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We show how both the tridiagonal and bidiagonal QR algorithms can be restructured so that they be-come rich in operations that can achieve near-peak performance on a modern processor. The key is anovel, cache-friendly algorithm for applying multiple sets of Givens rotations to the eigenvector/singularvector matrix. This algorithm is then implemented with optimizations that (1) leverage vector instructionunits to increase floating-point throughput, and (2) fuse multiple rotations to decrease the total number ofmemory operations. We demonstrate the merits of these new QR algorithms for computing the Hermitianeigenvalue decomposition (EVD) and singular value decomposition (SVD) of dense matrices when all eigen-vectors/singular vectors are computed. The approach yields vastly improved performance relative to thetraditional QR algorithms for these problems and is competitive with two commonly used alternatives—Cuppen’s Divide and Conquer algorithm and the Method of Multiple Relatively Robust Representations—while inheriting the more modest O(n) workspace requirements of the original QR algorithms. Since thecomputations performed by the restructured algorithms remain essentially identical to those performed bythe original methods, robust numerical properties are preserved

Restructuring the Tridiagonal and Bidiagonal QR Algorithms for Performance