A comparison of several methods for inverting large symmetric positive definite matrices

Abstract

number of parameters the accumulation of round-off error during the course of the computation, resulting in a loss of significance, is one of the most serious restrictions. Since the evaluation of a large number of parameters and their variances by a least squares procedure on a high speed computer involves the inversion of a symmetric, positive-definite matrix it becomes important to choose an inversion scheme in which the effects of the accumulation of round-off error are minimized. This is especially true if limited memory space for storage precludes the use of double precision arithmetic. In this paper a comparison of several direct methods for inverting such matrices is given. An attempt has been made to consider the effects of both condition and order of the matrix to be inverted on the closeness of the computed inverse to the exact inverse. The matrix inversion methods compared are the Gauss-Jordan [1], Choleski [2], congruent transformation [3], and rank annihilation [4] schemes. To give a fair comparison of the methods each was programmed in IBM 7090 FORTRAN II (Version 2) using only single precision arithmetic (good to about 8 decimal digits). The error indicators were then computed using double precision arithmetic (good to about 16 decimal digits) so that the latter calculation was not a limiting factor. In the following discussion the symbols for matrices will be underlined while other symbols will not be.