we enter 1 then the all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[2, 0], [1, 0]],<br />
[[-1, 0], [-2, 0]],<br />
[[-2, 0], [-1, 0]],<br />
[[1, 0], [2, 0]]<br />
]<br />
<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 4 solutions the first solution is [[2, 0], [1, 0]] <br />
-- and we read it as x=2+0i, y=1+0i <br />
<br />
</example><br />
<example><br />
-- An example of zero dimensional Homogeneous Solving using the classical linear homotopy.<br />
-- We want to solve zero dimensional homogeneous system x^2-y^2=0, xy-y^2=0.<br />
<br />
Use S ::= QQ[x,y]; <br />
M := [x^2-y^2, xy-y^2];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.HSolve(M,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- If we enter 2 then the all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000], <br />
[20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000]], <br />
[[0, 0], [0, 0]],<br />
[[0, 0], [0, 0]], <br />
[[-60689558229793541/10000000000000000000000000, 245542879738863/2000000000000000000000], <br />
[-3034482281801981/500000000000000000000000, 3069286290270979/25000000000000000000000]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 4 solutions the first solution is <br />
-- [[20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000], <br />
-- [20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000]] <br />
-- and we read it as x=20597740658111043/500000000000000000000000 - 74403123210058537/1000000000000000000000000i,<br />
-- y = 20597740658111043/500000000000000000000000 - 74403123210058537/1000000000000000000000000i <br />
<br />
<br />
------------------------------------<br />
</example><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to solve zero dimensional non-homogeneous system x[1]^2-1=0, x[1]x[2]-1=0. <br />
<br />
Use QQ[x[1..2]]; <br />
P := [x[1]^2-1, x[1]x[2]-1];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.HSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- We enter 1 because we want to use polyhedral homotopy.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[1, 0], [1, 0]],<br />
[[-1, 0], [-1, 0]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 2 solutions the first solution is [[1, 0], [1, 0]] <br />
-- and we read it as x[1]=1+0i, x[2]=1+0i <br />
<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Hom.LRSolve</see><br />
<see>Hom.SRSolve</see><br />
</seealso><br />
<br />
<key>hsolve</key><br />
<key>hom4ps.hsolve</key><br />
<key>hom.hsolve</key><br />
<key>solve zero dimensional polynomial system</key><br />
<wiki-category>Package_hom4ps</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Hom.HSolve&diff=12234ApCoCoA-1:Hom.HSolve2011-08-18T08:41:46Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>Hom.HSolve</title><br />
<short_description>Solves a zero dimensional square homogeneous or non-homogeneous polynomial system of equations.</short_description><br />
<syntax><br />
Hom.HSolve(P:LIST,HomTyp:INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function can do two kinds of different computations depending on the input that you provide in ApCoCoAServer during execution. After passing the command <tt>Hom.HSolve(P)</tt> in ApCoCoA you need to interact with ApCoCoAServer. At this stage ApCoCoAServer asks you to enter 1 for the polyhedral homotopy and to enter 2 for the classical linear homotopy. As a result this function provides all isolated solutions of a zero dimensional system of polynomial equations. The system of polynomials may be homogeneous or non-homogeneous.<br />
<itemize><br />
<item>@param <em>P</em>: List of polynomials of the given system.</item><br />
<item>@param <em>HomTyp</em>: set it to 1 for polyhedral homotopy and to 2 for classical linear homotopy.</item><br />
<item>@return A list of lists containing the finite solutions of the system P.</item><br />
<br />
</itemize><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0. <br />
<br />
Use S ::= QQ[x,y]; <br />
P := [x^2+y^2-5, xy-2];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.HSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- If we enter 1 then the all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[2, 0], [1, 0]],<br />
[[-1, 0], [-2, 0]],<br />
[[-2, 0], [-1, 0]],<br />
[[1, 0], [2, 0]]<br />
]<br />
<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 4 solutions the first solution is [[2, 0], [1, 0]] <br />
-- and we read it as x=2+0i, y=1+0i <br />
<br />
</example><br />
<example><br />
-- An example of zero dimensional Homogeneous Solving using the classical linear homotopy.<br />
-- We want to solve zero dimensional homogeneous system x^2-y^2=0, xy-y^2=0.<br />
<br />
Use S ::= QQ[x,y]; <br />
M := [x^2-y^2, xy-y^2];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.HSolve(M,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- If we enter 2 then the all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000], <br />
[20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000]], <br />
[[0, 0], [0, 0]],<br />
[[0, 0], [0, 0]], <br />
[[-60689558229793541/10000000000000000000000000, 245542879738863/2000000000000000000000], <br />
[-3034482281801981/500000000000000000000000, 3069286290270979/25000000000000000000000]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 4 solutions the first solution is <br />
-- [[20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000], <br />
-- [20597740658111043/500000000000000000000000, -74403123210058537/1000000000000000000000000]] <br />
-- and we read it as x=20597740658111043/500000000000000000000000 - 74403123210058537/1000000000000000000000000i,<br />
-- y = 20597740658111043/500000000000000000000000 - 74403123210058537/1000000000000000000000000i <br />
<br />
<br />
------------------------------------<br />
</example><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to solve zero dimensional non-homogeneous system x[1]^2-1=0, x[1]x[2]-1=0. <br />
<br />
Use QQ[x[1..2]]; <br />
P := [x[1]^2-1, x[1]x[2]-1];<br />
<br />
-- Then we compute the solution with<br />
Hom.HSolve(P);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- We enter 1 because we want to use polyhedral homotopy.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[1, 0], [1, 0]],<br />
[[-1, 0], [-1, 0]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 2 solutions the first solution is [[1, 0], [1, 0]] <br />
-- and we read it as x[1]=1+0i, x[2]=1+0i <br />
<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Hom.LRSolve</see><br />
<see>Hom.SRSolve</see><br />
</seealso><br />
<br />
<key>hsolve</key><br />
<key>hom4ps.hsolve</key><br />
<key>hom.hsolve</key><br />
<key>solve zero dimensional polynomial system</key><br />
<wiki-category>Package_hom4ps</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Hom.SRSolve&diff=12233ApCoCoA-1:Hom.SRSolve2011-08-18T08:38:58Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>Hom.SRSolve</title><br />
<short_description>Solves a non-square zero dimensional homogeneous or non-homogeneous polynomial system of equations.</short_description><br />
<syntax><br />
Hom.SRSolve(P:LIST,HomTyp:INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function computes isolated solutions of a non-square polynomial system by using the idea of randomization. Consider the non-square polynomial system <tt>F:=[ x^2-1, xy-1, x^2-x ]</tt>. After randomization this system will be converted into a system <tt>G := [ x^2-1+as, xy-1+bs, x^2+x+cs ]</tt>, where a, b and c are complex numbers having absolute value near one and s is a slack variable. The system G is a randomization of system F. Some times the system G may have more solutions than F. This function first randomizes the given system to make it square and then call HOM4PS to solve it. This function and the function <ref>Hom.LRSovle</ref> do the same job but with a different technique of randomization. <br />
<br />
This function provides two different kinds of computation depending on the input that you provide in ApCoCoAServer during execution. After passing the command <tt>Hom.SRSolve(P,HomTyp)</tt> in ApCoCoA you need to interact with ApCoCoAServer. At this stage ApCoCoAServer asks you to enter 1 for the polyhedral homotopy and enter 2 for the classical linear homotopy. As a result this function provides all isolated solutions of a zero dimensional system of polynomial equations. The system of polynomials may be homogeneous or non-homogeneous.<br />
<itemize><br />
<item>@param <em>P</em>: List of polynomials of the given system.</item><br />
<item>@param <em>HomTyp</em>: set it to 1 for polyhedral homotopy and to 2 for classical linear homotopy.</item><br />
<item>@return A list of lists containing the finite solutions of the system P.</item><br />
<br />
</itemize><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the classical linear homotopy.<br />
-- We want to find isolated solutions of the following system. <br />
<br />
Use QQ[x[1..3]]; <br />
P := [<br />
x[1]x[2]x[3] - x[1]x[2]-15, <br />
3x[1]x[2]-x[1]+5, <br />
7x[1]x[3] - x[1],<br />
24x[1]x[2]+x[3] - 3x[1]x[3] - 1, <br />
x[1]^2 - x[1] <br />
];<br />
<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.SRSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- Since we want to use the classical linear homotopy therefore we enter 2.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[-2902316230611307/1250000000000000, -27857718907640603/10000000000000000],<br />
[-5674959881126967/500000000000000000, -8937804463608219/25000000000000000],<br />
[46724750476097837/100000000000000000, -38716232173770071/100000000000000000]],<br />
[[65920011696250427/1000000000000000, -1372011739419221/12500000000000], <br />
[84785163919836641/10000000000000000, -10345947531705213/25000000000000000],<br />
[12838340038652873/1000000000000000, 1179955721096759/156250000000000]],<br />
[[-11200479653149161/50000000000000000000, -1847541166671739/4000000000000000000],<br />
[-398464307671313/250000000000, -83690324485203917/100000000000000],<br />
[27034776057405041/100000000000000000, -16633293038412589/1000000000000000]],<br />
[[1283479859536169/400000000000000, 226533750215299/80000000000000],<br />
[-739601189373527/40000000000000000, 10474383104189437/50000000000000000],<br />
[-4621056904224851/25000000000000000, 6394233590549897/25000000000000000]],<br />
[[4510213996339667/12500000000000000000, 551766683622709/4000000000000000000],<br />
[-26177318181795687/50000000000000, -3490003097264787/2000000000000],<br />
[31612616513119707/10000000000000000, 21347082990880249/1000000000000000]]<br />
]<br />
<br />
-- The smallest list represents a complex number.<br />
<br />
</example><br />
<br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to find isolated solutions of non-homogeneous polynomial system x[1]^2-1=0, x[1]x[2]-1=0, x[1]^2-x[1]=0. <br />
<br />
Use QQ[x[1..2]]; <br />
P := [x[1]^2-1, x[1]x[2]-1,x[1]^2-x[1]];<br />
HomTyp:=1;<br />
<br />
-- Then we compute the solution with<br />
Hom.SRSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- Since we want to use polyhedral homotopy therefore we enter 1.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[-51917361941691031/100000000000000000, -1846796377886887/4000000000000000],<br />
[-14765467180940843/10000000000000000, 23807586810196137/10000000000000000]],<br />
[[1, 0], [1, 0]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 2 solutions the second solution is [[1, 0], [1, 0]] <br />
-- and we read it as x=2+0i, y=1+0i. Since imaginary part is zero therefore its a real solution. <br />
<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Hom.HSolve</see><br />
<see>Hom.LRSolve</see><br />
</seealso><br />
<br />
<key>srsolve</key><br />
<key>hom.srsolve</key><br />
<key>hom4ps.srsolve</key><br />
<key>solve zero dimensional polynomial system</key><br />
<wiki-category>Package_hom4ps</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Hom.SRSolve&diff=12232ApCoCoA-1:Hom.SRSolve2011-08-18T08:37:28Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>Hom.SRSolve</title><br />
<short_description>Solves a non-square zero dimensional homogeneous or non-homogeneous polynomial system of equations.</short_description><br />
<syntax><br />
