On a New Way of Solving the Linear Equations that Arise in the Method
of Least Squares
On a New Way of Solving the Linear Equations that Arise in the Method
of Least Squares
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Date
1998-10-15
Authors
Jacobi, C. G.
Stewart, G. W.
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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)