Computing AAlpha, log(A), and Related Matrix Functions by Contour Integrals

New methods are proposed for the numerical evaluation of f (A) or f (A)b, where f (A) is a function such as A 1/2 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.