In the 1990's, Dhillon and Parlett devised the algorithm of multiple relatively robust representations (MRRR) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with O(n 2 ) cost. While previous publications related to MRRR focused on theoretical aspects of the algorithm, a documentation of software issues has been missing. In this article, we discuss the design and implementation of the new MRRR version STEGR that will be included in the next LAPACK release. By giving an algorithmic description of MRRR and identifying governing parameters, we hope to make STEGR more easily accessible and suitable for future performance tuning. Furthermore, this should help users understand design choices and tradeoffs when using the code.
Abstract. We compare four algorithms from the latest LAPACK 3.1 release for computing eigenpairs of a symmetric tridiagonal matrix. These include QR iteration, bisection and inverse iteration (BI), the Divide-and-Conquer method (DC), and the method of Multiple Relatively Robust Representations (MR).Our evaluation considers speed and accuracy when computing all eigenpairs, and additionally subset computations. Using a variety of carefully selected test problems, our study includes a variety of today's computer architectures.Our conclusions can be summarized as follows. (1)
Abstract. During the last ten years, Dhillon and Parlett devised a new algorithm (multiple relatively robust representations (MRRR)) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with O(n 2 ) cost. It has been incorporated into LAPACK version 3.0 as routine stegr.We have discovered that the MRRR algorithm can fail in extreme cases. Sometimes eigenvalues agree to working accuracy and MRRR cannot compute orthogonal eigenvectors for them. In this paper, we describe and analyze these failures and various remedies. 1. Introduction. Starting in the mid-90s, Dhillon and Parlett developed the algorithm of multiple relatively robust representations (MRRR) that computes numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with O(n 2 ) cost [13,15,16,17,7]. This algorithm was tested on a large and challenging set of matrices and has been incorporated into LAPACK version 3.0 as routine stegr. The algorithm is described, in the detail we need here, in section 2. For a more detailed description see also [8]. An example of MRRR in action is given in section 3.In 2003, one of us (Vömel) came to Berkeley to assist in the modification of stegr to compute a subset of k eigenpairs with O(kn) operations. When testing stegr on more and more challenging matrices, he discovered cases of failure.Investigation of these cases brought to light assumptions made in stegr that hold in exact arithmetic and in the majority of cases in finite precision arithmetic, but that can fail. These assumptions, why they are reasonable, and how they can fail are the subject of section 4.In section 5, we propose and analyze various remedies for the aforementioned shortcomings and show how to incorporate them into MRRR. We also look at the cost of these modifications. We select as most suitable an approach that is based on small random perturbations and introduces artificial roundoff effects. This approach preserves the complexity of the original algorithm.
eliaus (éa 3679 upvd) and Guillaume Melquiond lip (umr 5668 cnrs-éns Lyon-inria) Gappa is a tool designed to formally verify the correctness of numerical softwares and hardwares. It uses interval arithmetic and forward error analysis to bound mathematical expressions that involve rounded as well as exact operators. It then generates a theorem and its proof for each verified enclosure. This proof can be automatically checked with a proof assistant, such as Coq or HOL Light. It relies on the facts of a large companion library we have developed. This Coq library provides theorems dealing with addition, multiplication, division, and square root, for both fixedand floating-point arithmetics. Gappa uses multiple-precision dyadic fractions for the endpoints of intervals and performs forward error analysis on rounded operators when necessary. When asked, Gappa reports the best bounds it is able to reach for a given expression in a given context. This feature can be used to identify where the set of facts and automatic techniques implemented in Gappa becomes insufficient. Gappa handles seamlessly additional properties expressed as interval properties or rewriting rules in order to establish more intricate bounds. Recent work showed that Gappa is suited to discharge proof obligations generated for small pieces of software. They may be produced by third-party tools and the first applications of Gappa use proof obligations written by designers or obtained from traces of execution.
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