Hale, Nicholas; Higham, Nicholas J.; Trefethen, Lloyd N. Computing \(A^\alpha, \log(A)\), and related matrix functions by contour integrals. (English) Zbl 1176.65053 SIAM J. Numer. Anal. 46, No. 5, 2505-2523 (2008). Summary: 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. Cited in 4 ReviewsCited in 80 Documents MSC: 65F30 Other matrix algorithms (MSC2010) 65D30 Numerical integration Keywords:Cauchy integral; conformal map; contour integral; matrix function; quadrature; rational approximation; trapezoid rule; parallel computation; numerical examples Software:Schwarz-Christoffel PDFBibTeX XMLCite \textit{N. Hale} et al., SIAM J. Numer. Anal. 46, No. 5, 2505--2523 (2008; Zbl 1176.65053) Full Text: DOI