Solve and Big Memory needed

*To*: mathgroup at smc.vnet.net*Subject*: [mg110405] Solve and Big Memory needed*From*: leigh pascoe <leigh at evry.inserm.fr>*Date*: Wed, 16 Jun 2010 05:42:07 -0400 (EDT)

Dear Experts/Adam Strebonski, I have been trying for several years to solve a set of simultaneous polynomial equations. Since the algorithms used by Mathematica are guaranteed to find solutions to these types of equations it should be a slam-dunk. However I have been frustrated by memory limitations (Windows XP/quad core with 4 Go) - Mathematica crashes out with a memory exceeded error. Having seen the post "Big Memory needed" and the welcome development that V8 of Mathematica is much better at these problems, I am hoping that someone with access to V8 and a large capacity machine may be able to help me. The equations are given below and occur in the theory of population genetics. The variables of interest are denoted x1, x2, x3, x4 First define the variables d := x2*x3 - x1*x4 c := r*w14*d and some constants (symmetric matrix) w21 := w12 w31 := w13 w41 := w14 w23 := w14 w42 := w24 w43 := w34 w32 := w14 We use the following intermediate functions to simplify the writing of the equations: w1 := x1*w11 + x2*w12 + x3*w13 + x4*w14 w2 := x1*w12 + x2*w22 + x3*w23 + x4*w24 w3 := x1*w13 + x2*w23 + x3*w33 + x4*w34 w4 := x1*w14 + x2*w24 + x3*w34 + x4*w44 Now the equations can be written eq1 = x1 + x2 + x3 + x4 == 1 eq2 = x1*(x1*w1 + x2*w2 + x3*w3 + x4*w4) == x1*w1 + c eq3 = x2*(x1*w1 + x2*w2 + x3*w3 + x4*w4) == x2*w2 - c eq4 = x3*(x1*w1 + x2*w2 + x3*w3 + x4*w4) == x3*w3 - c These are three equations of 3rd degree in x's and a linear constraint. I attempt to solve them with the Solve command FullSimplify[Solve[{eq1, eq2, eq3, eq4}, {x1, x2, x3, x4}]] No more memory available. Mathematica kernel has shut down. Try quitting other applications and then retry. If anyone can help me to know the number of solutions and their formulae, I would be greatly appreciative. Eventually I would also like to subject the solutions to the restriction that all the xi's are in the closed interval [0,1], which of course depends on the values of the constants. Mathematica is able to find the solutions of the special case when c=0 eq5 = x1 + x2 + x3 + x4 == 1 eq6 = x1*(x1*w1 + x2*w2 + x3*w3 + x4*w4) == x1*w1 eq7 = x2*(x1*w1 + x2*w2 + x3*w3 + x4*w4) == x2*w2 eq8 = x3*(x1*w1 + x2*w2 + x3*w3 + x4*w4) == x3*w3 Timing[FullSimplify[Solve[{eq5, eq6, eq7, eq8}, {x1, x2, x3, x4}]]] The 15 solutions to the simplified case are found in about 8 secs and occupy several screens. However these solutions can be found by hand and written much more succintly than the Mathematica result. Thanks in advance for any help. LP