Re: Can I solve this system of nonlinear equations?

*To*: mathgroup at smc.vnet.net*Subject*: [mg125274] Re: Can I solve this system of nonlinear equations?*From*: Ray Koopman <koopman at sfu.ca>*Date*: Sat, 3 Mar 2012 06:56:03 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jil5ld$rrm$1@smc.vnet.net>

On Feb 29, 4:28 am, Andy <andy7... at gmail.com> 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~ To see if there is an exact solution to the problem, make all the coefficients exact and use Solve: x = { 500(c - a) - (995018769272803*^-15 + h*b), 500(d - b) - (990074756047929*^-15 + h*c), 500(e - c) - (985167483257382*^-15 + h*d), 500(f - d) - (980296479563062*^-15 + h*e), 500(g - e) - (975461279165159*^-15 + h*f), (-1*e + 8*d - 8*b + a)1000/12 - (990074756047929*^-15 + h*c), (-1*f + 8*e - 8*c + b)1000/12 - (985167483257382*^-15 + h*d), (-1*g + 8*f - 8*d + c)1000/12 - (980296479563062*^-15 + h*e)}; Solve[Thread[x == 0],{a,b,c,d,e,f,g,h}] {} To get an approximate solution, minimize some measure of the differences between the two sides of the equations. The sum of the squared differences is convenient and not obviously improper; that is, there is nothing to suggest that the equations are on vastly different scales. TimeConstrained[Minimize[x.x,{a,b,c,d,e,f,g,h}],300] aborted. Both Minimize[N[x.x],{a,b,c,d,e,f,g,h}] and NMinimize[x.x,{a,b,c,d,e,f,g,h}] finished in < 2 sec and gave identical results: {3.7850320584543324`*^-11, {a -> 1.78322069825487`, b -> 1.7856155567256693`, c -> 1.7880252569409105`, d -> 1.79041402958568`, e -> 1.7928176848807527`, f -> 1.7952004996719013`, g -> 1.7975982365756977`, h -> 0.7881096531282696`}} The fit is a little better than Stephen Luttrell got, but the parameter estimates are far what he got. x /. %[[2]] {-2.832840262367853`*^-7, 1.7089251762580915`*^-6, 3.9068509285478115`*^-6, 1.6397939603951528`*^-6, -2.7478474384778906`*^-7, -1.6998965117753784`*^-6, -3.348694662008711`*^-6, -1.6487347667126784`*^-6} It looks like only the first 5 digits of the 15-digit constants are accurate.