Incorporating Origin Shifts into the QR
Algorithm for Symmetric Tridiagonal Matrices

The QR iteration for the eigenvalues of a symmetric tridiagonal matrix can 
be accelerated by incorporating a sequence of origin shifts.  The origin shift
may be either subtracted directly from the diagonal elements of
the matrix or incorporated by means of an implicit algorithm.  Both
methods have drawbacks: the direct method can unnecessarily degrade small 
eigenvalues, while the implicit method can effectively loose the shift and 
thereby retard the convergence.  This paper presents a new method which has 
neither drawback.

CACM June, 1970

Stewart, G. W.

eigenvalues, eigenvectors, QR method, origin
shifts, symmetric tridiagonal matrix

5.14

CA700608 JB February 13, 1978  11:02 AM

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