Solution of Systems of Linear Equations by Minimized Iterations1

In an earlier publication [14] a method was described which generated the eigenvalues and eigenvectors of a matrix by a successive algorithm based on minimizations by least squares. The advantage of this method consists in the fact that the successive iterations are constantly employed with maximum efficiency which guarantees fastest convergence for a given number of iterations. Moreover, with the proper care the accumulation of rounding errors can be avoided. The resulting high precision is of great advantage if the separation of closely bunched eigenvalues and eigenvectors is demanded [16]. It was pointed out in [14, p. 256] that the inversion of a matrix, and thus the solution of simultaneous systems of linear equations, is contained in the general procedure as a special case. However, in view of the great importance associated with the solution of large systems of linear equations, this problem deserved more than passing attention. It is the purpose of the present discussion to adopt the general principles of the previous investigation to the specific demands that arise if we are not interested in the complete analysis of a matrix but only in the more special problem of obtaining the solution of a given set of linear equations