Volume 39, pp. 414-436, 2012.

Computation of the matrix $p$th root and its Fréchet derivative by integrals

João R. Cardoso

Abstract

We present new integral representations for the matrix $p$th root and its Fréchet derivative and then investigate the computation of these functions by numerical quadrature. Three different quadrature rules are considered: composite trapezoidal, Gauss-Legendre and adaptive Simpson. The problem of computing the matrix $p$th root times a vector without the explicit evaluation of the $p$th root is also analyzed and bounds for the norm of the matrix $p$th root and its Fréchet derivative are derived.

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Key words

matrix $p$th root, Fréchet derivative, quadrature, composite trapezoidal rule, Gauss-Legendre rule, adaptive Simpson rule

AMS subject classifications

65F60, 65D30

Links to the cited ETNA articles

[6]	Vol. 38 (2011), pp. 202-217 João R. Cardoso: Evaluating the Fréchet derivative of the matrix $p$th root

ETNA articles which cite this article

Vol. 54 (2021), pp. 558-580 Fuminori Tatsuoka, Tomohiro Sogabe, Yuto Miyatake, Tomoya Kemmochi, and Shao-Liang Zhang: Computing the matrix fractional power with the double exponential formula