Hom.SRSolve(P:LIST,HomTyp)<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function computes isolated solutions of a non-square polynomial system by using the idea of randomization. Consider the non-square polynomial system <tt>F:=[ x^2-1, xy-1, x^2-x ]</tt>. After randomization this system will be converted into a system <tt>G := [ x^2-1+as, xy-1+bs, x^2+x+cs ]</tt>, where a, b and c are complex numbers having absolute value near one and s is a slack variable. The system G is a randomization of system F. Some times the system G may have more solutions than F. This function first randomizes the given system to make it square and then call HOM4PS to solve it. This function and the function <ref>Hom.LRSovle</ref> do the same job but with a different technique of randomization. <br />
<br />
This function provides two different kinds of computation depending on the input that you provide in ApCoCoAServer during execution. After passing the command <tt>Hom.SRSolve(P,HomTyp)</tt> in ApCoCoA you need to interact with ApCoCoAServer. At this stage ApCoCoAServer asks you to enter 1 for the polyhedral homotopy and enter 2 for the classical linear homotopy. As a result this function provides all isolated solutions of a zero dimensional system of polynomial equations. The system of polynomials may be homogeneous or non-homogeneous.<br />
<itemize><br />
<item>@param <em>P</em>: List of polynomials of the given system.</item><br />
<item>@param <em>HomTyp</em>: set it to 1 for polyhedral homotopy and to 2 for classical linear homotopy.</item><br />
<item>@return A list of lists containing the finite solutions of the system P.</item><br />
<br />
</itemize><br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the classical linear homotopy.<br />
-- We want to find isolated solutions of the following system. <br />
<br />
Use QQ[x[1..3]]; <br />
P := [<br />
x[1]x[2]x[3] - x[1]x[2]-15, <br />
3x[1]x[2]-x[1]+5, <br />
7x[1]x[3] - x[1],<br />
24x[1]x[2]+x[3] - 3x[1]x[3] - 1, <br />
x[1]^2 - x[1] <br />
];<br />
<br />
-- Then we compute the solution with<br />
Hom.SRSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- Since we want to use the classical linear homotopy therefore we enter 2.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[-2902316230611307/1250000000000000, -27857718907640603/10000000000000000],<br />
[-5674959881126967/500000000000000000, -8937804463608219/25000000000000000],<br />
[46724750476097837/100000000000000000, -38716232173770071/100000000000000000]],<br />
[[65920011696250427/1000000000000000, -1372011739419221/12500000000000], <br />
[84785163919836641/10000000000000000, -10345947531705213/25000000000000000],<br />
[12838340038652873/1000000000000000, 1179955721096759/156250000000000]],<br />
[[-11200479653149161/50000000000000000000, -1847541166671739/4000000000000000000],<br />
[-398464307671313/250000000000, -83690324485203917/100000000000000],<br />
[27034776057405041/100000000000000000, -16633293038412589/1000000000000000]],<br />
[[1283479859536169/400000000000000, 226533750215299/80000000000000],<br />
[-739601189373527/40000000000000000, 10474383104189437/50000000000000000],<br />
[-4621056904224851/25000000000000000, 6394233590549897/25000000000000000]],<br />
[[4510213996339667/12500000000000000000, 551766683622709/4000000000000000000],<br />
[-26177318181795687/50000000000000, -3490003097264787/2000000000000],<br />
[31612616513119707/10000000000000000, 21347082990880249/1000000000000000]]<br />
]<br />
<br />
-- The smallest list represents a complex number.<br />
<br />
</example><br />
<br />
<br />
<example><br />
-- An example of zero dimensional Non-Homogeneous Solving using the polyhedral homotopy.<br />
-- We want to find isolated solutions of non-homogeneous polynomial system x[1]^2-1=0, x[1]x[2]-1=0, x[1]^2-x[1]=0. <br />
<br />
Use QQ[x[1..2]]; <br />
P := [x[1]^2-1, x[1]x[2]-1,x[1]^2-x[1]];<br />
<br />
-- Then we compute the solution with<br />
Hom.SRSolve(P,HomTyp);<br />
<br />
-- Now you have to interact with ApCoCoAServer<br />
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.<br />
-- Since we want to use polyhedral homotopy therefore we enter 1.<br />
-- The all finite solutions are:<br />
<br />
----------------------------------------<br />
[<br />
[[-51917361941691031/100000000000000000, -1846796377886887/4000000000000000],<br />
[-14765467180940843/10000000000000000, 23807586810196137/10000000000000000]],<br />
[[1, 0], [1, 0]]<br />
]<br />
<br />
<br />
-- The smallest list represents a complex number. For example above system has 2 solutions the second solution is [[1, 0], [1, 0]] <br />
-- and we read it as x=2+0i, y=1+0i. Since imaginary part is zero therefore its a real solution. <br />
<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Hom.HSolve</see><br />
<see>Hom.LRSolve</see><br />
</seealso><br />
<br />
<key>srsolve</key><br />
<key>hom.srsolve</key><br />
<key>hom4ps.srsolve</key><br />
<key>solve zero dimensional polynomial system</key><br />
<wiki-category>Package_hom4ps</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMBBasisF2&diff=12023ApCoCoA-1:CharP.IMBBasisF22011-04-28T13:59:37Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.IMBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.IMBBasisF2(F:LIST):LIST<br />
CharP.IMBBasisF2(F:LIST, NSol: INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses improved mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item><br />
<item>@return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute a Border Basis with<br />
CharP.IMBBasisF2(F);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Total No. of Mutants are = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=15<br />
Total No. of Mutants are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Total No. of Mutants are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=15<br />
Total No. of Mutants are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=17<br />
No. of Columns=15<br />
Total No. of Mutants are = 1<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=17<br />
No. of Columns=15<br />
Total No. of Mutants are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=18<br />
No. of Columns=15<br />
Total No. of Mutants are = 0<br />
<br />
[x[4] + 1, x[3], x[2] + 1, x[1]]<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
NSol:=3;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
-- Compute the solution with<br />
CharP.IMBBasisF2(F,NSol);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=14<br />
Total No. of Mutants are = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=14<br />
[x[3]x[4] + x[4], x[1]x[4] + x[1], x[1]x[3] + x[1], x[1]x[2] + x[1], x[2]x[3]x[4] + x[4], x[1]x[2]x[4] + x[1]]<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MBBasisF2</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMBBasisF2&diff=12022ApCoCoA-1:CharP.IMBBasisF22011-04-28T13:38:42Z<p>132.231.10.53: New page: <command> <title>CharP.IMBBasis</title> <short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description> <syntax> CharP.IMBBasisF2(F:LIST):LIST ...</p>
<hr />
<div><command><br />
<title>CharP.IMBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.IMBBasisF2(F:LIST):LIST<br />
CharP.IMBBasisF2(F:LIST, NSol: INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item><br />
<item>@return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute a Border Basis with<br />
CharP.MBBasisF2(F);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
No. of mutants found =1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
No. of mutants found =2<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=11<br />
No. of mutants found =0<br />
The size of Matrix is:<br />
No. of Rows=31<br />
No. of Columns=15<br />
No. of mutants found =0<br />
<br />
[x[4] + 1, x[3], x[2] + 1, x[1]]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
NSol:=3;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
-- Compute the solution with<br />
CharP.MBBasisF2(F,NSol);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=15<br />
[x[3]x[4] + x[4], x[1]x[4] + x[1], x[1]x[3] + x[1], x[1]x[2] + x[1], x[2]x[3]x[4] + x[4], x[1]x[2]x[4] + x[1]]<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.IMBBasisF2</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MBBasisF2&diff=12021ApCoCoA-1:CharP.MBBasisF22011-04-28T13:37:20Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.MBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.MBBasisF2(F:LIST):LIST<br />
CharP.MBBasisF2(F:LIST, NSol: INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item><br />
<item>@return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute a Border Basis with<br />
CharP.MBBasisF2(F);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
No. of mutants found =1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
No. of mutants found =2<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=11<br />
No. of mutants found =0<br />
The size of Matrix is:<br />
No. of Rows=31<br />
No. of Columns=15<br />
No. of mutants found =0<br />
<br />
[x[4] + 1, x[3], x[2] + 1, x[1]]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
NSol:=3;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
-- Compute the solution with<br />
CharP.MBBasisF2(F,NSol);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=15<br />
[x[3]x[4] + x[4], x[1]x[4] + x[1], x[1]x[3] + x[1], x[1]x[2] + x[1], x[2]x[3]x[4] + x[4], x[1]x[2]x[4] + x[1]]<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.IMBBasisF2</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MBBasisF2&diff=12020ApCoCoA-1:CharP.MBBasisF22011-04-28T13:26:42Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.MBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.MBBasisF2(F:LIST):LIST<br />
CharP.MBBasisF2(F:LIST, NSol: INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item><br />
<item>@return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute a Border Basis with<br />
CharP.MBBasisF2(F);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
No. of mutants found =1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
No. of mutants found =2<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=11<br />
No. of mutants found =0<br />
The size of Matrix is:<br />
No. of Rows=31<br />
No. of Columns=15<br />
No. of mutants found =0<br />
<br />
[x[4] + 1, x[3], x[2] + 1, x[1]]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
NSol:=3;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
-- Compute the solution with<br />
CharP.MBBasisF2(F,NSol);<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=15<br />
[x[3]x[4] + x[4], x[1]x[4] + x[1], x[1]x[3] + x[1], x[1]x[2] + x[1], x[2]x[3]x[4] + x[4], x[1]x[2]x[4] + x[1]]<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.IMXLSolve</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MBBasisF2&diff=12019ApCoCoA-1:CharP.MBBasisF22011-04-28T13:22:18Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.MBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.MBBasisF2(F:LIST):LIST<br />
CharP.MBBasisF2(F:LIST, NSol: INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item><br />
<item>@return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute a Border Basis with<br />
CharP.MBBasisF2(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
No. of mutants found =1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
No. of mutants found =2<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=11<br />
No. of mutants found =0<br />
The size of Matrix is:<br />
No. of Rows=31<br />
No. of Columns=15<br />
No. of mutants found =0<br />
<br />
[x[4] + 1, x[3], x[2] + 1, x[1]]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MBBasisF2(F,NSol);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.IMXLSolve</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MBBasisF2&diff=12018ApCoCoA-1:CharP.MBBasisF22011-04-28T13:18:08Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.MBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.MBBasisF2(F:LIST):LIST<br />
CharP.MBBasisF2(F:LIST, NSol: INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span.<br />
<br />
If you want to use the second version with the parameter NSol, you need to provide the exact number of <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions.<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item><br />
<item>@return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.MBBasisF2(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MBBasisF2(F,NSol);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.IMXLSolve</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MBBasisF2&diff=12017ApCoCoA-1:CharP.MBBasisF22011-04-28T12:50:03Z<p>132.231.10.53: New page: <command> <title>CharP.MBBasis</title> <short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description> <syntax> CharP.MBBasisF2(F:LIST):LIST </...</p>
