A comparison of methods for computing the eigenvalues and eigenvectors of real symmetric matrix

Abstract

II. Methods Tested. Three codes were selected from SHARE which represented three different methods. A. Jacobi. SHARE distribution 705 by MIT computing lab (FORTRAN). This is the original version of the Jacobi method [1] in which plane rotations are used to produce zeros in all off-diagonal positions using the maximum off-diagonal element as a pivot at each step. This is probably slower but more accurate than the cserial version which pivots on the off-diagonal elemenlts in sequence whether or not they are large. B. Givens. SHARE distribution 664 (AN F202) by the AEC Computing and Applied Mathematics Center, NYU (FORTRAN). This is the method devised by W. Givens in 1954 at the Oak Ridge National Laboratory [2] in which plane rotations are used to produce a tri-diagonal matrix with the same eigenvalues as the original matrix. The roots of the matrix are computed by the use of Sturm's sequence derived from the tri-diagonal nmatrix. C. Householder. SHARE distribution # 1385 (AN 202) by the AEC Computing and Applied Mathematics Center, NYU (FORTRAN). This is a variation of the Givens method in which the tri-diagonal matrix is produced by an orthogonal transformation that does not depend on plane rotations. It is described in detail in [3] by J. H. Wilkinson. The Sturm sequences are used as before to obtain the roots from the tri-diagonal matrix. Series overflow and/or underflow problems occurred in the earlier experiments because a scaling device was not incorporated into the codes of methods B and C. Both methods were modified to incorporate this feature in the decks used in the experiments described below. Modified listings are available from the authors.