Taylor's theorem for matrix functions with applications to condition number estimation

Edvin Deadman, Samuel Relton

    Research output: Contribution to journalArticlepeer-review


    We derive an explicit formula for the remainder term of a Taylor polynomial of a matrix function. This formula generalizes a known result for the remainder of the Taylor polynomial for an analytic function of a complex scalar. We investigate some consequences of this result, which culminate in new upper bounds for the level-1 and level-2 condition numbers of a matrix function in terms of the pseudospectrum of the matrix. Numerical experiments show that, although the bounds can be pessimistic, they can be computed much faster than the standard methods. This makes the upper bounds ideal for a quick estimation of the condition number whilst a more accurate (and expensive) method can be used if further accuracy is required. They are also easily applicable to more complicated matrix functions for which no specialized condition number estimators are currently available.
    Original languageUndefined
    Pages (from-to)354-371
    Number of pages18
    JournalLinear Algebra and its Applications
    Early online date18 Apr 2016
    Publication statusPublished - 1 Sept 2016

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