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 | + | <short_description>Solves a zero dimensional non-homogeneous polynomial system using multi-homogenization and default configurations.</short_description> |
<syntax> | <syntax> | ||
| − | Bertini.BMSolve( | + | 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> | + | <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.</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[ | + | 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 | -- Then we compute the solution with | ||
| − | Bertini.BMSolve( | + | Bertini.BMSolve(P); |
| − | -- And we achieve a list of lists containing finite solutions | + | -- And we achieve a list of lists containing finite solutions. |
---------------------------------------- | ---------------------------------------- | ||
| − | [[Vector( | + | [ |
| − | + | [ | |
| − | + | Vector(-1754775022937541/1000000000000000000000000000, -6761671559595563/10000000000000000000000000000), | |
| − | + | Vector(947843957587963/25000000000000000000000000000, 623113227620389/5000000000000000000000000000) | |
| − | + | ], | |
| − | + | [ | |
| − | + | Vector(-85573832182963743719/50000000000000000000000000000000, -89829012439528360233/250000000000000000000000000000000), | |
| − | + | Vector(-230164951873451072943/2500000000000000000000000000000000, 298328875801698252183/10000000000000000000000000000000000) | |
| − | --For | + | ], |
| + | [ | ||
| + | Vector(-1479267029218781/1000000000000000000000000000, -5565180110034249/10000000000000000000000000000), | ||
| + | Vector(-4881416330105221/50000000000000000000000000000, 856957743028027/5000000000000000000000000000) | ||
| + | ] | ||
| + | ] | ||
| + | |||
| + | |||
| + | --For Bertini output files refer to ApCoCoA directory/Bertini. | ||
------------------------------------------ | ------------------------------------------ | ||
</example> | </example> | ||
| + | |||
</description> | </description> | ||
| Line 43: | Line 88: | ||
<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> | ||
| 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
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