MATHEMATICS OF COMPUTATION, VOLUME 26, NUMBER 120, OCTOBER 1972 Modification Methods for Inverting Matrices and Solving Systems of Linear Algebraic Equations By D. Goldfarb* Abstract. Modification methods for inverting matrices and solving systems of linear algebraic equations are developed from Broyden's rank-one modification formula. Several algorithms are presented that take as few, or nearly as few, arithmetic operations as Gaussian elimination and are well suited for the handling of data. The effect of rounding errors is discussed briefly. Some of these algorithms are essentially equivalent to, or "compact" forms of, such known methods as Sherman and Morrison's modification method, Hestenes' biorthogonalization method, Gauss-Jordan elimination, Aitken's below-the-line elimination method, Purcell's vector method, and its equivalent, Pietrzykowski's projection method, and the bordering method. These methods are thus shown to be directly related to each other. Iterative methods and methods for inverting symmetric matrices are also given, as are the results of some computational experiments. 1. Introduction. This paper is concerned with the related problems of inverting matrices and solving linear nonhomogeneous algebraic systems of equations by a class of direct methods which we shall call modification methods. The adjective modification is used to describe these methods, for they are all based upon the modifica- tion of a matrix and its inverse by a matrix of rank one. Most of these methods are direct—that is, a solution to the problem is obtained by using a finite number of elementary arithmetic operations. The general formula that these methods are based upon, however, is iterative in nature, and iterative methods are also given. Modification methods have been developed and published by other authors. These include the methods of Sherman and Morrison [30], [31], and [32], and com- putational schemes and extensions based upon their work due to Woodbury [37], Wilf [34], [35], Zielke [38], Ershov [10] (as reported in Faddeev and Faddeeva [11]), and Kron [23]. Methods developed by the author are found in Section 2 (for in- verting matrices), in Section 6 (for inverting symmetric matrices), in Section 7 (iterative methods), and in Sections 8, 9, and 10 (for solving simultaneous linear equations). At first it was thought that all of these methods were new. Subsequent comparisons with the literature, however, revealed that, although the theoretical development underlying these methods was new, most of them were essentially equivalent to, or were a compact form of, known techniques. These techniques include Sherman and Morrison's modification method [31], Hestenes' biorthogonali- zation method [17], [18], Gauss-Jordan elimination, Aitken's below-the-line elimina- Received April 3, 1971. A MS 1969 subject classifications. Primary 6535, 1510, 1515; Secondary 6510, 6580, 1525. * This research was supported in part by City University Faculty Research Grant No. 1085 and the IBM New York Scientific Center. Copyright © 1972, American Mathematical Society 829 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use