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Numarical integration fails to work... Help, please!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122743] Numarical integration fails to work... Help, please!
  • From: FrozenRiver <yfywan at hotmail.com>
  • Date: Wed, 9 Nov 2011 06:24:54 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

I came across this problem in my calculation. 

I have G(x, y) which is a very complicated matrix function of x and y. The integrand of my problem, denoted by f(G(x, y)), is a function of the eigenvalues and eigenvectors of G(x, y), and so naturally is a function of x and y. 

G(x, y) is formally so complicated that it's impractical to be diagonalized symbolically on the level of x and y. When I tried indeed to do so, I get an error message like
Eigenvectors::eivec0: Unable to find all eigenvectors. >>

 So I thought numerical integration should be the way to go. The procedure that I had in mind was: write down G(x, y) numerically for each (x0, y0) point which appeared in the numerical integration, diagonalize the numerical matrix, calculate the eigenvectors and eigenvalues, and then calculate f(G(x0, y0)). In this way, at least in principle, the numerical integration could be performed. 
The code goes like
G[x_, y_]:=...
f[x_, y_]:=... 
NIntegrate[f[x, y],{x, xmin, xmax}, {y, ymin, ymax}] 

But it did not work out. The reason probably is, I suspect based on the same error message that I received, that mathematica evaluates the integrand at each (x0, y0) point using something like 
f(G(x, y))/{x->x0, y->y0}, 
which requires the explicit form of the integrand f(G(x, y)), which, as I have mentioned, is difficult to calculate, especially the eigenvectors. 

So, my question is, how should I proceed with this... It seemed such an innocent problem. 

Thanks a lot guys. 

Cheers,
GQ



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