Restructuring the QR Algorithm for Performance

We show how the QR algorithm can be restructured so that it becomes rich in operations that can achieve near-peak performance on a modern processor. The key is a novel algorithm for applying multiple sets of Givens rotations. We demonstrate the merits of this new QR algorithm for computing the Hermitian (symmetric) eigenvalue decomposition and singular value decomposition of dense matrices when all eigenvectors/singular vectors are computed. The approach yields vastly improved performance relative to the traditional QR algorithm 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 algorithm. Since the computations performed by the restructured algorithm remain essentially identical to those performed by the original method, robust numerical properties are preserved.