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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 eigen-values, while the implicit method can effectively loose the shift and thereby retard the convergence. This paper presents a new method which has neither drawback. [ABSTRACT FROM AUTHOR]