Scaling and Error Analysis for Matrix Inversion by Partitioning

Abstract

There is presently available a large number of techniques purporting to accomplish the inversion of matrices. While the purely mathematical aspects of this problem, on one hand, are thus well recognized, the computational ones, on the other hand, are not. The growth of the rounding error, in particular, may be so rapid as to make some inversion procedures altogether unstable. It is from this point of view that the partitioning method seems to be capable of yielding more accurate results than do other methods. By stopping, at any desired step, to improve the intermediate inverses until satisfactory accuracy is attained, the growth of the rounding error may be kept in check. The following sections, then, give a brief description of the partitioning method and treat in some detail an effective scaling scheme permitting the inversion routine to be carried out by high speed computing machinery. Next a careful examination is carried out of the accuracy attainable by the proposed scheme; together with an error squaring iteration procedure it is found capable of yielding accuracies sufficient for most practical purposes.