On a New Way of Solving the Linear Equations that Arise in the Method of Least Squares

Abstract

Translated by G. W. Stewart

This report contains a translation of a paper of C. G. J. Jacobi, ``Ueber eine neue Aufl\"osungsart der bei der Methode der kleinsten Quadrate vorkommenden line\"aren Gleichungen,'' which appeared in {\it Astronomische Nachrichten\/} {\bf 22} (1845). In the paper Jacobi shows how to use rotations to increase the diagonal dominance of symmetric linear systems, which he then solves by what we today call the point Jacobi method. This preconditioner is none other than Jacobi's method for diagonalizing a symmetric matrix. Although Jacobi points out his method can be used to find eigenvalues, he reserves a fuller exposition for a later paper [Journal f\"ur die reine und angewandte Mathematik, {\bf 30} (1846), 51--s94], which is now generally cited as the source of the method. A variant for unsymmetric equations is also considered. (Also cross-referenced as UMIACS-TR-92-42)