Computing $A^\alpha, \log(A)$, and Related Matrix Functions by Contour Integrals

Authors:
Nicholas Hale;Nicholas J. Higham;Lloyd N. Trefethen
Affiliations:
nich@comlab.ox.ac.uk and http://www.comlab.ox.ac.uk/nick.hale/ and LNT@comlab.ox.ac.uk and http://web.comlab.ox.ac.uk/nick.trefethen/;higham@ma.man.ac.uk and http://www.ma.man.ac.uk/$\'sim$higham/;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2008

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Abstract

New methods are proposed for the numerical evaluation of $f(\mathbf{A})$ or $f(\mathbf{A}) b$, where $f(\mathbf{A})$ is a function such as $\mathbf{A}^{1/2}$ or $\log (\mathbf{A})$ with singularities in $(-\infty,0]$ and $\mathbf{A}$ is a matrix with eigenvalues on or near $(0,\infty)$. The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of $f(\mathbf{A})b$ is typically reduced to one or two dozen linear system solves, which can be carried out in parallel.