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

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.