We present a new deflation criterion for the multishift QR algorithm motivated by convergence analysis for the basic QR algorithm. The performance of the criterion is illustrated by numerical experiments using the LAPACK implementation of the double-shift QR algorithm

Francoise Tisseur

Mario Ahues

We present a new deflation criterion for the multishift QR algorithm motivated by convergence analysis for the basic QR algorithm. The performance of the criterion is illustrated by numerical experiments using the LAPACK implementation of the double-shift QR algorithm. 1 Introduction  The practical QR algorithm computes the eigenvalues of a full matrix A 2 C  n\Thetan  by first reducing A to upper Hessenberg form by a unitary similarity transformation and then iterating on the Hessenberg form to reduce it to upper triangular form. On each iteration a deflation test is used to determine whether a subdiagonal element can be neglected and the original problem replaced by two subproblems of smaller size. Let  H denote an upper Hessenberg matrix QR iterate. If h i;i\Gamma1 is &quot;sufficiently small&quot;, for some 2  i  n, then we replace h i;i\Gamma1 by zero, obtaining  `  H 11 H 12  0 H 22  &apos;  i\Gamma1 n\Gammai+1 i\Gamma1 n\Gammai+1  ; we say that the problem decouples into two problems of smalle..

Françoise Tisseur

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A New Deflation Criterion for the QR Algorithm

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A New Deflation Criterion for the QR Algorithm

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