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

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{{Version|1}}
 
<command>
 
<command>
 
<title>Bertini.BMSolve</title>
 
<title>Bertini.BMSolve</title>
−
<short_description>Solves zero dimensional non-homogeneous polynomial system using mult-homogenization 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.BMSolve(M:LIST)
+
Bertini.BMSolve(P:LIST):LIST
 
</syntax>
 
</syntax>
 
<description>
 
<description>
 
<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.
 
<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/>
 +
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>
 
<itemize>
−
<item>@param <em>M</em>: List of polynomials in the system.</item>
+
<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>
 
</itemize>
 
   
 
   
 
<example>
 
<example>
−
-- We want to solve the non-homogenous system x[1]^2+x[2]^2-5=0, x[1]x[2]-2=0, using multi-homogenization.  
+
-- 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>
 +
 
 +
<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[x[1..2]];            --  Define appropriate ring
+
Use S ::= QQ[z[1..2]];             
−
M := [x[1]^2+x[2]^2-5, x[1]x[2]-2];
+
P := [(29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2];
  
 
-- Then we compute the solution with
 
-- Then we compute the solution with
−
Bertini.BMSolve(M);
+
Bertini.BMSolve(P);
  
−
-- And we achieve a list of lists containing finite solutions:
+
-- 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(-1754775022937541/1000000000000000000000000000, -6761671559595563/10000000000000000000000000000),
−
Vector(500000000000001/500000000000000, -619892334722183/25000000000000000000000000000)],
+
Vector(947843957587963/25000000000000000000000000000, 623113227620389/5000000000000000000000000000)
−
[Vector(-2, 1724810333092189/1000000000000000000000000000000),
+
],
−
Vector(-500000000000001/500000000000000, -355984244774691/200000000000000000000000000000)],
+
[
−
[Vector(-9999999999999971/10000000000000000, -4053926086793577/1000000000000000000000000000000),
+
Vector(-85573832182963743719/50000000000000000000000000000000, -89829012439528360233/250000000000000000000000000000000),  
−
Vector(-1999999999999999/1000000000000000, -3669041992638223/5000000000000000000000000000000)]]
+
Vector(-230164951873451072943/2500000000000000000000000000000000, 298328875801698252183/10000000000000000000000000000000000)
−
--For other Bertini output files refer to Bertini directory (../ApCoCoA-1.2/Bertini)
+
],
 +
[
 +
Vector(-1479267029218781/1000000000000000000000000000, -5565180110034249/10000000000000000000000000000),  
 +
Vector(-4881416330105221/50000000000000000000000000000, 856957743028027/5000000000000000000000000000)
 +
]
 +
]
 +
 
 +
 
 +
--For Bertini output files refer to ApCoCoA directory/Bertini.
 
------------------------------------------
 
------------------------------------------
 
</example>
 
</example>
 +
  
 
</description>
 
</description>
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<seealso>
 
<seealso>
−
  <see>Bertini.BCMSolve</see>
+
  <see>ApCoCoA-1:Bertini.BCMSolve|Bertini.BCMSolve</see>
−
  <see>Bertini.BSolve</see>
+
  <see>ApCoCoA-1:Bertini.BSolve|Bertini.BSolve</see>
−
  <see>Bertini.BUHSolve</see>
+
  <see>ApCoCoA-1:Bertini.BUHSolve|Bertini.BUHSolve</see>
 
</seealso>
 
</seealso>
  
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<key>bertini.bmsolve</key>
 
<key>bertini.bmsolve</key>
 
<key>solve zero dimensional Non-homogeneous polynomial system using mult-homogenization</key>>
 
<key>solve zero dimensional Non-homogeneous polynomial system using mult-homogenization</key>>
−
<wiki-category>Package_bertini</wiki-category>
+
<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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