<hr />
<div><command><br />
<title>CharP.MBBasis</title><br />
<short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description><br />
<syntax><br />
CharP.MBBasisF2(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
Let <tt>f_1</tt>, ... , <tt>f_m</tt> is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials.<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials.</item><br />
<item>@return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.MXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[0, 1, 0, 1]<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.IMXLSolve</see> <br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mxlsolve</key><br />
<key>mxlsolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMNLASolve&diff=12016ApCoCoA-1:CharP.IMNLASolve2011-04-28T11:42:52Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.IMNLASolve</title><br />
<short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.IMNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2</tt>. It uses <tt>I</tt>mproved <tt>M</tt>utant <tt>NLA</tt>-Algorithm to find the unique zero. The Improved Mutant <tt>NLA</tt>-Algorithm generates a sequence of linear systems to solve the given system. The Improved Mutant <tt>NLA</tt>-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Improved Mutant NLA-Algorithm is the NLA-Algorithm with improved mutant strategy. It uses <ref>LinAlg.EF</ref> for gaussian elimination.<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return Possibly the unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.IMNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 1<br />
The total No. of Mutants found are = 1<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=10<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.IMNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=8<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=8<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=14<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 2<br />
The total No. of Mutants found are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=14<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=14<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=13<br />
No. of Columns=10<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=10<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=24<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=24<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.MNLASolve</see><br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.imnlasolve</key><br />
<key>imnlasolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMNLASolve&diff=12015ApCoCoA-1:CharP.IMNLASolve2011-04-28T11:40:56Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.IMNLASolve</title><br />
<short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.IMNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2</tt>. It uses <tt>I</tt>mproved <tt>M</tt>utant <tt>NLA</tt>-Algorithm to find the unique zero. The Improved Mutant <tt>NLA</tt>-Algorithm generates a sequence of linear systems to solve the given system. The Improved Mutant <tt>NLA</tt>-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Improved Mutant NLA-Algorithm is the NLA-Algorithm with improved mutant strategy. It uses <ref>LinAlg.EF</ref> for gaussian elimination.<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return Possibly the unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.IMNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Applying Gaussian Elimination for finding Mutants...<br />
Gaussian Elimination Compeleted.<br />
No. of New Mutants found = 1<br />
The total No. of Mutants found are = 1<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=10<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.IMNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 2<br />
The total No. of Mutants found are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=10<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=10<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=22<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.MNLASolve</see><br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.imnlasolve</key><br />
<key>imnlasolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MNLASolve&diff=12014ApCoCoA-1:CharP.MNLASolve2011-04-28T11:28:21Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.MNLASolve</title><br />
<short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA-Algorithm to find the unique zero. The Mutant NLA-Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA-Algorithm is the NLA-Algorithm with mutant strategy. It uses <ref>LinAlg.EF</ref> for gaussian elimination.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=10<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
<br />
[0, 1, 0, 1]<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=16<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=16<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=28<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=28<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=20<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=20<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mnlasolve</key><br />
<key>mnlasolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MNLASolve&diff=12013ApCoCoA-1:CharP.MNLASolve2011-04-28T11:24:51Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.MNLASolve</title><br />
<short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA-Algorithm to find the unique zero. The Mutant NLA-Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA-Algorithm is the NLA-Algorithm with mutant strategy. It uses <ref>LinAlg.EF</ref> for gaussian elimination.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=9<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Applying Gaussian Elimination finding Muatants...<br />
Gaussian Elimination Compeleted<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=12<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=10<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=5<br />
Applying Gaussian Elimination to check solution coordinate...<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
<br />
[0, 1, 0, 1]<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>poly_system</type><br />
</types><br />
<br />
<key>charP.mnlasolve</key><br />
<key>mnlasolve</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.RPCSolve&diff=12012ApCoCoA-1:GLPK.RPCSolve2011-04-27T14:15:35Z<p>132.231.10.53: New page: <command> <title>GLPK.RPCSolve</title> <short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description> <syntax> GLPK.RPC...</p>
<hr />
<div><command><br />
<title>GLPK.RPCSolve</title><br />
<short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.RPCSolve(F:LIST, QStrategy:INT, CStrategy:INT, MinMax:STRING)<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It uses Real Polynomial Conversion (RPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modeled mixed integer linear programming problem is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 2<br />
<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: LinearPartner, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 4<br />
<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: CubicParnterDegree2.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 1<br />
<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>rpcsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.l01pSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.IPCSolve&diff=12011ApCoCoA-1:GLPK.IPCSolve2011-04-27T14:08:20Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>GLPK.IPCSolve</title><br />
<short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.IPCSolve(F:LIST, QStrategy:INT, CStrategy:INT, MinMax:STRING)<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It uses Integer Polynomial Conversion (IPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modeled mixed integer linear programming problem is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 2<br />
<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: LinearPartner, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 4<br />
<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: CubicParnterDegree2.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 1<br />
<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>ipcsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.l01pSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.IPCSolve&diff=12010ApCoCoA-1:GLPK.IPCSolve2011-04-27T14:06:11Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>GLPK.IPCSolve</title><br />
<short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.IPCSolve(F:LIST, QStrategy:INT, CStrategy:INT, MinMax:STRING)<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It uses Integer Polynomial Conversion (IPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modeled mixed integer linear programming problem is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: LinearPartner, CStrategy: Standard.<br />
Model is ready to solve with GLPK...<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 4<br />
<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
Modelling the system as a mixed integer programming problem. <br />
QStrategy: Standard, CStrategy: CubicParnterDegree2.<br />
Model is ready to solve with GLPK...<br />
<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 1<br />
<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>ipcsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.l01pSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.IPCSolve&diff=12009ApCoCoA-1:GLPK.IPCSolve2011-04-27T13:24:45Z<p>132.231.10.53: New page: <command> <title>GLPK.IPCSolve</title> <short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description> <syntax> GLPK.IPC...</p>
<hr />
<div><command><br />
<title>GLPK.IPCSolve</title><br />
<short_description>Solve a system of polynomial equations over <tt>F_2</tt> for one solution in <tt>F_2^n</tt>.</short_description><br />
<syntax><br />
GLPK.IPCSolve(F:LIST, QStrategy:INT, CStrategy:INT, MinMax:STRING)<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function finds one solution in <tt>F_2^n</tt> of a system of polynomial equations over the field <tt>F_2</tt>. It uses Integer Polynomial Conversion (IPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modeled mixed integer linear programming problem is solved using glpk. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F</em>: A List containing the polynomials of the given system.</item> <br />
<item>@param <em>QStrategy</em>: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;</item><br />
<item>@param <em>CStrategy</em>: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;</item><br />
<item>@param <em>MinMax</em>: Optimization direction i.e. minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>).</item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
QStrategy:=0;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
[0, 1, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
<example><br />
Use S::=Z/(2)[x[1..5]];<br />
F:=[<br />
x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],<br />
x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,<br />
x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],<br />
x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,<br />
x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]<br />
];<br />
<br />
<br />
QStrategy:=1;<br />
CStrategy:=0;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
[1, 1, 1, 1, 0]<br />
-------------------------------<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x[1..3]];<br />
F := [ x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[3] +1,<br />
x[1]x[2]x[3] + x[1]x[2] + x[2]x[3] + x[1] + x[2],<br />
x[1]x[2] + x[2]x[3] + x[2]<br />
];<br />
<br />
<br />
QStrategy:=0;<br />
CStrategy:=1;<br />
MinMax:=<quotes>Max</quotes>;<br />
<br />
-- Then we compute the solution with<br />
<br />
GLPK.IPCSolve(F, QStrategy, CStrategy, MinMax);<br />
<br />
-- The result will be the following:<br />
<br />
[0, 0, 1]<br />
-------------------------------<br />
</example><br />
<br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
<type>poly_system</type><br />
</types><br />
<key>ipcsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.l01pSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=LWBugTrackerForTesting&diff=11899LWBugTrackerForTesting2010-12-17T07:54:46Z<p>132.231.10.53: </p>
<hr />
