The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance

Abstract

This paper presents a small-bulge multishift variation of the multishift QR algorithm that avoids the phenomenon of shift blurring, which retards convergence and limits the number of simultaneous shifts. It replaces the large diagonal bulge in the multishift QR sweep with a chain of many small bulges. The small-bulge multishift QR sweep admits nearly any number of simultaneous shifts-even hundreds-without adverse effects on the convergence rate. With enough simultaneous shifts, the small-bulge multishift QR algorithm takes advantage of the level 3 BLAS, which is a special advantage for computers with advanced architectures.

Structured Eigenvalue Problems

“…For other structures, such as skew-Hamiltonian and Hamiltonian matrices, this figure can be less dramatic [9]. Moreover, in view of recent progress made in improving the performance of general-purpose algorithms [21,22], it may require considerable implementation efforts to turn this reduction of flops into an actual reduction of computational time.…”

Book Review: Spectra and pseudospectra: The behavior of nonnormal matrices and operators

“…Here, A is factorized into the product of an orthogonal matrix (Q) and an upper triangular matrix (R). With appropriate refinements developed by Francis, such as an initial reduction to Hessenberg form and the use of shifts to accelerate convergence, together with more recent refinements that exploit modern machine architectures [1], [4], the QR algorithm is the state of the art for computing the complete eigensystem of a dense matrix. Nominated as one of the "10 algorithms with the greatest influence on the development and practice of science and engineering in the 20th century" [8], the QR algorithm has been the standard method for solving the eigenvalue problem for over 40 years.…”

Accelerating Computation of Eigenvectors in the Dense Nonsymmetric Eigenvalue Problem

“…Being based on similarity transformations, the eigenvalues of A are the same as the eigenvalues of T , which are simply the diagonal elements of T . The QR iteration method takes O(n 3 ) flops, but being an iterative method, the exact count depends heavily on the convergence rate and techniques such as aggressive early deflation [3,4]. It includes a mixture of Level 1 BLAS for applying Givens rotations and Level 3 BLAS for updating H and accumulating Q 2 .…”