The computation of eigenvalues and eigenvectors of very large sparse matrices

Several methods are avi1iible for computing elgenvalues and eigenveotors of large sparse matrices, but as yet no outstandingly good algorithm is generally known. For the synimetric matrix case one of the most elegant algorithms thetiretically is the method of m1rini1zed iterations developed by Lanczos in 1950 • This method reduces the origi1 matrix to tn-diagonal form from which the eigenaystem can easily be found. The method can be used iteratively, and here the convergence properties and different possible eigenvalue intervals are first considered assiinrtng infinite precision computation. Next rounding error pn1 yses are given for the method both with and without re-orthogonalization. It is shown that the method has been unjustly neglected, in fact a particular computational algorithm for the method without re-orthogoiiRl I zation is shown to have remarkably good error properties. As well as this the algorithm is very fast aM can be pronamined to require very little store compared with other comparable methods, and this suggests that this variant of the Lanczos process is likely to become an extremely useful algorithm for finding several extreme eigenvalues, and their eigenvectors if needed, of very large sparse symmetric matrices.