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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