Estimating the condition number of f(A)b

• Published in: Numerical algorithms Vol. 70; no. 2; pp. 287 - 308
• Main Author: Deadman, Edvin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2015; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computer Science; Estimation; Iterative methods; Mathematical analysis; Numeric Computing; Numerical Analysis; Original Paper; Theory of Computation; 65F60; Fréchet derivative; Block 1-norm estimator; Power iteration; Condition number estimation; 65F35; 15A60; Matrix function; Matrix exponential; Python
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-014-9947-4

New algorithms are developed for estimating the condition number of f ( A ) b , where A is a matrix and b is a vector. The condition number estimation algorithms for f ( A ) already available in the literature require the explicit computation of matrix functions and their Fr´echet derivatives and are therefore unsuitable for the large, sparse A typically encountered in f ( A ) b problems. The algorithms we propose here use only matrix-vector multiplications. They are based on a modified version of the power iteration for estimating the norm of the Fr´echet derivative of a matrix function, and work in conjunction with any existing algorithm for computing f ( A ) b . The number of matrix-vector multiplications required to estimate the condition number is proportional to the square of the number of matrix-vector multiplications required by the underlying f ( A ) b algorithm. We develop a specific version of our algorithm for estimating the condition number of e A b , based on the algorithm of Al-Mohy and Higham (SIAM J. Matrix Anal. Appl. 30 (4), 1639–1657, 2009 ). Numerical experiments demonstrate that our condition estimates are reliable and of reasonable cost.