Re: Can I solve this system of nonlinear equations?

*To*: mathgroup at smc.vnet.net*Subject*: [mg125267] Re: Can I solve this system of nonlinear equations?*From*: danl at wolfram.com*Date*: Sat, 3 Mar 2012 06:53:36 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jil5ld$rrm$1@smc.vnet.net>

On Wednesday, February 29, 2012 6:28:29 AM UTC-6, Andy wrote: > I'm dealing with systems of nonlinear equations that have 8 equations > and 8 unknowns. Here's an example: > > Solve[{(((c - a)/0.002) - (0.995018769272803 + h*b)) == 0, > (((d - b)/0.002) - (0.990074756047929 + h*c)) == 0, > (((e - c)/0.002) - (0.985167483257382 + h*d)) == 0, > (((f - d)/0.002) - (0.980296479563062 + h*e)) == 0, > (((g - e)/0.002) - (0.975461279165159 + h*f)) == 0, > (((-1*e + 8*d - 8*b + a)/(12*0.001)) - (0.990074756047929 + h*c)) == > 0, > (((-1*f + 8*e - 8*c + b)/(12*0.001)) - (0.985167483257382 + h*d)) == > 0, > (((-1*g + 8*f - 8*d + c)/(12*0.001)) - (0.980296479563062 + h*e)) == > 0}, {a, b, c, d, e, f, g, h}] > > Whenever I try this, Mathematica 7 just returns the empty set {}. How > can I tell if this is unsolvable? Shouldn't I at least be able to get > a numerical approximation with NSolve? I've tried using stochastic > optimization to get approximate answers but every method gives poor > results, and that's why I would like to at least approximately solve > this if possible. Thanks very much for any help~ It is not solvable. Or at least, a rationalized version has no solution. Here is one way to show this. polys = Apply[ Subtract, {(((c - a)/0.002) - (0.995018769272803 + h*b)) == 0, (((d - b)/0.002) - (0.990074756047929 + h*c)) == 0, (((e - c)/0.002) - (0.985167483257382 + h*d)) == 0, (((f - d)/0.002) - (0.980296479563062 + h*e)) == 0, (((g - e)/0.002) - (0.975461279165159 + h*f)) == 0, (((-1*e + 8*d - 8*b + a)/(12*0.001)) - (0.990074756047929 + h*c)) == 0, (((-1*f + 8*e - 8*c + b)/(12*0.001)) - (0.985167483257382 + h*d)) == 0, (((-1*g + 8*f - 8*d + c)/(12*0.001)) - (0.980296479563062 + h*e)) == 0}, 1]; p2 = Rationalize[polys, 0]; p3 = Expand[p2]; vars = Variables[p2]; For showing non-solvability, it suffices (by Hilbert's Nullstellensatz, apologies for pulling that bit of math jargon out of a hat) to find a way to write a constant as an algebraic sum of these polynomials. That is to say, we show the ideal defined by those polynomials contains 1. One way to do so uses a bit of code that I posted to MathGroup around a year ago. http://forums.wolfram.com/mathgroup/archive/2011/Mar/msg00362.html I'll just copy it here for completeness. moduleGroebnerBasis[polys_, vars_, cvars_, opts___] := Module[{newpols, rels, len = Length[cvars], gb, j, k, rul}, rels = Flatten[ Table[cvars[[j]]*cvars[[k]], {j, len}, {k, j, len}]]; newpols = Join[polys, rels]; gb = GroebnerBasis[newpols, Join[cvars, vars], opts]; rul = Map[(# :> {}) &, rels]; gb = Flatten[gb /. rul]; Collect[gb, cvars]] conversionMatrix[polys_, vars_] := Module[{aa, coords, pmat, len = Length[polys], newpolys, mgb, gb, convmat, fvar, rvars}, coords = Array[aa, len + 1]; fvar = First[coords]; rvars = Rest[coords]; pmat = Transpose[Join[{polys}, IdentityMatrix[len]]]; newpolys = pmat.coords; mgb = moduleGroebnerBasis[newpolys, vars, coords]; gb = mgb /. Join[{fvar -> 1}, Thread[rvars -> 0]] /. 0 :> Sequence[]; convmat = Select/. fvar -> 0; {gb, convmat /. Thread[rvars -> Table[UnitVector[len, j], {j, len}]]}] On your example we proceed as follows. In[40]:= {gb, cmat} = conversionMatrix[p3, vars] Out[40]= {{2321109176622379997592951205}, \ {{-2476806404274022328337996647266500, 19814451234192178626703973178132000, 29721676851288267940055959767198 h, \ -19814451234192178626703973178132000, 2476806404274022328337996647266500, \ -14860838425644133970027979883599000, \ -29721676851288267940055959767198 h, 14860838425644133970027979883599000}}} We check the result. Specifically we show that the conversion matrix, multiplied by the polynomial system input regarded as a vector, indeed gives a constant. In[42]:= Expand[cmat.p3] Out[42]= {2321109176622379997592951205} So there you have it: no solution for this system. Daniel Lichtblau Wolfram Research