Difference between revisions of "ApCoCoA-1:Bertini.BMSolve"

From ApCoCoAWiki
(New page: <command> <title>BSolve</title> <short_description>Solves zero dimensional Homogeneous or Non-Homogeneous polynomial system with Default Configurations.</short_description> <syntax> Bertin...)
 
m (insert version info)
 
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 +
{{Version|1}}
 
<command>
 
<command>
−
<title>BSolve</title>
+
<title>Bertini.BMSolve</title>
−
<short_description>Solves zero dimensional Homogeneous or Non-Homogeneous polynomial system with Default Configurations.</short_description>
+
<short_description>Solves a zero dimensional non-homogeneous polynomial system using multi-homogenization and default configurations.</short_description>
 
<syntax>
 
<syntax>
−
Bertini.BSolve(M:LIST, SysTyp:STRING)
+
Bertini.BMSolve(P:LIST):LIST
 
</syntax>
 
</syntax>
 
<description>
 
<description>
−
{{ApCoCoAServer}}
+
<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.
−
 
+
<par/>
−
<em>M</em>: List of polynomials in the system to be solved.
+
This function solves a polynomial system of equations using multihomogeneous homotopy. The polynomial system of equations must be quadratic. If the system has <tt>N</tt> variables then multihomogeneous homotopy will introduce <tt>N</tt> homogeneous variables to solve the system. It uses total degree homotopy to find all isolated solutions and default configurations provided by Bertini. The system of polynomials should be non-homogeneous. The output will be the list of all finite solutions.  
−
 
+
<itemize>
−
<em>SysTyp</em>: Type of the system to be solved. Homogeneous ("hom") or nonhomogeneous ("Nhom").
+
<item>@param <em>P</em>: List of polynomials of the given system.</item>
−
 
+
<item>@return A list of lists containing the finite solutions of the polynomial system P.</item>
 +
</itemize>
 
   
 
   
 
<example>
 
<example>
−
-- Zero Dimensional Non-Homogeneous Solving
+
-- We want to solve the non-homogenous polynomial system x[1]^2+x[2]^2-5=0, x[1]x[2]-2=0, using multi-homogenization.  
−
-- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0.  
 
  
−
Use S ::= QQ[x,y];            --  Define appropriate ring
+
Use S ::= QQ[x[1..2]];             
−
M := [x^2+y^2-5, xy-2];
+
P := [x[1]^2+x[2]^2-5, x[1]x[2]-2];
−
SysTyp := "Nhom";
 
  
 
-- Then we compute the solution with
 
-- Then we compute the solution with
−
$Bertini.BSolve(M,SysTyp);
+
Bertini.BMSolve(P);
  
−
-- And we achieve:
+
-- And we achieve a list of lists containing finite solutions.
 
----------------------------------------
 
----------------------------------------
−
The number of real finite solutions are:
+
[
−
4     
+
[
−
The real finite solutions are:
+
Vector(1000000000000001/1000000000000000, -2305082859180703/100000000000000000000000000000),
−
                                       
+
Vector(1999999999999971/1000000000000000, 4135565953005217/100000000000000000000000000000)
−
2.000000000000052e+00 -1.207721921243940e-14
+
],
−
9.999999999999164e-01 1.727395183148409e-14
+
[
 +
Vector(1000000000000003/500000000000000, 2604577577014449/50000000000000000000000000000),
 +
Vector(500000000000001/500000000000000, -619892334722183/25000000000000000000000000000)
 +
],
 +
[
 +
Vector(-2, 1724810333092189/1000000000000000000000000000000),
 +
Vector(-500000000000001/500000000000000, -355984244774691/200000000000000000000000000000)
 +
],
 +
[
 +
Vector(-9999999999999971/10000000000000000, -4053926086793577/1000000000000000000000000000000),
 +
Vector(-1999999999999999/1000000000000000, -3669041992638223/5000000000000000000000000000000)
 +
]
 +
]
  
