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2014.13: Testing matrix function algorithms using identities

2014.13: Edvin Deadman and Nicholas J. Higham (2014) Testing matrix function algorithms using identities.

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Abstract

Algorithms for computing matrix functions are typically tested by comparing the forward error with the product of the condition number and the unit roundoff. The forward error is computed with the aid of a reference solution, typically computed at high precision. An alternative approach is to use functional identities such as the ``round trip tests'' $e^{\log A} = A$ and $(A^{1/p})^p = A$, as are currently employed in a SciPy test module. We show how a linearized perturbation analysis for a functional identity allows the determination of a maximum residual consistent with backward stability of the constituent matrix function evaluations. Comparison of this maximum residual with a computed residual provides a necessary test for backward stability. We also show how the actual linearized backward error for these relations can be computed. Our approach makes use of Fr\'echet derivatives and estimates of their norms. Numerical experiments show that the proposed approaches are able both to detect instability and to confirm stability.

Item Type:MIMS Preprint
Additional Information:

To appear in ACM Transactions on Mathematical Software

Uncontrolled Keywords:matrix function, normwise relative error, functional identity, testing, forward error, backward error, SciPy, Python, MATLAB, NAG Library
Subjects:MSC 2000 > 65 Numerical analysis
MIMS number:2014.13
Deposited By:Dr Edvin Deadman
Deposited On:16 January 2015

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