Generating extreme-scale matrices with specified singular values or condition numbers

Fasi, Massimiliano and Higham, Nicholas J. (2020) Generating extreme-scale matrices with specified singular values or condition numbers. [MIMS Preprint] (Unpublished)

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Abstract

A widely used form of test matrix is the randsvd matrix constructed as the product A = USV*, where U and V are random orthogonal or unitary matrices from the Haar distribution and S is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution. The cost of forming an m-by-n randsvd matrix is m³ + n³ flops, which is prohibitively expensive at extreme scale; moreover, the randsvd construction requires a significant amount of communication, making it unsuitable for distributed memory environments. By dropping the requirement that U and V be Haar distributed and that both be random, we derive new algorithms for forming A that have cost linear in the number of matrix elements and require a low amount of communication and synchronization. We specialize these algorithms to generating matrices with specified 2-norm condition number. Numerical experiments show that the algorithms have excellent efficiency and scalability.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Mr Massimiliano Fasi
Date Deposited: 20 Oct 2020 10:59
Last Modified: 20 Oct 2020 10:59
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2786

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