−
-9.999999999999680e-01 -1.380221440309691e-14
+
--For Bertini output files refer to ApCoCoA directory/Bertini.
−
-2.000000000000005e+00 -8.389594590085023e-15
+
------------------------------------------
−
 
+
</example>
−
9.999999999999293e-01 2.686603221243866e-14
 
−
2.000000000000473e+00 4.530296702485832e-13
 
−
 
 
−
-2.000000000000031e+00 -1.809322618557695e-15
 
−
-9.999999999999383e-01 -2.558999563654189e-15
 
−
 
 
−
For summary of all solutions refer to ApCoCoAServer.
 
  
−
</example>
 
 
<example>
 
<example>
−
-- Zero Dimensional Homogeneous Solving
+
-- We want to solve the non-homogenous polynomial system (29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2, using multi-homogenization.  
−
-- We want to solve zero dimensional homogeneous system x^2-z^2=0, xy-z^2=0.
 
  
−
Use S ::= QQ[x,y,z];            --  Define appropriate ring
+
Use S ::= QQ[z[1..2]];             
−
M := [x^2-z^2, xy-z^2];
+
P := [(29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2];
−
SysTyp := "hom";
 
  
 
-- Then we compute the solution with
 
-- Then we compute the solution with
−
$Bertini.BSolve(M,SysTyp);
+
Bertini.BMSolve(P);
  
−
-- And we achieve:
+
-- And we achieve a list of lists containing finite solutions.
 
----------------------------------------
 
----------------------------------------
−
The number of real solutions are:
+
[
−
4     
+
[
−
The real solutions are:
+
Vector(-1754775022937541/1000000000000000000000000000, -6761671559595563/10000000000000000000000000000),
−
                                       
+
Vector(947843957587963/25000000000000000000000000000, 623113227620389/5000000000000000000000000000)
 +
],
 +
[
 +
Vector(-85573832182963743719/50000000000000000000000000000000, -89829012439528360233/250000000000000000000000000000000),
 +
Vector(-230164951873451072943/2500000000000000000000000000000000, 298328875801698252183/10000000000000000000000000000000000)
 +
],
 +
[
 +
Vector(-1479267029218781/1000000000000000000000000000, -5565180110034249/10000000000000000000000000000),
 +
Vector(-4881416330105221/50000000000000000000000000000, 856957743028027/5000000000000000000000000000)
 +
]
 +
]
  
−
4.750270171019972e-01 7.277175694441498e-01
 
−
4.750270171019972e-01 7.277175694441498e-01
 
−
4.750270171019972e-01 7.277175694441498e-01
 
  
−
-1.161874166440340e+00 -1.121939725361908e+00
+
--For Bertini output files refer to ApCoCoA directory/Bertini.
−
-1.161874166440340e+00 -1.121939725361908e+00
+
------------------------------------------
−
1.161874166440340e+00 1.121939725361908e+00
+
</example>
−
 
 
−
-1.213218743783253e-14 9.540042296620362e-14
 
−
1.297490331797821e+00 -3.349764345312171e-01
 
−
-9.696352192508132e-14 -3.162549982974766e-13
 
−
 
 
−
 
 
−
-2.845295858183006e-14 1.079961801218032e-13
 
−
1.297490331797885e+00 -3.349764345312022e-01
 
−
-9.799048563439788e-14 -3.558617333271439e-13
 
−
 
 
−
For summary of all solutions refer to ApCoCoAServer
 
  
−
------------------------------------
 
−
</example>
 
  
 
</description>
 
</description>
 
<types>
 
<types>
−
   <type>cocoaserver</type>
+
   <type>apcocoaserver</type>
 +
  <type>poly_system</type>
 
