Computing A^α‚ log(A) and related matrix functions by contour integrals

Abstract

New methods are proposed for the numerical evaluation of f(A) or f(A) b, where f(A) is a function such as A^½ or log(A) with singularities in (-∞,0] and A is a matrix with eigenvalues on or near (0,∞). 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(A)b is typically reduced to one or two dozen linear system solves, which can be carried out in parallel.