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I am attempting to write a program that will be able to numerically solve a homogeneous Fredholm Integral Equation of Second Kind, with a Symmetric Kernel. I have been looking through textbooks and websites for some time, but most of literature are concerned with the derivation for continuous functions. Since the program will have to be discrete, I was wondering if anybody knew of any ways to solve this integral equation discretely. My plan was to be able to generate an arbitrary symmetric Kernel and let the program solve for the eigenvalues and eigenfunctions. If anybody can help me out with this, and hopefully provide me with a reference it would be greatly appreciated.

Thank you very much.

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The discrete version of the homogeneous Fredholm integral equation of the second kind is simply a linear algebra eigenvalue problem: $\phi = \lambda K \phi$ where $K$ is an $n \times n$ matrix and $\phi$ an $n$-component vector (so here the eigenvalue is $\lambda^{-1}$). Your favourite numerical linear algebra library will have methods for finding eigenvalues and eigenvectors of real symmetric matrices.

If this is to be a discretized approximation of a continuous problem, you may want to construct the matrix using, say, a Newton-Cotes rule or Gaussian quadrature.

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  • $\begingroup$ Thank you very much for your answer, this makes sense. Just one additional question. With this method, should I also expect to get n eigenvalues? Therefore resulting in a set of n eigenvectors? $\endgroup$ – user137038 Mar 21 '14 at 14:51

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