<div>To report a bug, add a line describing the problem and how to reproduce it in the "Open" section.<br />
<br />
==Open==<br />
* Alias for the package CharP: I have checked with the latest nightly build the Alias for the package CharP is still not working. EU<br />
* ApCoCoAServer throws the error/warning: ''Trying to launch application: /home/garak/ApCoCoA-1.6.0/plugins/org.apcocoa.contribution.linux.x86.gtk.fragment_1.6.0.201012151632/os/linux/apcocoa/Glpk/bin/glpsol in path ' ' '' (x86 Linux) SN<br />
** ST-2010-12-16: Postponed.<br />
** ST-2010-12-16: This is the usual info when launching stuff by the server. Maybe the empty path is irritating. Are there any errors after this line?<br />
* The input line ''<->'' closes the ApCoCoA Session after printing two parse errors. (x86 Linux Moccha) SN<br />
** ST-2010-12-16: Postponed.<br />
* ApCoCoAServer & ApCoCoAQT crashed with 'invalid pointer' when I tried to use (missing) GNUPlot: ''Trying to launch application: /home/garak/ApCoCoA-1.6.0/gnuplot/bin/wgnuplot in path' ' No path is given.'' (x86 Linux QT) SN<br />
** ST-2010-12-16: Postponed.<br />
** ST-2010-12-16: We have to discuss a standard way of handling missing external binaries (which we do not want to supply for some platforms, like gnuplot on linux).<br />
<br />
==Assigned==<br />
* ''Latte tests fail on Linux/Mac''. Latte build seems to be broken. Linux 32/64, Mac, SK<br />
** SK-2010-12-15: Update: the tests only fail if called from Moccha.<br />
** SK-2010-12-15: Update: fixed on Linux platforms. Mac platform will be fixed shortly (I guess).<br />
*** ST-2010-12-16: Postponed.<br />
*** ST-2010-12-16: Hmm, with the MacOS 10.5 vs 10.6 issue this might take some time. There are currently more important issues I'll fix first. <br />
* Find a better way to handle changed default interpreter install after update, see also below ''Unpacking win standalone...''<br />
** ST-2010-12-16: Postponed.<br />
<br />
==Resolved==<br />
* Moccha uses some Java-1.6 specifics s.a. String.isEmpty() which shoule be avoided due to 1.5 compatibility.<br />
** JL/ST-2010-12-16: Jan tested most important GUI features on a virtual machine with Java 1.5.<br />
** ST-2010-12-15: Fixed for the problem in question, however we need to check all files for this issue. However, we need a pure 1.5 installation for this.<br />
* Unpacking win standalone, starting and reusing already existing projec raises error "ApCoCoA could not be started. Reason: The given ApCoCoA executable does not exist.". In the preferences, the interpreter install is deleted when opening the preferences dialog. Also every time you start it you need a new workspace. EU<br />
** ST-2010-12-16: Right now there is a workaround: If ApCoCoA cannot be started, you are redirected to the preferences page and can execute a search via the Search button (which now should find the installed contribution). We have to find a proper way to handle interpreters after an update or if different GUI installations use the same workspace. But that's for the next release.<br />
* Project wizard states "New DLTK Project", which is a bit confusing. Better would be "New ApCoCoA Project".<br />
** ST-2010-12-16: OK<br />
* ''ApCoCoA default interpreter missing''. After an update via the Eclipse update mechanism the default ApCoCoA interpreter is missing and it is not possible to add a new interpreter other than 'cocoa_text'. Linux 32 (Update), SK<br />
** SK/ST-2010-12-16: OK<br />
** ST-2010-12-16: Problem is a bug in the DLTK, which forces us to use lower case only names for the ApCoCoA executable.<br />
** SK-2010-12-15: Temporary workaround to get it working: rename 'ApCoCoAText' to 'cocoa_text' and select it as interpreter.<br />
** (?) Some issue with the downloaded version (ApCoCoA-1.6.0-20101215-1631-linux.gtk.x86.tar.gz) SN<br />
* ''Opening of library files broken''. Add new project, go to libraries, try to open. Instead of the content, an error message is displayed in the editor window. MacOS X, ST<br />
** ST-2010-12-16: OK<br />
* Naming of "Default Interpreter".<br />
** ST-2010-12-16: OK<br />
* ''Latte tests fail on Standalone'': Error massage is: TestLatte06(): FAILURE! TODO: Some error message!. EU<br />
* The test suit is not running successfully in standalone at WindowsXP. When I run $apcocoa/ts/aunit.RunTests(); I get the following error message: TestTemplate01(): FAILURE! Unknown operator, TestTemplate02(): SUCCESS!. EU<br />
** Did you follow the instructions and execute '''both''' commands <pre>$apcocoa/ts/aunit.RegisterAllTests();$apcocoa/ts/aunit.RunTests();</pre><br />
* Two functions IMXLSolve and IMNLASolve are missing in the documentation of the package CharP. <br />
** TS-2010-12-16: The two functions will be added today.<br />
* The Alias for the package CharP is also not working The error message displayed is "ERROR: Non local alias CharP CONTEXT: CharP.XLSolve(F).". EU<br />
** ST-2010-12-16: Fixed, alias was still Char2.<br />
* "My" functions ''G2V.G2VAlgorithm(...)'' is missing in tab completion. Am I missing something in my .cpkg? SN<br />
** ST-2010-12-16: Corresponding entry in aliases.coc was missing, it's fixed, please check if it is working.<br />
* ''Inconvenience While restarting ApCoCoA with Moccha''. While working in Eclips GUI(Moccha) when we restart ApCoCoA with restart button then it asks for saving the unsaved file if there is any. It would be nice if it do not asks for it. Moccha(Windows), EU<br />
** ST-2010-12-14: There is an option ''Save required dirty editors before launching'' under ''Preferences > Run/Debug > Launchin'', which you can set to ''Always'', ''Never'' or ''Prompt''.<br />
* '''#######VERSIONSTRING#######' not replaced''. The string '#######VERSIONSTRING#######' is not being replaced but should be replaced with current version number. Linux 32 (Update), SK<br />
* ApCoCoAServer doesn't lauch correctly for ApCoCoA GUI in Mac.<br />
** ST-2010-12-15: ApCoCoAServer was compiled on 10.6, the problem occured on a 10.5 machine. After recompiling needed libraries and server with cxx-flags "-isysroot /Developer/SDKs/MacOSX10.5.sdk -mmacosx-version-min=10.5" it is now exectuable again on the 10.5 machine.<br />
* ''GLPK tests fail on Linux with GLPK > 4.34''. Due to a change in the input file format for glpsol. Linux 32/64, SK<br />
** SK-2010-12-15: Will be fixed after the release.<br />
* The Linux *.zip-files can not be extracted. Renaming to *.tar.gz solves the problem, TS<br />
** ST-2010-12-15: Fixed, files are now named correctly.<br />
* Wrong about text "1.5.1"<br />
** ST-2010-12-15: OK.<br />
* In the ApCoCoA preferences dialog there is a typo in "Choose method to start CoCoA". It states "spawner withot PTY".<br />
** ST-2010-12-15: OK.<br />
* At line 770 in the package CharP there is a typo error i.e. "EndIf;8". Please remove 8 from there. EU.<br />
** SK-2010-12-15: Fixed.<br />
* The function name BB.BBscheme(); should be renamed to BB.BBScheme(); (just a suggestion !) RA<br />
** SK-2010-12-16: Will be done after the release as a part of the cleanup of the borderbasis package.<br />
* The function Weyl.WGB() is although giving a correct output, but Server displays a following warning message after completion<br />
** JL: This error message appears if some memory leakage occurs. Please check your code for non-freed memory.<br />
<br />
***IMMINENT DISASTER*** CoCoA::GlobalManager being destroyed while CoCoA objects still live!<br />
For example, in the Ring A::=QQ[x,y], run the command Weyl.WGB(Ideal(x,y)); RA<br />
<br />
==Closed==</div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=LWBugTrackerForTesting&diff=11883LWBugTrackerForTesting2010-12-16T10:24:54Z<p>132.231.10.53: </p>
<hr />
<div>To report a bug, add a line describing the problem and how to reproduce it in the "Open" section.<br />
<br />
==Open==<br />
* ''Latte tests fail on Standalone'': Error massage is: TestLatte06(): FAILURE! TODO: Some error message!. EU<br />
* Unpacking win standalone, starting and reusing already existing projec raises error "ApCoCoA could not be started. Reason: The given ApCoCoA executable does not exist.". In the preferences, the interpreter install is deleted when opening the preferences dialog. Also every time you start it you need a new workspace. EU<br />
* ApCoCoAServer throws the error/warning: ''Trying to launch application: /home/garak/ApCoCoA-1.6.0/plugins/org.apcocoa.contribution.linux.x86.gtk.fragment_1.6.0.201012151632/os/linux/apcocoa/Glpk/bin/glpsol in path ' ' '' (x86 Linux) SN<br />
** ST-2010-12-16: This is the usual info when launching stuff by the server. Maybe the empty path is irritating. Are there any errors after this line?<br />
* The input line ''<->'' closes the ApCoCoA Session after printing two parse errors. (x86 Linux Moccha) SN<br />
* ApCoCoAServer & ApCoCoAQT crashed with 'invalid pointer' when I tried to use (missing) GNUPlot: ''Trying to launch application: /home/garak/ApCoCoA-1.6.0/gnuplot/bin/wgnuplot in path' ' No path is given.'' (x86 Linux QT) SN<br />
** ST-2010-12-16: We have to discuss a standard way of handling missing external binaries (which we do not want to supply for some platforms, like gnuplot on linux).<br />
<br />
==Assigned==<br />
* ''Latte tests fail on Linux/Mac''. Latte build seems to be broken. Linux 32/64, Mac, SK<br />
** SK-2010-12-15: Update: the tests only fail if called from Moccha.<br />
** SK-2010-12-15: Update: fixed on Linux platforms. Mac platform will be fixed shortly (I guess).<br />
* Naming of "Default Interpreter".<br />
* ''ApCoCoA default interpreter missing''. After an update via the Eclipse update mechanism the default ApCoCoA interpreter is missing and it is not possible to add a new interpreter other than 'cocoa_text'. Linux 32 (Update), SK<br />
** ST-2010-12-16: Problem is a bug in the DLTK, which forces us to use lower case only names for the ApCoCoA executable.<br />
** SK-2010-12-15: Temporary workaround to get it working: rename 'ApCoCoAText' to 'cocoa_text' and select it as interpreter.<br />
** (?) Some issue with the downloaded version (ApCoCoA-1.6.0-20101215-1631-linux.gtk.x86.tar.gz) SN<br />
* Moccha uses some Java-1.6 specifics s.a. String.isEmpty() which shoule be avoided due to 1.5 compatibility.<br />
** ST-2010-12-15: Fixed for the problem in question, however we need to check all files for this issue. However, we need a pure 1.5 installation for this.<br />
* ''Opening of library files broken''. Add new project, go to libraries, try to open. Instead of the content, an error message is displayed in the editor window. MacOS X, ST<br />
<br />
<br />
==Resolved==<br />
* The test suit is not running successfully in standalone at WindowsXP. When I run $apcocoa/ts/aunit.RunTests(); I get the following error message: TestTemplate01(): FAILURE! Unknown operator, TestTemplate02(): SUCCESS!. EU<br />
** Did you follow the instructions and execute '''both''' commands <pre>$apcocoa/ts/aunit.RegisterAllTests();$apcocoa/ts/aunit.RunTests();</pre><br />
* Two functions IMXLSolve and IMNLASolve are missing in the documentation of the package CharP. <br />
** TS-2010-12-16: The two functions will be added today.<br />
* The Alias for the package CharP is also not working The error message displayed is "ERROR: Non local alias CharP CONTEXT: CharP.XLSolve(F).". EU<br />
** ST-2010-12-16: Fixed, alias was still Char2.<br />
* "My" functions ''G2V.G2VAlgorithm(...)'' is missing in tab completion. Am I missing something in my .cpkg? SN<br />
** ST-2010-12-16: Corresponding entry in aliases.coc was missing, it's fixed, please check if it is working.<br />
* ''Inconvenience While restarting ApCoCoA with Moccha''. While working in Eclips GUI(Moccha) when we restart ApCoCoA with restart button then it asks for saving the unsaved file if there is any. It would be nice if it do not asks for it. Moccha(Windows), EU<br />
** ST-2010-12-14: There is an option ''Save required dirty editors before launching'' under ''Preferences > Run/Debug > Launchin'', which you can set to ''Always'', ''Never'' or ''Prompt''.<br />
* '''#######VERSIONSTRING#######' not replaced''. The string '#######VERSIONSTRING#######' is not being replaced but should be replaced with current version number. Linux 32 (Update), SK<br />
* ApCoCoAServer doesn't lauch correctly for ApCoCoA GUI in Mac.<br />
** ST-2010-12-15: ApCoCoAServer was compiled on 10.6, the problem occured on a 10.5 machine. After recompiling needed libraries and server with cxx-flags "-isysroot /Developer/SDKs/MacOSX10.5.sdk -mmacosx-version-min=10.5" it is now exectuable again on the 10.5 machine.<br />
* ''GLPK tests fail on Linux with GLPK > 4.34''. Due to a change in the input file format for glpsol. Linux 32/64, SK<br />
