Estimating the condition number of f(A)b

Deadman, Edvin

doi:10.1007/s11075-014-9947-4

Cited by 5 publications

(2 citation statements)

References 28 publications

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“…Using the adjoint, the value γ ≈ K f (A, b) 2 is estimated from the following algorithm, given by Deadman [5].…”

Section: Algorithmmentioning

confidence: 99%

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A Block Krylov Method to Compute the Action of the Fréchet Derivative of a Matrix Function on a Vector with Applications to Condition Number Estimation

Kandolf¹,

Relton²

2017

SIAM J. Sci. Comput.

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Abstract. We design a block Krylov method to compute the action of the Fréchet derivative of a matrix function on a vector using only matrix-vector products, i.e., the derivative of f (A)b when A is subject to a perturbation in the direction E. The algorithm we derive is especially effective when the direction matrix E in the derivative is of low rank, while there are no such restrictions on A. Our results and experiments are focused mainly on Fréchet derivatives with rank 1 direction matrices. Our analysis applies to all functions with a power series expansion convergent on a subdomain of the complex plane which, in particular, includes the matrix exponential. We perform an a priori error analysis of our algorithm to obtain rigorous stopping criteria. Furthermore, we show how our algorithm can be used to estimate the 2-norm condition number of f (A)b efficiently. Our numerical experiments show that our new algorithm for computing the action of a Fréchet derivative typically requires a small number of iterations to converge and (particularly for single and half precision accuracy) is significantly faster than alternative algorithms. When applied to condition number estimation, our experiments show that the resulting algorithm can detect ill-conditioned problems that are undetected by competing algorithms.

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“…Using the adjoint, the value γ ≈ K f (A, b) 2 is estimated from the following algorithm, given by Deadman [5].…”

Section: Algorithmmentioning

confidence: 99%

“…Application to condition number estimation. In this section we use our new algorithm for computing L f (A, E)b to design an algorithm for estimating the condition number of f (A)b, building upon work by Deadman [5]. In particular, Deadman proves the following bounds on cond(f, A, b), which we defined in (1.3).…”

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confidence: 99%