a b s t r a c tThe computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.
This paper provides a streamlined and modular presentation of the MR 3 algorithm for computing selected eigenpairs of symmetric tridiagonal matrices, thus disentangling the principles driving MR 3 and the (recursive) "core" algorithm from the specific (e.g., twisted) decompositions used to represent the matrices at different recursion depths and from the (dqds) transformations converting between them. Our approach allows a modular full proof for the correctness of the MR 3 algorithm. This proof is based on five requirements concerning the interplay between the core algorithm and its subcomponents. These requirements can also guide in implementing the algorithm, because they expose quantities that can and should be monitored at runtime. Our new implementation XMR, which is based on the above analysis, is described and compared to xSTEMR from Lapack 3.2.2. Numerical experiments comparing the robustness and performance of both implementations are given.1. Introduction. The established strategy for solving a dense real symmetric or complex Hermitian eigenproblem numerically is to reduce the matrix to real tridiagonal form by an orthogonal similarity transformation [1,16] and then solve the tridiagonal problem. Before there was MR 3 , usually one of the following three standard algorithms was employed for the second part.QR iteration has been the workhorse since the 1960's [14,15]. The current implementation xSTEQR in Lapack [1] is rock solid. At the time of this writing, the best method for computing all eigenvalues is the dqds algorithm [13,31], which can be considered to be a specially tailored QR iteration. But for computing eigenvectors, QR exhibits a true O(n 3 ) complexity, making it very slow in practice. In fact, for the dense symmetric problem, the solution of the tridiagonal problem using QR can outweigh the reduction phase by far [9]. On the other hand, QR is the only method that allows accumulating the eigenvectors of the original dense problem in place, thus saving a final matrix-matrix multiplication. Together with recently proposed high-performance realizations of sequences of rotations [21,33], this can make QR (almost) competitive with the approaches discussed below if most or all eigenvectors are required.Divide and conquer (DC) has been known quite a long time [3], but it took nearly 15 years until a stable implementation was found [18]. Complexity is O(n 3 ) in theory, but in practice DC behaves more like O(n 2.5 ) on average [4, 6], due to effective use of BLAS3-operations and a large amount of work that can be saved via deflation.
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