Re: Can I solve this system of nonlinear equations?

*To*: mathgroup at smc.vnet.net*Subject*: [mg125293] Re: Can I solve this system of nonlinear equations?*From*: "Stephen Luttrell" <steve at _removemefirst_stephenluttrell.com>*Date*: Mon, 5 Mar 2012 01:04:10 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jil5ld$rrm$1@smc.vnet.net> <jit112$e7i$1@smc.vnet.net> <jivd29$n60$1@smc.vnet.net>

"Ray Koopman" <koopman at sfu.ca> wrote in message news:jivd29$n60$1 at smc.vnet.net... > On Mar 3, 3:58 am, Ray Koopman <koop... at sfu.ca> wrote: <SNIPPED> > Not only is there no exact solution, there is also no finite > approximate solution. To see why, note that if the last unknown, h, > is treated as a fixed constant then the system is linear in the > seven other unknowns, a,...,g. Hence for any given value of h we > can get a closed-form expression for a,...,g that minimizes x.x . > The conditionally minimized value of x.x turns out to be > > 8782274623591681 > ------------------------------------------ . > 320000000000000000000000*(6312500 + 9*h^2) > > The corresponding conditional values of {a,...,g} are > > {(-57239352511710937500000 + h*(7888018985908781250000 + > h*(-1578205521876264250000 + 9*(1249663134566125 - > 249999762898476*h)*h)))/(250000000000000*h^3*(6312500 + 9*h^2)), > (-228957410046843750000000 + 3*h*(10441039511196093750000 + > h*(-2093794831545500750000 + 3*(4962147772987500 - > 995018769272803*h)*h)))/(1000000000000000*h^3*(6312500 + 9*h^2)), > (-228957410046843750000000 + h*(31094161123541437500000 + > h*(-6250175850437880437500 + 9*(4925643007710500 - > 990074756047929*h)*h)))/(1000000000000000*h^3*(6312500 + 9*h^2)), > (3*((-610553093458250000 + (82307209902652250 - > 16583652634832597*h)*h)/h^3 - 66361078251875/(6312500 + 9*h^2)))/ > 50500000000000000, > (-114478705023421875000000 + h*(15318123151723875000000 + > h*(-3094222721505445625000 + 9*(2426551023055750 - > 490148239781531*h)*h)))/(500000000000000*h^3*(6312500 + 9*h^2)), > (-228957410046843750000000 + 3*h*(10135762964466968750000 + > h*(-2052641226594174625000 + 3*(4817065849789500 - > 975461279165159*h)*h)))/(1000000000000000*h^3*(6312500 + 9*h^2)), > (-228957410046843750000000 + h*(30178331483354062500000 + > h*(-6127630865224089437500 + 9*(4781029653467500 - > 970662347863483*h)*h)))/(1000000000000000*h^3*(6312500 + 9*h^2))}. > > All those go to 0 as h -> Infinity. > I agree with and like your result. If I reevaluate everything (in my previous comment) starting with h=10^6 (i.e. a large value) and minimise w.r.t. {a,b,c,d,e,f,g} (i.e. omitting h), then I get Hessian eigenvalues {4.*10^12, 4.*10^12, 4.*10^12, 2.*10^12, 2.*10^12, 6958.34, 6930.56} where are still 2 (relatively) small eigenvalues whose eigenvectors are {{1., -0.000499972, 0.0000419583, -4.87781*10^-8, 7.10513*10^-9, -4.33947*10^-8, -0.0000867811}, {0.0000867811, -4.33864*10^-8, 7.07126*10^-9, -4.8444*10^-8, -0.000041375, 0.000500028, 1.}} which approximate {1,0,0,0,0,0,0} and {0,0,0,0,0,0,1}, so (for large h) variables a and g are much more weakly constrained by the minimisation than variables {b, c, d, e, f}. Looking at the equations, I see that variables a and g are different from the rest, because the structure of the first 5 equations suggests that variables a and g are associated with "boundaries" of some sort. This begs the question of where the equations came from in the first place. Maybe h is a parameter, and "solving" for {a, b, c, d, e, f, g} is actually an optimisation problem. The equations given would then define an unattainable "ideal" solution. -- Stephen Luttrell West Malvern, UK