Fast computation of spectral projectors of banded matrices

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algo- rithm based on the QDWH algorithm, along the works by Nakatsukasa et al. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary Matlab implementation becomes faster than eig already for matrix sizes of a few thousand.Comment: 27 pages, 10 figure

A Parallel Structured Divide-and-Conquer Algorithm for Symmetric Tridiagonal Eigenvalue Problems

© 2021 IEEE. Personal use of this material is permitted. Permissíon from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertisíng or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.[EN] In this article, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [16] and only scaled to 300 processes for the same matrices. Comparing with PDSTEDC in ScaLAPACK, PSDC is always faster and achieves 1.4x-1.6x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.The authors would like to thank the referees for their valuable comments which greatly improve the presentation of this article. This work was supported by National Natural Science Foundation of China (No. NNW2019ZT6-B20, NNW2019ZT6B21, NNW2019ZT5-A10, U1611261, 61872392, and U1811461), National Key RD Program of China (2018YFB0204303), NSF of Hunan (No. 2019JJ40339), NSF of NUDT (No. ZK18-03-01), Guangdong Natural Science Foundation (2018B030312002), and the Program for Guangdong Introducing Innovative and Entrepreneurial Teams under Grant 2016ZT06D211. The work of Jose E. Roman was supported by the Spanish Agencia Estatal de Investigacion (AEI) under project SLEPc-DA (PID2019-107379RB-I00).Liao, X.; Li, S.; Lu, Y.; Román Moltó, JE. (2021). A Parallel Structured Divide-and-Conquer Algorithm for Symmetric Tridiagonal Eigenvalue Problems. IEEE Transactions on Parallel and Distributed Systems. 32(2):367-378. https://doi.org/10.1109/TPDS.2020.3019471S36737832

Orthogonal matrix polynomials and applications

AbstractOrthogonal matrix polynomials, on the real line or on the unit circle, have properties which are natural generalizations of properties of scalar orthogonal polynomials, appropriately modified for the matrix calculus. We show that orthogonal matrix polynomials, both on the real line and on the unit circle, appear at various places and we describe some of them. The spectral theory of doubly infinite Jacobi matrices can be described using orthogonal 2×2 matrix polynomials on the real line. Scalar orthogonal polynomials with a Sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line. Orthogonal matrix polynomials on the unit circle are related to unitary block Hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing. Finally we show how orthogonal matrix polynomials can be used for Gaussian quadrature of matrix-valued functions. Parallel algorithms for this purpose have been implemented (using PVM) and some examples are worked out

MRRR-based Eigensolvers for Multi-core Processors and Supercomputers

The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to tridiagonal form. For its solution, the algorithm of Multiple Relatively Robust Representations (MRRR or MR3 in short) - introduced in the late 1990s - is among the fastest methods. To compute k eigenpairs of a real n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in contrast, all the other practical methods require O(k^2 n) or O(n^3) operations in the worst case. This thesis centers around the performance and accuracy of MRRR.Comment: PhD thesi