Geodesic
Computation of the matrix \(p\)th root and its Fréchet derivative by integrals
Electronic transactions on numerical analysis, Tome 39 (2012), pp. 414-436
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.
Classification : 65F60, 65D30
Keywords: matrix $p$th root, Fréchet derivative, quadrature, composite trapezoidal rule, Gauss-Legendre rule, adaptive simpson rule 
@article{ETNA_2012__39__a2,
     author = {Cardoso,  Jo\~ao R.},
     title = {Computation of the matrix \(p\)th root and its {Fr\'echet} derivative by integrals},
     journal = {Electronic transactions on numerical analysis},
     pages = {414--436},
     year = {2012},
     volume = {39},
     zbl = {1287.65035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2012__39__a2/}
}
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AU  - Cardoso,  João R.
TI  - Computation of the matrix \(p\)th root and its Fréchet derivative by integrals
JO  - Electronic transactions on numerical analysis
PY  - 2012
SP  - 414
EP  - 436
VL  - 39
UR  - http://geodesic.mathdoc.fr/item/ETNA_2012__39__a2/
LA  - en
ID  - ETNA_2012__39__a2
ER  - 
%0 Journal Article
%A Cardoso,  João R.
%T Computation of the matrix \(p\)th root and its Fréchet derivative by integrals
%J Electronic transactions on numerical analysis
%D 2012
%P 414-436
%V 39
%U http://geodesic.mathdoc.fr/item/ETNA_2012__39__a2/
%G en
%F ETNA_2012__39__a2
Cardoso,  João R. Computation of the matrix \(p\)th root and its Fréchet derivative by integrals. Electronic transactions on numerical analysis, Tome 39 (2012), pp. 414-436. http://geodesic.mathdoc.fr/item/ETNA_2012__39__a2/