ApCoCoA-1:Bertini.BMSolve: Difference between revisions

From ApCoCoAWiki
← Older edit
No edit summary
m insert version info
 
(14 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{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>
Line 42: Line 88:


<seealso>
<seealso>
  <see>Bertini.BCMSolve</see>
  <see>ApCoCoA-1:Bertini.BCMSolve|Bertini.BCMSolve</see>
  <see>Bertini.BCSolve</see>
  <see>ApCoCoA-1:Bertini.BSolve|Bertini.BSolve</see>
<see>Bertini.BSolve</see>
  <see>ApCoCoA-1:Bertini.BUHSolve|Bertini.BUHSolve</see>
  <see>Bertini.BUHSolve</see>
</seealso>
</seealso>


Line 51: Line 96:
<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



>


Retrieved from "https://app.apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Bertini.BMSolve&oldid=15417"