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I believe mathematica uses some techniques beyond our book. According to the text book it is not possible to solve eigenvalues of which the matrix has not completely eigenvectors, or of those matrix has cluster eigenvalues and multiple matrix yet not symmetric. How does mathematica solve that? On mathematica book they say they use interpolation for the characteristic polynomials, yet there are no thesis pertaining to this. I want to know the algorithm, how can I do so?

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mathematica do not provide the source code,you are not supposed to know the technical details –  tintin Sep 16 '13 at 6:53
    
Standard reference (and as source code is available, the best one for most things) is Some Notes on Internal Implementation in Mathematica documentation. –  kirma Sep 16 '13 at 10:43
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If you are really interested in guessing a list of algorithms that M might be using, you would like to study some numerical methods theoretically too. All M does comes from there. –  Rorschach Sep 16 '13 at 11:06
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For approximate numeric matrices Mathematica uses Lapack (machine numbers) and an Lapack-like implementation for bignums. For exact case it finds the nullspaces for each eigenvalue, with eigenvalues computed from the characteristic polynomial. –  Daniel Lichtblau Sep 16 '13 at 13:33
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@tintin WRI support is pretty good about supplying references though, if you ask them. Personally I find references more useful than reading source code. –  Szabolcs Sep 16 '13 at 17:07
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