randsvdfast

Fast generation of random test matrices with pre-assigned singular values or 2-norm condition number.

The MATLAB randsvdfast test matrix

This repository contains a MATLAB function to generates a matrix with specified singular values or 2-norm condition number and the corresponding unit tests.

The function is named after the randsvd matrix in the MATLAB gallery, as it provides similar functionalities but uses a faster algorithm. The method was designed to generate test matrices for extreme-scale benchmarks such as the High-performance Linpack Benchmark (HPL) or the HPL-AI Mixed-Precision Benchmark.

Usage

The command

A = randsvdfast(n, kappa, mode, method, matrix, classname, realout)

generates a matrix A of class classname with condition number kappa and singular values distributed according to mode. The function generates a matrix of order n if n is a positive integer, and of size n(1) by n(2) if n is a vector of length 2. By default, n and kappa are both set to 10.

The functions provides functionalities similar to those of the MATLAB function galley('randsvd', ...). The most notable difference is that this routine allows the user to specify a custom distribution of the singular values (see below), but does not implement the reduction to banded form.

The singular values can have one of the following distributions:

• mode = 0: one large singular value and one small singular value,
• mode = 1: one large singular value,
• mode = 2: one small singular value,
• mode = 3 (default): geometric distribution,
• mode = 4: arithmetic distribution,
• mode = 5: random singular values with uniformly distributed magnitude,
• mode = 6: the vector kappa contains the singular values.

The parameter method selects the algorithm that will be used to generate the test matrix. It can take any of the following values:

• method = 1 (default): [Alg. 3.1, 1],
• method = 2: [Alg. 3.2, 1],
• method = 3: [Alg. 4.1, 1] (only mode = 0, 1, 2),
• method = 4: [Alg. 4.2, 1] (only mode = 0, 1, 2).

This function is faster for method = 3 or 4 than for method = 1 or 2.

The algorithm uses an orthogonal matrix Q that depends on the value of the parameter matrix, which can take the following values:

• matrix = 0 (default): Q is a Haar distributed random unitary generated as the Q factor of the QR decomposition of the matrix randn(n(1),n(2)).
• matrix = an integer from 1 to 7: Q is the matrix gallery('orthog',n,matrix).
• matrix is the function handle of a two-argument function that generates an n(1)-by-n(2) matrix with orthonormal columns.

The output matrix will be of class classname where classname is either 'single' or 'double'. Constants are computed in double precision, whereas operations at the scalar level are performed in precision classname. The entries of A will be real if realout is true, and complex otherwise. By default the function generates a real matrix of doubles.

Tests

The class-based unit tests for the randsvdfast function can be ran with the command test_run.

Anymatrix

The repository can be downloaded as a remote group into the extensible matrix collection Anymatrix with

anymatrix('g', 'randsvdfast', 'mfasi/randsvdfast-matlab')

and the test matrix can be generated with

anymatrix('randsvdfast/randsvdfast', n, kappa, mode, method, matrix, classname, realout)

Reference

[1] M. Fasi & N. J. Higham. Generating extreme-scale matrices with specified singular values or condition numbers. SIAM J. Sci. Comput., 43(1), 663–684, 2021.

License

The code is distributed under the terms of the 2-Clause BSD License, see license.txt