Laguerre’s method applied to the matrix eigenvalue problem

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 -zl) and any number of derivatives in an accurate and stable way. However each evaluation requires approximately n real multiplications and n real additions for annXn 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

A multishift QR iteration without computation of the shifts

“…Hyman's method, which constitutes the basis for our work, is the best of the latter for its simplicity, efficiency, stability, and accuracy. It was effectively used by Parlett [13] with Laguerre's iteration to produce eigenvalue solvers outperformed only by their QR competitors. While it is generally thought of as a scalar algorithm, its extension to the vector form (2.4) is straight-forward, as demonstrated below.…”

Time Domain Techniques in the Singularity Expansion Method.

“…To obtain the natural frequencies and natural modes, we are interested in ecuation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) in the absence of a forcing function, i.e., where v, denotes the natural mode vector corresponding to a pole of multiplicity in.,.…”

“…A simple method known as Fourier stability analysis may be used to determine the stability criterion for an uncom-* pressed difference scheme (e.g., equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) or (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) instead of equation (3-29) …”

On optimal parameter of Laguerre's family of zero-finding methods

“…[6][7][8]17,22]. A number of numerical experiments have shown that convergence breakdown is extremely rare solving polynomial equations, making this method very close to being globally convergent.…”

“…A number of numerical experiments have shown that convergence breakdown is extremely rare solving polynomial equations, making this method very close to being globally convergent. Extensive studies of Laguerre's method can be found in [9,13,16] (see, also, [4,8,12,17,18,20]). …”