ApCoCoA-1:Hom.LRSolve
Hom.HSolve
Solves a zero dimensional square homogeneous or non-homogeneous polynomial system of equations.
Syntax
Hom.HSolve(P: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.
The use of this function is two folds depending on the input that you provide in ApCoCoAServer during execution. After passing the command HSolve(P) in CoCoA 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 nonhomogeneous.
@param P: List of polynomials of the given system.
@return A list of lists containing the finite solutions of the system P.
Example
-- We want to solve the given system for Gamma=0.2+1.2*I and using default configurations. -- The start solution for the homotopy is [ [0.0, 0.0 ], [1.0, 0.0], [0.0, 0.0], [1.0, 0.0] ]. -- The start system for the homotopy is [ x[1],x[2]-1,x[3],x[4]-1,x[1]^2 - x[1], x[2]^2 - x[2], x[3]^2 - x[3], x[4]^2 - x[4] ]. Use S ::= QQ[x[1..4]]; P := [ 186x[1]x[2]x[3]x[4] - 42x[1]x[2]x[3] - 24x[1]x[2]x[4] - 18x[1]x[3]x[4] - 48x[2]x[3]x[4] + 9x[1]x[2] - 6x[1]x[3] + 9x[2]x[3] + 15x[2]x[4] + 9x[3]x[4] + 15x[1] + 15x[3] - 15, 186x[1]x[2]x[3]x[4] - 42x[1]x[2]x[3] - 24x[1]x[2]x[4] - 48x[1]x[3]x[4] - 18x[2]x[3]x[4] + 9x[1]x[2] + 9x[1]x[3] - 6x[2]x[3] + 15x[1]x[4] + 9x[3]x[4] + 15x[2] + 15x[3] - 15, 186x[1]x[2]x[3]x[4] - 48x[1]x[2]x[3] - 18x[1]x[2]x[4] - 42x[1]x[3]x[4] - 24x[2]x[3]x[4] + 9x[1]x[2] + 9x[1]x[3] + 15x[2]x[3] - 6x[1]x[4] + 9x[3]x[4] + 15x[1] + 15x[4] - 15, 24x[1]x[2]x[3]x[4] - 3x[1]x[3] - 3x[2]x[3] - 3x[1]x[4] - 3x[2]x[4] + 3, x[1]^2 - x[1], x[2]^2 - x[2], x[3]^2 - x[3], x[4]^2 - x[4] ]; SSys := [ x[1],x[2]-1,x[3],x[4]-1,x[1]^2 - x[1], x[2]^2 - x[2], x[3]^2 - x[3], x[4]^2 - x[4] ]; Gamma := <quotes>0.2+1.2*I</quotes>; SSol := [ [ [<quotes>0.0</quotes>, <quotes>0.0</quotes>], [<quotes>1.0</quotes>, <quotes>0.0</quotes>], [<quotes>0.0</quotes>, <quotes>0.0</quotes>], [<quotes>1.0</quotes>, <quotes>0.0</quotes>] ] ]; ConfigSet := [<quotes>USERHOMOTOPY: 1</quotes>]; -- Then we compute the solution with Bertini.BUHSolve(P, SSys, Gamma, SSol, ConfigSet); -- And we achieve a list of lists containing real solutions. ---------------------------------------- [[0, 1, 0, 1]] --For Bertini output files please refer to ApCoCoA directory/Bertini.
See also