** SK-2010-12-15: Will be fixed after the release.<br />
* The Linux *.zip-files can not be extracted. Renaming to *.tar.gz solves the problem, TS<br />
** ST-2010-12-15: Fixed, files are now named correctly.<br />
* Wrong about text "1.5.1"<br />
** ST-2010-12-15: OK.<br />
* In the ApCoCoA preferences dialog there is a typo in "Choose method to start CoCoA". It states "spawner withot PTY".<br />
** ST-2010-12-15: OK.<br />
* At line 770 in the package CharP there is a typo error i.e. "EndIf;8". Please remove 8 from there. EU.<br />
** SK-2010-12-15: Fixed.<br />
<br />
==Closed==</div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=LWBugTrackerForTesting&diff=11877LWBugTrackerForTesting2010-12-16T08:42:30Z<p>132.231.10.53: </p>
<hr />
<div>To report a bug, add a line describing the problem and how to reproduce it in the "Open" section.<br />
<br />
==Open==<br />
* The test suit is not running successfully in standalone at WindowsXP. When I run $apcocoa/ts/aunit.RunTests(); I get the following error message: TestTemplate01(): FAILURE! Unknown operator, TestTemplate02(): SUCCESS!. EU<br />
* Two functions IMXLSolve and IMNLASolve are missing in the documentation of the package CharP. The Alias for the package CharP is also not working The error message displayed is "ERROR: Non local alias CharP CONTEXT: CharP.XLSolve(F).". EU<br />
* ''Latte tests fail on Linux/Mac''. Latte build seems to be broken. Linux 32/64, Mac, SK<br />
** SK-2010-12-15: Update: the tests only fail if called from Moccha.<br />
** SK-2010-12-15: Update: fixed on Linux platforms. Mac platform will be fixed shortly (I guess).<br />
* Naming of "Default Interpreter".<br />
* Moccha uses some Java-1.6 specifics s.a. String.isEmpty() which shoule be avoided due to 1.5 compatibility.<br />
** ST-2010-12-15: Fixed for the problem in question, however we need to check all files for this issue. However, we need a pure 1.5 installation for this.<br />
* ''Opening of library files broken''. Add new project, go to libraries, try to open. Instead of the content, an error message is displayed in the editor window. MacOS X, ST<br />
* ''ApCoCoA default interpreter missing''. After an update via the Eclipse update mechanism the default ApCoCoA interpreter is missing and it is not possible to add a new interpreter other than 'cocoa_text'. Linux 32 (Update), SK<br />
** SK-2010-12-15: Temporary workaround to get it working: rename 'ApCoCoAText' to 'cocoa_text' and select it as interpreter.<br />
** (?) Some issue with the downloaded version (ApCoCoA-1.6.0-20101215-1631-linux.gtk.x86.tar.gz) SN<br />
** Unpacking win standalone, starting and reusing already existing projec raises error "ApCoCoA could not be started. Reason: The given ApCoCoA executable does not exist.". In the preferences, the interpreter install is deleted when opening the preferences dialog. Also every time you start it you need a new workspace. EU<br />
* "My" functions ''G2V.G2VAlgorithm(...)'' is missing in tab completion. Am I missing something in my .cpkg? SN<br />
* ApCoCoAServer throws the error/warning: ''Trying to launch application: /home/garak/ApCoCoA-1.6.0/plugins/org.apcocoa.contribution.linux.x86.gtk.fragment_1.6.0.201012151632/os/linux/apcocoa/Glpk/bin/glpsol in path ' ' '' (x86 Linux) SN<br />
* ApCoCoAServer & ApCoCoAQT crashed with 'invalid pointer' when I tried to use (missing) GNUPlot: ''Trying to launch application: /home/garak/ApCoCoA-1.6.0/gnuplot/bin/wgnuplot in path' ' No path is given.'' (x86 Linux QT) SN<br />
* The input line ''<->'' closes the ApCoCoA Session after printing two parse errors. (x86 Linux Moccha) SN<br />
<br />
==Resolved==<br />
* ''Inconvenience While restarting ApCoCoA with Moccha''. While working in Eclips GUI(Moccha) when we restart ApCoCoA with restart button then it asks for saving the unsaved file if there is any. It would be nice if it do not asks for it. Moccha(Windows), EU<br />
** ST-2010-12-14: There is an option ''Save required dirty editors before launching'' under ''Preferences > Run/Debug > Launchin'', which you can set to ''Always'', ''Never'' or ''Prompt''.<br />
* '''#######VERSIONSTRING#######' not replaced''. The string '#######VERSIONSTRING#######' is not being replaced but should be replaced with current version number. Linux 32 (Update), SK<br />
* ApCoCoAServer doesn't lauch correctly for ApCoCoA GUI in Mac.<br />
** ST-2010-12-15: ApCoCoAServer was compiled on 10.6, the problem occured on a 10.5 machine. After recompiling needed libraries and server with cxx-flags "-isysroot /Developer/SDKs/MacOSX10.5.sdk -mmacosx-version-min=10.5" it is now exectuable again on the 10.5 machine.<br />
* ''GLPK tests fail on Linux with GLPK > 4.34''. Due to a change in the input file format for glpsol. Linux 32/64, SK<br />
** SK-2010-12-15: Will be fixed after the release.<br />
* The Linux *.zip-files can not be extracted. Renaming to *.tar.gz solves the problem, TS<br />
** ST-2010-12-15: Fixed, files are now named correctly.<br />
* Wrong about text "1.5.1"<br />
** ST-2010-12-15: OK.<br />
* In the ApCoCoA preferences dialog there is a typo in "Choose method to start CoCoA". It states "spawner withot PTY".<br />
** ST-2010-12-15: OK.<br />
* At line 770 in the package CharP there is a typo error i.e. "EndIf;8". Please remove 8 from there. EU.<br />
** SK-2010-12-15: Fixed.</div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=LWBugTrackerForTesting&diff=11863LWBugTrackerForTesting2010-12-15T17:05:28Z<p>132.231.10.53: </p>
<hr />
<div>To report a bug, add a line describing the problem and how to reproduce it in the "Open" section.<br />
<br />
==Open==<br />
* Two functions IMXLSolve and IMNLASolve are missing in the documentation of the package CharP. The Alias for the package CharP is also not working The error message displayed is "ERROR: Non local alias CharP CONTEXT: CharP.XLSolve(F).". EU<br />
* Unpacking win standalone, starting and reusing already existing projec raises error "ApCoCoA could not be started. Reason: The given ApCoCoA executable does not exist.". In the preferences, the interpreter install is deleted when opening the preferences dialog. Also every time you start it you need a new workspace. EU<br />
* ''Latte tests fail on Linux''. Latte build seems to be broken. Linux 32/64, SK<br />
** SK-2010-12-15: Update: the tests only fail if called from Moccha.<br />
* Naming of "Default Interpreter".<br />
* Moccha uses some Java-1.6 specifics s.a. String.isEmpty() which shoule be avoided due to 1.5 compatibility.<br />
** ST-2010-12-15: Fixed for the problem in question, however we need to check all files for this issue. However, we need a pure 1.5 installation for this.<br />
* ''Opening of library files broken''. Add new project, go to libraries, try to open. Instead of the content, an error message is displayed in the editor window. MacOS X, ST<br />
* ''ApCoCoA default interpreter missing''. After an update via the Eclipse update mechanism the default ApCoCoA interpreter is missing and it is not possible to add a new interpreter other than 'cocoa_text'. Linux 32 (Update), SK<br />
** SK-2010-12-15: Temporary workaround to get it working: rename 'ApCoCoAText' to 'cocoa_text' and select it as interpreter.<br />
<br />
==Resolved==<br />
* ''Inconvenience While restarting ApCoCoA with Moccha''. While working in Eclips GUI(Moccha) when we restart ApCoCoA with restart button then it asks for saving the unsaved file if there is any. It would be nice if it do not asks for it. Moccha(Windows), EU<br />
** ST-2010-12-14: There is an option ''Save required dirty editors before launching'' under ''Preferences > Run/Debug > Launchin'', which you can set to ''Always'', ''Never'' or ''Prompt''.<br />
* '''#######VERSIONSTRING#######' not replaced''. The string '#######VERSIONSTRING#######' is not being replaced but should be replaced with current version number. Linux 32 (Update), SK<br />
* ApCoCoAServer doesn't lauch correctly for ApCoCoA GUI in Mac.<br />
** ST-2010-12-15: ApCoCoAServer was compiled on 10.6, the problem occured on a 10.5 machine. After recompiling needed libraries and server with cxx-flags "-isysroot /Developer/SDKs/MacOSX10.5.sdk -mmacosx-version-min=10.5" it is now exectuable again on the 10.5 machine.<br />
* ''GLPK tests fail on Linux with GLPK > 4.34''. Due to a change in the input file format for glpsol. Linux 32/64, SK<br />
** SK-2010-12-15: Will be fixed after the release.<br />
* The Linux *.zip-files can not be extracted. Renaming to *.tar.gz solves the problem, TS<br />
** ST-2010-12-15: Fixed, files are now named correctly.<br />
* Wrong about text "1.5.1"<br />
** ST-2010-12-15: OK.<br />
* In the ApCoCoA preferences dialog there is a typo in "Choose method to start CoCoA". It states "spawner withot PTY".<br />
** ST-2010-12-15: OK.</div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=LWBugTrackerForTesting&diff=11862LWBugTrackerForTesting2010-12-15T16:56:02Z<p>132.231.10.53: </p>
<hr />
<div>To report a bug, add a line describing the problem and how to reproduce it in the "Open" section.<br />
<br />
==Open==<br />
* Unpacking win standalone, starting and reusing already existing projec raises error "ApCoCoA could not be started. Reason: The given ApCoCoA executable does not exist.". In the preferences, the interpreter install is deleted when opening the preferences dialog. Also every time you start it you need a new workspace. EU<br />
* ''Latte tests fail on Linux''. Latte build seems to be broken. Linux 32/64, SK<br />
** SK-2010-12-15: Update: the tests only fail if called from Moccha.<br />
* Naming of "Default Interpreter".<br />
* Moccha uses some Java-1.6 specifics s.a. String.isEmpty() which shoule be avoided due to 1.5 compatibility.<br />
** ST-2010-12-15: Fixed for the problem in question, however we need to check all files for this issue. However, we need a pure 1.5 installation for this.<br />
* ''Opening of library files broken''. Add new project, go to libraries, try to open. Instead of the content, an error message is displayed in the editor window. MacOS X, ST<br />
* ''ApCoCoA default interpreter missing''. After an update via the Eclipse update mechanism the default ApCoCoA interpreter is missing and it is not possible to add a new interpreter other than 'cocoa_text'. Linux 32 (Update), SK<br />
** SK-2010-12-15: Temporary workaround to get it working: rename 'ApCoCoAText' to 'cocoa_text' and select it as interpreter.<br />
<br />
==Resolved==<br />
* ''Inconvenience While restarting ApCoCoA with Moccha''. While working in Eclips GUI(Moccha) when we restart ApCoCoA with restart button then it asks for saving the unsaved file if there is any. It would be nice if it do not asks for it. Moccha(Windows), EU<br />
** ST-2010-12-14: There is an option ''Save required dirty editors before launching'' under ''Preferences > Run/Debug > Launchin'', which you can set to ''Always'', ''Never'' or ''Prompt''.<br />
* '''#######VERSIONSTRING#######' not replaced''. The string '#######VERSIONSTRING#######' is not being replaced but should be replaced with current version number. Linux 32 (Update), SK<br />
* ApCoCoAServer doesn't lauch correctly for ApCoCoA GUI in Mac.<br />
** ST-2010-12-15: ApCoCoAServer was compiled on 10.6, the problem occured on a 10.5 machine. After recompiling needed libraries and server with cxx-flags "-isysroot /Developer/SDKs/MacOSX10.5.sdk -mmacosx-version-min=10.5" it is now exectuable again on the 10.5 machine.<br />
* ''GLPK tests fail on Linux with GLPK > 4.34''. Due to a change in the input file format for glpsol. Linux 32/64, SK<br />
** SK-2010-12-15: Will be fixed after the release.<br />
* The Linux *.zip-files can not be extracted. Renaming to *.tar.gz solves the problem, TS<br />
** ST-2010-12-15: Fixed, files are now named correctly.<br />
* Wrong about text "1.5.1"<br />
** ST-2010-12-15: OK.<br />
* In the ApCoCoA preferences dialog there is a typo in "Choose method to start CoCoA". It states "spawner withot PTY".<br />
** ST-2010-12-15: OK.</div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11820ApCoCoA-1:LinAlg.EF2010-12-14T15:14:07Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>LinAlg.EF</title><br />
<short_description>Computes a row echelon form of a matrix over <tt>F_2</tt> with record keeping.</short_description><br />
<br />
<syntax><br />
