Geodesic
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
Electronic transactions on numerical analysis, Tome 29 (2008)
Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of , where is a negative semidefinite matrix and is the exponential function or one of the related $\sterling $########

${\S}$

###$\copyright \ddot $###$ \sterling $" functions" such as . Building on previous work by Trefethen and Gutknecht, Minchev,

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§$\ddot $###§! #"%$!\ddot '\( and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems $£$########$©ddot$can be solved efficiently, e.g. by a sparse direct solver. The first method is based on best rational $ddot546$87 approximations to on the negative real axis computed via the Carath\'eodory-Fej\'er procedure. Rather than using $£$optimal poles we approximate the functions in a set of common poles, which speeds up typical computations by a factor of to . The second method is an application of the trapezoid rule on a Talbot-type contour.$
Classification : 65L05, 41A20, 30E20
Keywords: matrix exponential, exponential integrators, stiff semilinear parabolic pdes, rational uniform approximation, Hankel contour, numerical quadrature 
@article{ETNA_2008__29__a13,
     author = {Schmelzer,  Thomas and Trefethen,  Lloyd N.},
     title = {Evaluating matrix functions for exponential integrators via {Carath\'eodory-Fej\'er} approximation and contour integrals},
     journal = {Electronic transactions on numerical analysis},
     year = {2008},
     volume = {29},
     zbl = {1186.65092},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2008__29__a13/}
}
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AU  - Trefethen,  Lloyd N.
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JO  - Electronic transactions on numerical analysis
PY  - 2008
VL  - 29
UR  - http://geodesic.mathdoc.fr/item/ETNA_2008__29__a13/
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%0 Journal Article
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%A Trefethen,  Lloyd N.
%T Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
%J Electronic transactions on numerical analysis
%D 2008
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%U http://geodesic.mathdoc.fr/item/ETNA_2008__29__a13/
%G en
%F ETNA_2008__29__a13
Schmelzer,  Thomas; Trefethen,  Lloyd N. Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals. Electronic transactions on numerical analysis, Tome 29 (2008). http://geodesic.mathdoc.fr/item/ETNA_2008__29__a13/