       Re: Re: Cannot NSolve a system of equations

• To: mathgroup at smc.vnet.net
• Subject: [mg88984] Re: [mg88944] Re: Cannot NSolve a system of equations
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Fri, 23 May 2008 03:05:30 -0400 (EDT)
• Reply-to: hanlonr at cox.net

```Note that

Rationalize[0.74615385]

0.746154

As with your other numbers, use second parameter.

phi = Rationalize[0.74615385, 0];

And when you defined zet using N[] you reverted back to machine precision. Instead use the second parameter to set extended precision.

zet = N[5 taubar/Total[y^(1 + phi)], 50];

The precision used in NSolve need not be so large (try 100 vice 500).

sol = NSolve[eqns1, vars, 100]

Bob Hanlon

---- murat.koyuncu at gmail.com wrote:
> I sincerely thank you all for your help. It cleared up a lot of things
> for me. I had tried Rationalize but didn't know about precision
> issues.
>
> Now I have a follow-up question. I still cannot trust my results 100%
> because when I slightly change one of my parameters, solution set
> changes drastically. My theory says that resulting variables should
> between 0 and 1, so I am only interested in that range. And sometimes
> I don't even get any such solutions.
>
> For example, here is my model (very much polished, thanks to your
> suggestions):
>
> Unprotect[In, Out]; Clear[In, Out]; ClearAll["Global`*"];
> phi = Rationalize[0.74615385];
> eta = 7/4;
> alpha = 16/25;
> taubar = 13/100;
> {y1, y2, y3, y4, y5} =
>   Rationalize[{0.235457064, 0.512465374, 0.781779009, 1.109572176,
>     2.360726377}, 0];
> zet = N[5 taubar/(y1^(1 + phi) + y2^(1 + phi) + y3^(1 + phi) + y4^(
>        1 + phi) + y5^(1 + phi))];
> tau1 = zet y1^phi;
> tau2 = zet y2^phi;
> tau3 = zet y3^phi;
> tau4 = zet y4^phi;
> tau5 = zet y5^phi;
> a1 = (1 + phi) tau1;
> a2 = (1 + phi) tau2;
> a3 = (1 + phi) tau3;
> a4 = (1 + phi) tau4;
> a5 = (1 + phi) tau5;
> eqns1 = {x1 == (roverw (1 - tau1) + ((roverw + (1 - x)) y1 -
>           1) ((1 - taubar + (1 - abar)/eta) x + (taubar - tau1)
>            roverw - (1 - taubar)))/(roverw (1 -
>           tau1 + (1 - a1)/eta) - ((1 - taubar + (1 - abar)/eta)
>           x + (taubar - tau1) roverw - (1 - taubar))),
>    x2 == (roverw (1 - tau2) + ((roverw + (1 - x)) y2 -
>           1) ((1 - taubar + (1 - abar)/eta) x + (taubar - tau2)
>            roverw - (1 - taubar)))/(roverw (1 -
>           tau2 + (1 - a2)/eta) - ((1 - taubar + (1 - abar)/eta)
>           x + (taubar - tau2) roverw - (1 - taubar))),
>    x3 == (roverw (1 - tau3) + ((roverw + (1 - x)) y3 -
>           1) ((1 - taubar + (1 - abar)/eta) x + (taubar - tau3)
>            roverw - (1 - taubar)))/(roverw (1 -
>           tau3 + (1 - a3)/eta) - ((1 - taubar + (1 - abar)/eta)
>           x + (taubar - tau3) roverw - (1 - taubar))),
>    x4 == (roverw (1 - tau4) + ((roverw + (1 - x)) y4 -
>           1) ((1 - taubar + (1 - abar)/eta) x + (taubar - tau4)
>            roverw - (1 - taubar)))/(roverw (1 -
>           tau4 + (1 - a4)/eta) - ((1 - taubar + (1 - abar)/eta)
>           x + (taubar - tau4) roverw - (1 - taubar))),
>    x5 == (roverw (1 - tau5) + ((roverw + (1 - x)) y5 -
>           1) ((1 - taubar + (1 - abar)/eta) x + (taubar - tau5)
>            roverw - (1 - taubar)))/(roverw (1 -
>           tau5 + (1 - a5)/eta) - ((1 - taubar + (1 - abar)/eta)
>           x + (taubar - tau5) roverw - (1 - taubar))),