LinAlg.EF(M:LIST, L1:LIST, L2:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function computes a row echelon form of a matrix <tt>M</tt> defined over the field <tt>F_2</tt>. It allows to keep record of the order of rows inside the matrix using the parameter <tt>L1</tt>. If the matrix contains some rows which are already in echelon form then the parameter <tt>L2</tt> represent them with 0 and all others with 1. <br />
<par/><br />
<br />
<br />
<itemize><br />
<item>@param <em>M:</em> A List of Lists whose row echelon form to compute.</item><br />
<item>@param <em>L1:</em> List of integers. For example, the integers could represent the order of lists in the list M. </item><br />
<item>@param <em>L2:</em> List of integers. For example, the integers could be 0<tt>-</tt>1 to represent the lists already reduced(0) and to be reduced(1).</item><br />
<item>@return A row echelon form of <tt>M</tt> together with adjusted lists L1 and L2.</item><br />
</itemize><br />
<br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[1, 1, 0, 1, 0],<br />
[0, 1, 1, 0, 1],<br />
[1, 0, 1, 0, 0], <br />
[1, 1, 1, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[ [[1, 1, 0, 1, 0], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]], [1, 2, 4, 3], [0, 0, 1, 1]]<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[0, 1, 0, 1, 0],<br />
[0, 1, 0, 0, 1],<br />
[1, 0, 1, 1, 0], <br />
[1, 1, 0, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[[[1, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]], [3, 1, 4, 2], [1, 0, 1, 0]]<br />
-------------------------------<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
<br />
</example><br />
<br />
<br />
<br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>EF</key><br />
<key>LinAlg.EF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=LWBugTrackerForTesting&diff=11763LWBugTrackerForTesting2010-12-13T17:37:10Z<p>132.231.10.53: </p>
<hr />
<div>To report a bug, add a line describing the problem and how to reproduce it in the "Open" section.<br />
<br />
==Open==<br />
* ''Opening of library files broken''. Add new project, go to libraries, try to open. Instead of the content, an error message is displayed in the editor window. MacOS X, ST<br />
* ''ApCoCoA default interpreter missing''. After an update via the Eclipse update mechanism the default ApCoCoA interpreter is missing and it is not possible to add a new interpreter other than 'cocoa_text'. Linux 32 (Update), SK<br />
* '''#######VERSIONSTRING#######' not replaced''. The string '#######VERSIONSTRING#######' is not being replaced but should be replaced with current version number. Linux 32 (Update), SK<br />
* ''Inconvenience While restarting ApCoCoA with Maccha''. While working in Eclips GUI(Maccha) when we restart ApCoCoA with restart button then it asks for saving the unsaved file if there is any. It woould be nice if it do not asks for it. Maccha(Windows), EU<br />
<br />
==Resolved==</div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11445ApCoCoA-1:LinAlg.EF2010-12-07T12:34:18Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>LinAlg.EF</title><br />
<short_description>Computes a row echelon form of a matrix with record keeping.</short_description><br />
<br />
<syntax><br />
LinAlg.EF(M:LIST, L1:LIST, L2:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function allows you to compute a row echelon form of a matrix <tt>M</tt> defined over the field <tt>F_2</tt>. It alows to keep record of the order of rows inside the matrix using the parameter L1. If the matrix contains some rows which are already in echlon form then the parameter L2 represent them with 0 and all others with 1. <br />
<par/><br />
<br />
<br />
<itemize><br />
<item>@param <em>M:</em> A List of Lists whose row echelon form to compute.</item><br />
<item>@param <em>L1:</em> List of integers. For example, the integers could represent the order of lists in the list M. </item><br />
<item>@param <em>L2:</em> List of integers. For example, the integers could be 0<tt>-</tt>1 to represent the lists already reduced(0) and to be reduced(1).</item><br />
<item>@return A row echelon form of <tt>M</tt> together with adjusted lists L1 and L2.</item><br />
</itemize><br />
<br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[1, 1, 0, 1, 0],<br />
[0, 1, 1, 0, 1],<br />
[1, 0, 1, 0, 0], <br />
[1, 1, 1, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[ [[1, 1, 0, 1, 0], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]], [1, 2, 4, 3], [0, 0, 1, 1]]<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[0, 1, 0, 1, 0],<br />
[0, 1, 0, 0, 1],<br />
[1, 0, 1, 1, 0], <br />
[1, 1, 0, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[[[1, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]], [3, 1, 4, 2], [1, 0, 1, 0]]<br />
-------------------------------<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
<br />
</example><br />
<br />
<br />
<br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>REF</key><br />
<key>LinAlg.REF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11444ApCoCoA-1:LinAlg.EF2010-12-07T12:32:26Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>LinAlg.EF</title><br />
<short_description>Computes a row echelon form of a matrix with record keeping.</short_description><br />
<br />
<syntax><br />
LinAlg.EF(M:LIST, L1:LIST, L2:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function allows you to compute a row echelon form of a matrix <tt>M</tt> defined over the field <tt>F_2</tt>. It alows to keep record of the order of rows inside the matrix using the parameter L1. If the matrix contains some rows which are already in echlon form then the parameter L2 represent them with 0 and all others with 1. <br />
<par/><br />
<br />
<br />
<itemize><br />
<item>@param <em>M:</em> A List of Lists whose row echelon form to compute.</item><br />
<item>@param <em>L1:</em> List of integers. For example, the integers could represent the order of lists in the list M. </item><br />
<item>@param <em>L2:</em> List of integers. For example, the integers could be 0<tt>-</tt>1 to represent the lists already reduced(0) and to be reduced(1).</item><br />
<item>@return A row echelon form of <tt>M</tt> together with lists L1 and L2.</item><br />
</itemize><br />
<br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[1, 1, 0, 1, 0],<br />
[0, 1, 1, 0, 1],<br />
[1, 0, 1, 0, 0], <br />
[1, 1, 1, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[ [[1, 1, 0, 1, 0], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]], [1, 2, 4, 3], [0, 0, 1, 1]]<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[0, 1, 0, 1, 0],<br />
[0, 1, 0, 0, 1],<br />
[1, 0, 1, 1, 0], <br />
[1, 1, 0, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[[[1, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]], [3, 1, 4, 2], [1, 0, 1, 0]]<br />
-------------------------------<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
<br />
</example><br />
<br />
<br />
<br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>REF</key><br />
<key>LinAlg.REF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11443ApCoCoA-1:LinAlg.EF2010-12-07T12:24:24Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>LinAlg.EF</title><br />
<short_description>Computes a row echelon form of a matrix with record keeping.</short_description><br />
<br />
<syntax><br />
LinAlg.EF(M:LIST, L1:LIST, L2:LIST):MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function allows you to compute a row echelon form of a matrix <tt>M</tt> defined over the field <tt>F_2</tt>. <br />
<par/><br />
<br />
<br />
<itemize><br />
<item>@param <em>M:</em> A List of Lists whose row echelon form to compute.</item><br />
<item>@param <em>L1:</em> List of integers. For example, the integers could represent the order of lists in the list M. </item><br />
<item>@param <em>L2:</em> List of integers. For example, the integers could be 0<tt>-</tt>1 to represent the lists already reduced(0) and to be reduced(1).</item><br />
<item>@return A row echelon form of <tt>M</tt> together with lists L1 and L2.</item><br />
</itemize><br />
<br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[1, 1, 0, 1, 0],<br />
[0, 1, 1, 0, 1],<br />
[1, 0, 1, 0, 0], <br />
[1, 1, 1, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[ [[1, 1, 0, 1, 0], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]], [1, 2, 4, 3], [0, 0, 1, 1]]<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[0, 1, 0, 1, 0],<br />
[0, 1, 0, 0, 1],<br />
[1, 0, 1, 1, 0], <br />
[1, 1, 0, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[[[1, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]], [3, 1, 4, 2], [1, 0, 1, 0]]<br />
-------------------------------<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
<br />
</example><br />
<br />
<br />
<br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>REF</key><br />
<key>LinAlg.REF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11442ApCoCoA-1:LinAlg.EF2010-12-07T12:23:15Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>LinAlg.EF</title><br />
<short_description>Computes a row echelon form of a matrix with recond keeping.</short_description><br />
<br />
<syntax><br />
LinAlg.EF(M:MAT, L1:LIST, L2:LIST):MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function allows you to compute a row echelon form of a matrix <tt>M</tt> defined over the field <tt>F_2</tt>. <br />
<par/><br />
<br />
The parameter <tt>CompRREF</tt> lets you specify if you want to compute a row echelon form or the reduced row echelon form of <tt>M</tt>. If <tt>CompRREF</tt> is set to <tt>TRUE</tt>, the reduced row echelon form will be computed, and if it is set to <tt>FALSE</tt>, a row echelon form where all pivot elements are equal to one will be computed.<br />
<br />
<br />
<itemize><br />
<item>@param <em>M:</em> A List of Lists whose row echelon form to compute.</item><br />
<item>@param <em>L1:</em> List of integers. For example, the integers could represent the order of lists in the list M. </item><br />
<item>@param <em>L2:</em> List of integers. For example, the integers could be 0<tt>-</tt>1 to represent the lists already reduced(0) and to be reduced(1).</item><br />
<item>@return A row echelon form of <tt>M</tt> together with lists L1 and L2.</item><br />
</itemize><br />
<br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[1, 1, 0, 1, 0],<br />
[0, 1, 1, 0, 1],<br />
[1, 0, 1, 0, 0], <br />
[1, 1, 1, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[ [[1, 1, 0, 1, 0], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]], [1, 2, 4, 3], [0, 0, 1, 1]]<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
</example><br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := [<br />
[0, 1, 0, 1, 0],<br />
[0, 1, 0, 0, 1],<br />
[1, 0, 1, 1, 0], <br />
[1, 1, 0, 0, 1]<br />
];<br />
-- order of lists in M<br />
L1:=[1, 2, 3, 4]; <br />
<br />
-- 0 for lists which are already in echelon form and 1 for those to be reduced.<br />
L2:=[0, 0, 1, 1];<br />
<br />
LinAlg.EF(M, L1, L2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[[[1, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]], [3, 1, 4, 2], [1, 0, 1, 0]]<br />
-------------------------------<br />
<br />
-- The last two lists represent the new order of lists in M.<br />
<br />
</example><br />
<br />
<br />
<br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>REF</key><br />
<key>LinAlg.REF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11441ApCoCoA-1:LinAlg.EF2010-12-07T11:53:20Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>LinAlg.EF</title><br />
<short_description>Computes a row echelon form of a matrix with recond keeping.</short_description><br />
<br />
<syntax><br />
LinAlg.EF(M:MAT, L1:LIST, L2:LIST):MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function allows you to compute a row echelon form of a matrix <tt>M</tt> defined over the field <tt>F_2</tt>. <br />
<par/><br />
<br />
The parameter <tt>CompRREF</tt> lets you specify if you want to compute a row echelon form or the reduced row echelon form of <tt>M</tt>. If <tt>CompRREF</tt> is set to <tt>TRUE</tt>, the reduced row echelon form will be computed, and if it is set to <tt>FALSE</tt>, a row echelon form where all pivot elements are equal to one will be computed.<br />
<br />
<br />
<itemize><br />
<item>@param <em>M:</em> A matrix whose row echelon form to compute.</item><br />
<item>@param <em>L1:</em> List of integers.</item><br />
<item>@param <em>L2:</em> List of integers.</item><br />
