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Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices

Part of: Numerical linear algebra Nontrigonometric harmonic analysis Computer aspects of numerical algorithms Special matrices Error analysis and interval analysis

Published online by Cambridge University Press:  03 July 2015

Niall Emmart ,
Yang Chen  and
Charles C. Weems
Niall Emmart*
Affiliation:
College of Information and Computer Science, University of Massachusetts, Amherst, MA 01002, USA
Yang Chen
Affiliation:
Department of Mathematics, University of Macau, Macau, China
Charles C. Weems
Affiliation:
College of Information and Computer Science, University of Massachusetts, Amherst, MA 01002, USA
*
*Corresponding author. Email addresses: nemmart@yrrid.com (N. Emmart), yayangchen@umac.mo (Y. Chen), weems@cs.umass.edu (C. C.Weems)
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Abstract

This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned. Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic. Surprisingly, we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approach combines message passing and shared memory, achieving near-perfect scalability and high tolerance for network latency. We are thus able to find solutions for much larger matrices than previously possible, with the potential for extending this work to systems with greater levels of parallelism. The contributions of this work are in three areas: determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations; the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster, and the numerical results themselves.

Keywords

Parallel eigensolverHankel matricesextremely ill-conditioned matrices

MSC classification

Secondary: 15B52: Random matrices 15B57: Hermitian, skew-Hermitian, and related matrices 42C05: Orthogonal functions and polynomials, general theory 65F15: Eigenvalues, eigenvectors 65G30: Interval and finite arithmetic 65Y05: Parallel computation
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 104 - 124
Copyright
Copyright © Global-Science Press 2015 

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