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Estimating the condition number of f(A)b

Abstract

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.

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Correspondence to Edvin Deadman.

Additional information

This work was supported by European Research Council Advanced Grant MATFUN (267526).

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Deadman, E. Estimating the condition number of f(A)b . Numer Algor 70, 287–308 (2015). https://doi.org/10.1007/s11075-014-9947-4

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Keywords

  • Matrix function
  • Matrix exponential
  • Condition number estimation
  • Fréchet derivative
  • Power iteration
  • Block 1-norm estimator
  • Python

Mathematics Subject Classifications (2010)

  • 15A60
  • 65F35
  • 65F60