<item>@return A row echelon form of <tt>M</tt> together with lists L1 and L2.</item><br />
</itemize><br />
<br />
<br />
<example><br />
Use ZZ/(2)[x,y];<br />
M := Mat([[ 1/2, 1/3, 2], [200, 3000, 1], [2, 5, 17], [1, 1, 1]]);<br />
LinAlg.REF(M, FALSE);<br />
<br />
</example><br />
<example><br />
Use QQ[x,y];<br />
M := Mat([[ 1, 1, 2], [200, 3000, 1], [2, 5, 17], [1, 1, 1]]);<br />
LinAlg.REF(M, 17, TRUE);<br />
<br />
</example><br />
<br />
<br />
<br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>REF</key><br />
<key>LinAlg.REF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:LinAlg.EF&diff=11440ApCoCoA-1:LinAlg.EF2010-12-07T11:36:15Z<p>132.231.10.53: New page: <command> <title>LinAlg.REF</title> <short_description>Computes a row echelon form of a matrix.</short_description> <syntax> LinAlg.REF(M:MAT, CompRREF:BOOL, BACKEND:STRING):MAT Lin...</p>
<hr />
<div><command><br />
<title>LinAlg.REF</title><br />
<short_description>Computes a row echelon form of a matrix.</short_description><br />
<br />
<syntax><br />
LinAlg.REF(M:MAT, CompRREF:BOOL, BACKEND:STRING):MAT<br />
LinAlg.REF(M:MAT, P:INT, CompRREF:BOOL, BACKEND:STRING):MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function allows you to compute a (reduced) row echelon form of a matrix <tt>M</tt> defined over a (finite) field. If you want to use the first version without the parameter <tt>P</tt>, the components of the input matrix <tt>M</tt> must be castable to type <tt>RAT</tt> (<tt>BACKEND</tt> = <quotes>INTERNAL</quotes>) or <tt>ZMOD</tt> (<tt>BACKEND</tt> = <quotes>IML</quotes> or <tt>BACKEND</tt> = <quotes>LINBOX</quotes>) and your current working ring must be a finite field in the latter case. The second version of this function lets you compute a (reduced) row echelon form of <tt>M</tt> mod <tt>P</tt> and the components of <tt>M</tt> must be of type <tt>INT</tt>.<br />
<par/><br />
The parameter <tt>CompRREF</tt> lets you specify if you want to compute a row echelon form or the reduced row echelon form of <tt>M</tt>. If <tt>CompRREF</tt> is set to <tt>TRUE</tt>, the reduced row echelon form will be computed, and if it is set to <tt>FALSE</tt>, a row echelon form where all pivot elements are equal to one will be computed.<br />
<par/><br />
The optional parameter <tt>BACKEND</tt> lets you choose between an internal implementation (<tt>BACKEND</tt> = <quotes>INTERNAL</quotes>) or IML or LinBox driven computations (<tt>BACKEND</tt> = <quotes>IML</quotes> or <tt>BACKEND</tt> = <quotes>LINBOX</quotes>). The default value of <tt>BACKEND</tt> is <quotes>INTERNAL</quotes>.<br />
<itemize><br />
<item>@param <em>M</em> A matrix whose (reduced) row echelon form to compute. If parameter <tt>P</tt> is given, the components of <tt>M</tt> must be of type <tt>INT</tt>. Otherwise, they must be castable to type <tt>RAT</tt> or <tt>ZMOD</tt> (please see description above).</item><br />
<item>@param <em>CompRREF</em> Set to <tt>TRUE</tt> if you want to compute the reduced row echelon form of <tt>M</tt> or to <tt>FALSE</tt> otherwise.</item><br />
<item>@return A (reduced) row echelon form of <tt>M</tt>.</item><br />
</itemize><br />
The following parameters are optional.<br />
<itemize><br />
<item>@param <em>P</em> An integer value. If <tt>P</tt> is specified, the (reduced) row echelon form computation will be carried out over the ring <tt>Z/pZ</tt>.</item><br />
<item>@param <em>BACKEND</em> Allowed values are <quotes>IML</quotes>, <quotes>INTERNAL</quotes>, and <quotes>LINBOX</quotes>.</item><br />
</itemize><br />
<example><br />
Use QQ[x,y];<br />
M := Mat([[ 1/2, 1/3, 2], [200, 3000, 1], [2, 5, 17], [1, 1, 1]]);<br />
LinAlg.REF(M, FALSE);<br />
<br />
-------------------------------<br />
Mat([<br />
[1, 2/3, 4],<br />
[0, 1, -2397/8600],<br />
[0, 0, 1],<br />
[0, 0, 0]<br />
])<br />
-------------------------------<br />
</example><br />
<example><br />
Use QQ[x,y];<br />
M := Mat([[ 1, 1, 2], [200, 3000, 1], [2, 5, 17], [1, 1, 1]]);<br />
LinAlg.REF(M, 17, TRUE);<br />
<br />
-------------------------------<br />
Mat([<br />
[1, 0, 0],<br />
[0, 1, 0],<br />
[0, 0, 1],<br />
[0, 0, 0]<br />
])<br />
-------------------------------<br />
</example><br />
<example><br />
Use QQ[x,y];<br />
M := Mat([[ 1, 1, 2], [200, 3000, 1], [2, 5, 17], [1, 1, 1]]);<br />
LinAlg.REF(M, 17, TRUE, "IML");<br />
<br />
-------------------------------<br />
Mat([<br />
[1, 0, 0],<br />
[0, 1, 0],<br />
[0, 0, 1],<br />
[0, 0, 0]<br />
])<br />
-------------------------------<br />
</example><br />
<example><br />
Use ZZ/(239)[x];<br />
M := Mat([[1, 2, 3], [4, 5, 6], [7, 8, 9], [11, 12, 13]]);<br />
LinAlg.REF(M, FALSE, "LINBOX");<br />
<br />
-------------------------------<br />
Mat([<br />
[1 % 239, 2 % 239, 3 % 239],<br />
[0 % 239, 1 % 239, 2 % 239],<br />
[0 % 239, 0 % 239, 0 % 239],<br />
[0 % 239, 0 % 239, 0 % 239]<br />
])<br />
</example><br />
</description><br />
<see>Introduction to CoCoAServer</see><br />
<see>IML.REF</see><br />
<see>LinBox.REF</see><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type>><br />
</types><br />
<key>REF</key><br />
<key>LinAlg.REF</key><br />
<key>row echelon form</key><br />
<wiki-category>Package_linalg</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMNLASolve&diff=11439ApCoCoA-1:CharP.IMNLASolve2010-12-07T11:28:26Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.IMNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Improved Mutant NLA<tt>-</tt>Algorithm to find the unique zero. The Improved Mutant NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The Improved Mutant NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Improved Mutant NLA<tt>-</tt>Algorithm is the NLA<tt>-</tt>Algorithm with improved mutant strategy. It uses <ref>linalg.EF</ref> for gaussian elimination.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.IMNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 1<br />
The total No. of Mutants found are = 1<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1] <br />
<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.IMNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 2<br />
The total No. of Mutants found are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=10<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=10<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=10<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=22<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMNLASolve&diff=11438ApCoCoA-1:CharP.IMNLASolve2010-12-07T11:21:26Z<p>132.231.10.53: New page: <command> <title>CharP.GBasisF2</title> <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> <synt...</p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA<tt>-</tt>Algorithm to find the unique zero. The Mutant NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA<tt>-</tt>Algorithm is the NLA<tt>-</tt>Algorithm with mutant strategy. It uses <ref>linalg.EF</ref> for gaussian elimination.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MNLASolve&diff=11437ApCoCoA-1:CharP.MNLASolve2010-12-07T11:18:18Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA<tt>-</tt>Algorithm to find the unique zero. The Mutant NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA<tt>-</tt>Algorithm is the NLA<tt>-</tt>Algorithm with mutant strategy. It uses <ref>linalg.EF</ref> for gaussian elimination.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MNLASolve&diff=11436ApCoCoA-1:CharP.MNLASolve2010-12-07T11:14:14Z<p>132.231.10.53: New page: <command> <title>CharP.GBasisF2</title> <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> <synt...</p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MNLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA<tt>-</tt>Algorithm to find the unique zero. The Mutant NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA<tt>-</tt>Algorithm is the NLA<tt>-</tt>Algorithm with mutant strategy. It uses <ref>linalg.EF</ref> for gaussian elimination.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=8<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=10<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=7<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=4<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MNLASolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=4<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=12<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=13<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination for finding Mutants...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Compeleted<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination to check solution coordinate...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMXLSolve&diff=11435ApCoCoA-1:CharP.IMXLSolve2010-12-07T10:41:03Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.IMXLSolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Improved Mutant XL<tt>-</tt>Algorithm to find the unique zero. The idea is to linearize the polynomial system by considering terms as indeterminates and then apply gaussian elimination to find a univariate polynomial. If no univariate polynomial is found then the system is extended by generating more polynomials in the ideal and gaussian elimination is applied again. In this way by appling gaussian elimination repeatedly we find the zero of the system. In fact Improved Mutant XL<tt>-</tt>Algorithm is the XL<tt>-</tt>Algorithm with improved mutant strategy. The Improved Mutant XL<tt>-</tt>Algorithm is impelemented only to find the unique zero. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.IMXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 1<br />
The total No. of Mutants found are = 1<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[0, 1, 0, 1]<br />
[0, 1, 0, 1]<br />
<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.IMXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 4<br />
The total No. of Mutants found are = 4<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=10<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The total No. of Mutants found are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=17<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=25<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=13<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution.<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.MXLSolve</see><br />
<see>CharP.GBasisF2</see><br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MXLSolve&diff=11434ApCoCoA-1:CharP.MXLSolve2010-12-07T10:37:43Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MXLSolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant XL<tt>-</tt>Algorithm to find the unique zero. The idea is to linearize the polynomial system by considering terms as indeterminates and then apply gaussian elimination to find a univariate polynomial. If no univariate polynomial is found then the system is extended by generating more polynomials in the ideal and gaussian elimination is applied again. In this way by appling gaussian elimination repeatedly we find the zero of the system. In fact Mutant XL<tt>-</tt>Algorithm is the XL<tt>-</tt>Algorithm with mutant strategy. The Mutant XL<tt>-</tt>Algorithm is impelemented only to find the unique zero. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.MXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[0, 1, 0, 1]<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.IMXLSolve</see><br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.XLSolve&diff=11433ApCoCoA-1:CharP.XLSolve2010-12-07T10:34:15Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.XLSolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses XL<tt>-</tt>Algorithm to find the unique zero. The idea is to linearize the polynomial system by considering terms as indeterminates and then apply gaussian elimination to find a univariate polynomial. If no univariate polynomial is found then the system is extended by generating more polynomials in the ideal and gaussian elimination is applied again. In this way by appling gaussian elimination repeatedly we find the zero of the system. The XL<tt>-</tt>Algorithm is impelemented only to find a unique solution. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.XLSolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The size of Matrix is:<br />
No. of Rows=16<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[0, 1, 0, 1]<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.XLSolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The size of Matrix is:<br />
