Borrowing from a book NUMERICAL RECIPES - The Art of Scientific Computing (Third Edition), the fastest method of all for the vast majority of matrices is the MRRR algorithm (Multiple Relatively Robust Representations):

Dhillon, I.S., and Parlett, B.N. 2004, “Multiple Representations to Compute Orthogonal Eigenvectors of Symmetric Tridiagonal Matrices,” Linear Algebra and Its Applications, vol. 387, pp. 1–28.

1. Power method is only good for computing few largest eigenvalues, and is not very efficient at that - better use scipy.sparse.linalg.eigs that implements theoretically optimal ARPACK.

2. Characteristic polynomial is a purely theoretical thing, very inefficient for large matrices.

The most important question is: do you really need all eigenvalues or just few extreme (smallest or largest)?

If you do need all of them, then use scipy.linalg.eig - for a general matrix nothing better exists so far as I can see.

If you only need few extreme ones, then use scipy.sparse.linalg.eig.

If your matrix is real symmetric or complex Hermitian, I could help more (have my own package RALEIGH for that on PyPI).

I mentioned here Python packages only because they are easy to use, portable and free.

Regards,

Evgueni

There are many methods and they are very well described in the literature. Personally I was using often the extremely stable Jacobi algorithm (well known already in the 1960's, https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm) and the more efficient but somewhat less stable Housholder algorithm (https://en.wikipedia.org/wiki/Householder_transformation) based on tri-diagonalizations. There are also special subversions of those algorithms for symmetric matrices.

On a modern PC I have personally often diagonalized matrices up to 6000x6000, but on bigger computers one can do dimensions up to 1M x 1M.

As for books, I reccomend The Algebraic Eigenvalue Problem by J.H.Wilkinson (http://www.amazon.com/dp/0198534183/ref=nosim/?tag=stanshub-20).

The methods that you mention would be considered naive today (no offence intended, I mean it as an official category).

One more thing. Efficient algebraic packages which include matrix diagonalization are today available in all programming languages; they are among the first thing that one writes when building a library.

These links are very helpful prof Anton Vrdoljak , thank for sharing.

Thank you Prof Stan Sykora and Evgueni Ovtchinnikov for your suggestions.

I do agree that the methods which I have mentioned are naive and it should not be recommended to use computationally.

I actually want to write a code to find all the eigen values, without using any packages. It's just for my own learning. As you both mentioned that for real symmetric or complex Hamiltonian matrix , things are easy . One can apply the approach of divide and conquer . For general matrix , even Gauss Elimination approach can not applied to find the eigen values. Also, other numerical methods like LR or QR methods does not guarantee the convergence.

Please suggest me to frame algorithm for the general matrix.

If you want to code yourself: reduce the matrix to three-diagonal (use Householder reflections) and apply QR making sure you only work with non-zeros.

For computational subtleties (using shifts, bulge chasing etc.) consult Golub, G. H.; Van Loan, C. F. (1996). Matrix Computations (3rd ed.). Baltimore: Johns Hopkins University Press. ISBN 0-8018-5414-8.

The question posed by Bharat Soni is rather of a philosophical nature.

There exist a vast literature on Matrix Theory, Numerical Analysis, and related disciplines of applied mathematics dedicated to the theory of eigenvalues and eigenspaces of matrices. Not particular matrices-but the whole classes of matrices.

I would like to recommend the book, Matrix Algorithms, Volume II: Eigensystems, by G.W. Stewart, published by SIAM in 2001, where the author carefully and meticulously shows why such an algorithm may not exist for all matrices. The subject of eigensystems became classical nowadays and it is difficult to expect the breaking-through results in the nearest future.

In addition to the helpful answers, I have a little bit to add. Mathematicians failed to find exact roots of polynomials with degree n > 4 ( Galois theory). Numerical approximations are a must. The study of autonomous dynamical systems or similar problems is necessarily based on the eigenvalues. But exact solutions are far-fetched, so they turn to study the qualitative behavior. Bounds of the eigenvalues are enough to predict the solution behavior of the whole system under discussion.