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Re: minimizing f(x,y) ???!!!!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg25476] Re: minimizing f(x,y) ???!!!!
*From*: rogalsky <rogalsky at faupt100.physik.uni-erlangen.de>
*Date*: Mon, 2 Oct 2000 22:26:55 -0400 (EDT)
*Organization*: University of Erlangen, Germany
*References*: <8r1b8a$ifn@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
twirt at hotmail.com wrote:
> I never used Mathematica before and want to find the minimum for the
> following function:
>
> h(a,b) := g(a,b) / f(a,b)
>
> a [0 .. pi]
> b [pi-a .. pi]
>
> f(a,b) := 1/3*(1+1/(4*pi)*(a+b-2*acos(-1-cos(a)- cos(b))))
> g(a,b) := 1/8*(2-cos(a)-cos(b))
Numerically or symbolic?
Numerically
f[a_, b_] = 1/3*(1 + 1/(4*Pi)*(a + b - 2*ArcCos[-1 - Cos[a] - Cos[b]]));
g[a_, b_] = 1/8*(2 - Cos[a] - Cos[b]);
h[a_, b_] = f[a, b]/g[a, b];
FindMinimum[h[a, b], {a, 2/3Pi, 0, Pi}, {b, 2/3Pi, 0, Pi}]
yields:
{0.925674, {a -> 2.53431, b -> 2.53431}}
You get a nice plot of your function with:
Plot3D[If[b > Pi - a, h[a, b], 1], {a, 0, Pi}, {b, 0, Pi}, PlotPoints ->
40]
A symbolic expression is hard to get. You need to solve dh(a,a)/da=0 and
it
seems to me, that there is no closed symbolic expression for that. But
you may investigate on that.
Olaf Rogalsky
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