Geodesic
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
Electronic transactions on numerical analysis, Tome 29 (2008)
Zbl   EuDML
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 $########

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###$\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 
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/
@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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