scispace - formally typeset
Search or ask a question

Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1964"


Journal ArticleDOI
TL;DR: In this paper, the authors present a new algorithm for the calculation of the eigenvalues of real square matrices of orders up to 100, which is directly applicable to complex matrices as well.
Abstract: 1. Introduction. We present a new algorithm for the calculation of the eigenvalues of real square matrices of orders up to 100. The basic method is directly applicable to complex matrices as well and, in both cases, with each eigenvalue X of A a vector v is produced for which (A - XI)v is null except for a small last element. This vector is not always an approximation to the eigenvector for X and this algorithm claims only to find eigenvalues. The main concern has been to use only single precision arithmetic although the effect of using an accumulated inner product procedure in one part of the program is shown in the results in Section 15. The method consists of two parts. Firstly the given matrix A is reduced to almost triangular (Hessenberg) form H by elementary similarity transformations. Direct reduction of H to sparser forms requires extra precision in practice and even then is not without difficulties. So the second stage is the iterative search for the eigenvalues of H. A natural extension of Hyman's method [13] may be used to evaluate p(z) = det (H - zI) and any number of derivatives in an accurate and stable way. However each evaluation requires approximately n2 real multiplications and n2 real additions for an n X n matrix and complex z. Thus the viability of this approach depends on finding each eigenvalue with a small number of evaluations. Results so far with the method developed here indicate an average of just less than 9 evaluations (3 iterations) per eigenvalue on a wide variety of matrices of orders from 8 to 100. Now the iterations of Muller, Newton, and Bairstow converge quickly once a fair approximation to an eigenvalue has been found. They do not seem so satisfactory at the beginning of a search. Laguerre's method [2], [5], [6] was designed for polynomials with real zeros and when these are distinct it gives strong convergence right from any starting value. The method can be extended to the complex plane. No longer is convergence certain for any starting value but, in practice, the complex iteration seems as powerful as the real one on all examples so far considered. One Laguerre step requires more calculation than one step of any of the methods mentioned above, but when there are no a priori approximation to the zeros available the reduction in the number of iterations with Laguerre more than compensates for the extra calculation for each step. In addition when an eigenvalue has been found there is enough information available to take one Newton step towards the next eigenvalue. This paper is mainly a detailed discussion of the practical application of the method and techniques for keeping the number of iterations to a minimum. A description of the program is given in ALGOL 60 [1], together with some results ob

96 citations





Journal ArticleDOI
TL;DR: This paper shows how it is possible to combine one algorithm from each class together into a “mixed” strategy for diagonalizing a real symmetric matrix.
Abstract: The algorithms described in this paper are essentially Jacobi-like iterative procedures employing Householder orthogonal similarity transformations and Jacobi orthogonal similarity transformations to reduce a real symmetrix matrix to diagonal form. The convergence of the first class of algorithms depends upon the fact that the algebraic value of one diagonal element is increased at each step in the iteration and the convergence of the second class of algorithms depends upon the fact that the absolute value of one off-diagonal element is increased at each step in the iteration. Then it is shown how it is possible to combine one algorithm from each class together into a “mixed” strategy for diagonalizing a real symmetric matrix.

6 citations


Journal ArticleDOI
01 Jan 1964

4 citations