>    x == (x1 + x2 + x3 + x4 + x5)/5,
>    abar == (a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5)/(5 x),
>    roverw == (1 - x) (1 - alpha)/alpha};
> vars = {x, x1, x2, x3, x4, x5, abar, roverw};
> sol = NSolve[N[eqns1, 500], vars];
> InputForm[N[sol]]
>
> One of the solutions I get is between 0 and 1:
> {x -> 0.5004885149810059, x1 -> 0.6893393376625371,
>   x2 -> 0.6337652704581314, x3 -> 0.5731109245369462,
>   x4 -> 0.4917632705087048, x5 -> 0.1144637717387098,
>   abar -> 0.1244737777525507, roverw -> 0.2809752103231842}
>
> So I am happy. But when I drop the last digit of the first parameter,
> i.e. phi = Rationalize[0.7461538];, the new set of solutions do not
> have any results compatible with my assumptions:
>
> {{x -> 0.2949617829869077, x1 -> -1.9491786143565832, x2 ->
> 10.96762641745164, x3 -> 96.0369893781559, x4 -> -73.18097932930044,
>   x5 -> -30.399648937016046, abar -> -5.824073507107748, roverw ->
> 0.39658399706986447},
>  {x -> 2.0689803436369942, x1 -> -8.65276583855146, x2 ->
> -18.39904428993495, x3 -> -33.55295774059405, x4 ->
> -69.9736576669751,
>   x5 -> 140.92332725424055, abar -> 2.4458078645586667, roverw ->
> -0.6013014432958093},
>  {x -> 0.6213605439915423, x1 -> -12.823085779076337, x2 ->
> 41.57610980587073, x3 -> -6.337885849417286, x4 ->
> -8.665252197121296,
>   x5 -> -10.64308326029811, abar -> -0.7583206767716181, roverw ->
> 0.21298469400475745},
>  {x -> 0.6854511826071373 - 0.04362902433398709*I, x1 ->
> 0.59964456896311 + 0.08549874839569205*I,
>   x2 -> 0.6411455792212575 + 0.04079939260558141*I, x3 ->
> 0.6760747168284579 - 0.003707834204090894*I,
>   x4 -> 0.7125762414563436 - 0.05913956075670094*I, x5 ->
> 0.7978148065665167 - 0.28159586771041706*I,
>   abar -> 0.17051392124951886 - 0.016129276403199968*I, roverw ->
> 0.17693370978348527 + 0.024541326187867737*I},
>  {x -> 0.6854511826071373 + 0.04362902433398709*I, x1 ->
> 0.59964456896311 - 0.08549874839569205*I,
>   x2 -> 0.6411455792212575 - 0.04079939260558141*I, x3 ->
> 0.6760747168284579 + 0.003707834204090894*I,
>   x4 -> 0.7125762414563436 + 0.05913956075670094*I, x5 ->
> 0.7978148065665167 + 0.28159586771041706*I,
>   abar -> 0.17051392124951886 + 0.016129276403199968*I, roverw ->
> 0.17693370978348527 - 0.024541326187867737*I},
>  {x -> 0.9999999999989753, x1 -> 0.9999999999982316, x2 ->
> 0.9999999999985612, x3 -> 0.999999999998809, x4 ->
> 0.9999999999991211,
>   x5 -> 1.0000000000001525, abar -> 0.16068192162412143, roverw ->
> 0.}}
>
> And this is just one of the instances that this happens. I solve the
> same set of equations for many parameter sets. Sometimes I get a
> solution in the [0,1] range, but it is so different than the previous
> one, it doesn't make sense. Or sometimes, I have to tweak parameter
> values (drop a digit here, add another one there) to get a meaningful
> solution. I need a set of results over different parameter values that
> I can compare and contrast, but it is hard to do that when my results
> suddenly disappear or jump to an improbable value.
>
> Now I know that this is probably the most Mathematica can do for me,
> but is there a way to make my system more "stable"? Maybe play with my
> equations a little bit? Or put another way, what makes my system so
> unstable?
>
> Again, any suggestions will be very much appreciated. Thanks!
>

```

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