No. of Rows=18<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The size of Matrix is:<br />
No. of Rows=13<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.GBasisF16</see><br />
<see>CharP.IMXLSolve</see><br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.NLASolve&diff=11432ApCoCoA-1:CharP.NLASolve2010-12-07T10:32:19Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.NLASolve(F:LIST, Sparse:BOOL):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses NLA<tt>-</tt>Algorithm to find the unique zero. The NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. You can also choose between sparse and dense implementation of gaussian elimination. In sparse case it uses <ref>slinalg.SEF</ref> and in dense case it uses <ref>linalg.EF</ref>.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@param <em>Sparse:</em> Sparsity of given polynomial system. Set Sparse to <em>True</em> if the given polynomial system is sparse. Set Sparse to <em>False</em> if the given polynomial system is dense.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
Sparse:=True;<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=17<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
Sparse:=False;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.NLASolve&diff=11431ApCoCoA-1:CharP.NLASolve2010-12-07T10:30:19Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.NLASolve(F:LIST, Sparse:BOOL):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses NLA<tt>-</tt>Algorithm to find the unique zero. The NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. You can also choose between sparse and dense implementation of gaussian elimination. In sparse case it uses <ref>slinalg.SEF</ref> and in dense case it uses <ref>linalg.EF</ref>.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@param <em>Sparse:</em> Sparsity of given polynomial system. Set Sparse to <em>True</em> if the given polynomial system is sparse. Set Sparse to <em>False</em> if the given polynomial system is dense.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
Sparse:=1;<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=17<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
Sparse:=0;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.NLASolve&diff=11430ApCoCoA-1:CharP.NLASolve2010-12-07T10:26:07Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.NLASolve(F:LIST, Sparse:BOOL):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses NLA-Algorithm to find the unique zero. The NLA-Algorithm generates a sequence of linear systems to solve the given system. The NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. You can also choose between sparse and dense implementation of gaussian elimination. In sparse case it uses <ref>slinalg.SEF</ref> and in dense case it uses <ref>linalg.EF</ref>.<br />
<br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@param <em>Sparse:</em> Sparsity of given polynomial system. Set Sparse to True if the given polynomial system is sparse. Set Sparse to False if the given polynomial system is dense.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
Sparse:=1;<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=17<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
Sparse:=0;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.NLASolve&diff=11429ApCoCoA-1:CharP.NLASolve2010-12-07T10:24:42Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.NLASolve(F:LIST, Sparse:BOOL):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses NLA-Algorithm to find the unique zero. The NLA-Algorithm generates a sequence of linear systems to solve the given system. The NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. You can also choose between sparse and dense implementation of gaussian elimination. In sparse case it uses <ref>slinalg.SEF</ref> and in dense case it uses <ref>linalg.EF</ref><br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@param <em>Sparse:</em> Sparsity of given polynomial system. Set Sparse to True if the given polynomial system is sparse. Set Sparse to False if the given polynomial system is dense.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
Sparse:=1;<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=17<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
Sparse:=0;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=9<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=57<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = NA<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution or it has<br />
no solution over the finite field F2.<br />
<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.NLASolve&diff=11428ApCoCoA-1:CharP.NLASolve2010-12-07T09:59:00Z<p>132.231.10.53: New page: <command> <title>CharP.GBasisF2</title> <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> <synt...</p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.NLASolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses NLA-Algorithm to find the unique zero. The NLA-Algorithm generates a sequence of linear systems to solve the given system. The NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@param <em>Sparse:</em> Sparsity of given polynomial system. Set Sparse to 0 if the given polynomial system is sparse. Set Sparse to 1 if the given polynomial system is dense.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
Sparse:=1;<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
Finding Variable: x[4]<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=21<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[4] = 1<br />
Finding Variable: x[3]<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=5<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=17<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[3] = 0<br />
Finding Variable: x[2]<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=4<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
x[2] = 1<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
Sparse:=0;<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.NLASolve(F,Sparse);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.MXLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF2</see><br />
<see>CharP.XLSolve</see><br />
<see>CharP.IMXLSolve</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.IMXLSolve&diff=11427ApCoCoA-1:CharP.IMXLSolve2010-12-07T09:15:25Z<p>132.231.10.53: New page: <command> <title>CharP.GBasisF2</title> <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> <synt...</p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.IMXLSolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Improved Mutant XL-Algorithm to find the unique zero. The idea is to linearize the polynomial system by considering terms as indeterminates and then apply gaussian elimination to find a univariate polynomial. If no univariate polynomial is found then the system is extended by generating more polynomials in the ideal and gaussian elimination is applied again. In this way by appling gaussian elimination repeatedly we find the zero of the system. In fact Improved Mutant XL Algorithm is the XL Algorithm with improved mutant strategy. The Improved Mutant XL-Algorithm is impelemented only to find the unique zero. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.IMXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 1<br />
The total No. of Mutants found are = 1<br />
The No. of Mutants of Minimum degree (Mutants used) are = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[0, 1, 0, 1]<br />
[0, 1, 0, 1]<br />
<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.IMXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=7<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 4<br />
The total No. of Mutants found are = 4<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=10<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The total No. of Mutants found are = 2<br />
The No. of Mutants of Minimum degree (Mutants used) are = 2<br />
The size of Matrix is:<br />
No. of Rows=15<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=17<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=25<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
No. of New Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=13<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now are:<br />
[x[1], x[2], x[3], x[4]]<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution.<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.IMNLASolve</see><br />
<see>CharP.MNLASolve</see><br />
<see>CharP.NLASolve</see><br />
<see>CharP.MXLSolve</see><br />
<see>CharP.GBasisF2</see><br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.MXLSolve&diff=11426ApCoCoA-1:CharP.MXLSolve2010-12-07T09:09:50Z<p>132.231.10.53: </p>
<hr />
<div><command><br />
<title>CharP.GBasisF2</title><br />
<short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description><br />
<syntax><br />
CharP.MXLSolve(F:LIST):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<br />
<par/><br />
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant XL-Algorithm to find the unique zero. The idea is to linearize the polynomial system by considering terms as indeterminates and then apply gaussian elimination to find a univariate polynomial. If no univariate polynomial is found then the system is extended by generating more polynomials in the ideal and gaussian elimination is applied again. In this way by appling gaussian elimination repeatedly we find the zero of the system. In fact Mutant XL Algorithm is the XL Algorithm with mutant strategy. The Mutant XL-Algorithm is impelemented only to find the unique zero. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. <br />
<br />
<br />
<itemize><br />
<item>@param <em>F:</em> List of polynomials of given system.</item><br />
<item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item><br />
</itemize><br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[<br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, <br />
x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, <br />
x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1<br />
];<br />
<br />
-- Then we compute the solution with<br />
CharP.MXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen together with the solution at the end.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=8<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 1<br />
The size of Matrix is:<br />
No. of Rows=11<br />
No. of Columns=11<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[0, 1, 0, 1]<br />
[0, 1, 0, 1]<br />
<br />
</example><br />
<br />
<br />
<example><br />
Use Z/(2)[x[1..4]];<br />
F:=[ <br />
x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], <br />
x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], <br />
x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]<br />
];<br />
<br />
-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions <br />
<br />
-- Then we compute the solution with<br />
CharP.MXLSolve(F);<br />
<br />
-- And we achieve the following information on the screen.<br />
----------------------------------------<br />
The size of Matrix is:<br />
No. of Rows=4<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The size of Matrix is:<br />
No. of Rows=3<br />
No. of Columns=9<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 4<br />
The size of Matrix is:<br />
No. of Rows=27<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=12<br />
No. of Columns=14<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=19<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
The No. of Mutants found = 0<br />
The size of Matrix is:<br />
No. of Rows=14<br />
No. of Columns=15<br />
Appling Gaussian Elimination...<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
Gaussian Elimination Completed.<br />
The variables found till now, if any are:<br />
[x[1], x[2], x[3], x[4]]<br />
Please Check the uniqueness of solution.<br />
The Given system of polynomials does not<br />
seem to have a unique solution.<br />
<br />
</example><br />
<br />
<br />
</description><br />
<seealso><br />
<see>CharP.XLSolve</see><br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>CharP.GBasisF4</see><br />
<see>CharP.GBasisF8</see><br />
<see>CharP.GBasisF16</see><br />
<see>CharP.IMXLSolve</see><br />
<br />
</seealso><br />
<br />
<types><br />
<type>apcocoaserver</type><br />
<type>ideal</type><br />
<type>groebner</type><br />
</types><br />
<br />
<key>charP.GBasisF2</key><br />
<key>GBasisF2</key><br />
<key>finite field</key><br />
<wiki-category>Package_charP</wiki-category><br />
</command></div>132.231.10.53