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The \({\cal {D Q R}}\) algorithm, basic theory, convergence, and conditional stability

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We describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on \(D\)–orthogonal reduction techniques where \(D= \mbox{ diag}(\pm1)\) is computed from the given matrix \(A\). It operates on a similarity reduction of a real matrix \(A_{nn}\) to general tridiagonal form \(T\) and computes all of \(A\)'s eigenvalues in \(\frac{4}{3} n^3 + O(n^2)\) operations, where the \(O(n^2)\) part of the operations is possibly performed over \({\Bbb C}\), instead of the 7–8 \(n^3\) real flops required by the \(QR\) eigenvalue algorithm. Potential breakdo wn of the \(DQR\) algorithm can occur in the reduction to tridiagonal form and in the \(D\)–orthogonal \(DQR\) reductions. Both, however, can be monitored during the computations. The former occurs rather rarely for dimensions \(n \leq 300\) and can essentially be bypassed, while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the \(DQR\) steps. We prove an implicit \(DQ\) theorem which allows implicit shifts, give a convergence proof for the \(DQR\) algorithm and show that \(DQR\) is conditionally stable for general balanced tridiagonal matrices \(T_{\rm b}\).

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Received April 25, 1995 / Revised version received February 9, 1996

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Uhlig, F. The \({\cal {D Q R}}\) algorithm, basic theory, convergence, and conditional stability. Numer. Math. 76, 515–553 (1997). https://doi.org/10.1007/s002110050275

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  • DOI: https://doi.org/10.1007/s002110050275

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