</types>
 
</types>
−
<key>bsolve</key>
+
 
−
<key>solve zero dimensional polynomial system</key>
+
<seealso>
−
<key>solve b</key>
+
<see>ApCoCoA-1:Bertini.BCMSolve|Bertini.BCMSolve</see>
−
<key>eullah</key>
+
<see>ApCoCoA-1:Bertini.BSolve|Bertini.BSolve</see>
−
<wiki-category>Package_bertini</wiki-category>
+
<see>ApCoCoA-1:Bertini.BUHSolve|Bertini.BUHSolve</see>
 +
</seealso>
 +
 
 +
<key>bmsolve</key>
 +
<key>bertini.bmsolve</key>
 +
<key>solve zero dimensional Non-homogeneous polynomial system using mult-homogenization</key>>
 +
<wiki-category>ApCoCoA-1:Package_bertini</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:52, 7 October 2020

This article is about a function from ApCoCoA-1.

Bertini.BMSolve

Solves a zero dimensional non-homogeneous polynomial system using multi-homogenization and default configurations.

Syntax

Bertini.BMSolve(P:LIST):LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function solves a polynomial system of equations using multihomogeneous homotopy. The polynomial system of equations must be quadratic. If the system has N variables then multihomogeneous homotopy will introduce N homogeneous variables to solve the system. It uses total degree homotopy to find all isolated solutions and default configurations provided by Bertini. The system of polynomials should be non-homogeneous. The output will be the list of all finite solutions.

  • @param P: List of polynomials of the given system.

  • @return A list of lists containing the finite solutions of the polynomial system P.

Example

-- We want to solve the non-homogenous polynomial system x[1]^2+x[2]^2-5=0, x[1]x[2]-2=0, using multi-homogenization. 

Use S ::= QQ[x[1..2]];             
P := [x[1]^2+x[2]^2-5, x[1]x[2]-2];

-- Then we compute the solution with
Bertini.BMSolve(P);

-- And we achieve a list of lists containing finite solutions.
----------------------------------------
[
[
 Vector(1000000000000001/1000000000000000, -2305082859180703/100000000000000000000000000000),
 Vector(1999999999999971/1000000000000000, 4135565953005217/100000000000000000000000000000)
],
[
 Vector(1000000000000003/500000000000000, 2604577577014449/50000000000000000000000000000),
 Vector(500000000000001/500000000000000, -619892334722183/25000000000000000000000000000)
],
[
 Vector(-2, 1724810333092189/1000000000000000000000000000000),
 Vector(-500000000000001/500000000000000, -355984244774691/200000000000000000000000000000)
],
[
 Vector(-9999999999999971/10000000000000000, -4053926086793577/1000000000000000000000000000000),
 Vector(-1999999999999999/1000000000000000, -3669041992638223/5000000000000000000000000000000)
]
]

--For Bertini output files refer to ApCoCoA directory/Bertini.
------------------------------------------

Example

-- We want to solve the non-homogenous polynomial system (29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2, using multi-homogenization. 

Use S ::= QQ[z[1..2]];             
P := [(29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2];

-- Then we compute the solution with
Bertini.BMSolve(P);

-- And we achieve a list of lists containing finite solutions.
----------------------------------------
[
[
Vector(-1754775022937541/1000000000000000000000000000, -6761671559595563/10000000000000000000000000000),
Vector(947843957587963/25000000000000000000000000000, 623113227620389/5000000000000000000000000000)
],
[
Vector(-85573832182963743719/50000000000000000000000000000000, -89829012439528360233/250000000000000000000000000000000), 
Vector(-230164951873451072943/2500000000000000000000000000000000, 298328875801698252183/10000000000000000000000000000000000)
],
[
Vector(-1479267029218781/1000000000000000000000000000, -5565180110034249/10000000000000000000000000000), 
Vector(-4881416330105221/50000000000000000000000000000, 856957743028027/5000000000000000000000000000)
]
]


--For Bertini output files refer to ApCoCoA directory/Bertini.
------------------------------------------



See also

Bertini.BCMSolve

Bertini.BSolve

Bertini.BUHSolve



>


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