Computing A, 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 A1/2 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.

[1]  B. Baillaud.  Sur le calcul numerique des integrales definies , 1886 .

[2]  Radakovič.  The theory of approximation , 1932 .

[3]  George Szekeres,et al.  Numerical evaluation of high-dimensional integrals , 1964 .

[4]  Leon M. Hall,et al.  Special Functions , 1998 .

[5]  D. Newman.  Rational approximation to | x , 1964 .

[6]  Frank Stenger,et al.  Integration Formulae Based on the Trapezoidal Formula , 1973 .

[7]  Masatake Mori,et al.  Quadrature formulas obtained by variable transformation , 1973 .

[8]  Masatake Mori,et al.  Double Exponential Formulas for Numerical Integration , 1973 .

[9]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .

[10]  F. Stenger.  Numerical Methods Based on Whittaker Cardinal, or Sinc Functions , 1981 .

[11]  L. Trefethen,et al.  The Carathéodory–Fejér Method for Real Rational Approximation , 1983 .

[12]  Masao Iri,et al.  On a certain quadrature formula , 1987 .

[13]  F. Stenger.  Numerical Methods Based on Sinc and Analytic Functions , 1993 .

[14]  A. Laub,et al.  The matrix sign function , 1995, IEEE Trans. Autom. Control..

[15]  G. Golub,et al.  Bounds for the Entries of Matrix Functions with Applications to Preconditioning , 1999 .

[16]  Jean-François Richard,et al.  Methods of Numerical Integration , 2000 .

[17]  Dongwoo Sheen,et al.  A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature , 2000, Math. Comput..

[18]  Ivan P. Gavrilyuk,et al.  Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations , 2001 .

[19]  H. V. D. Vorst,et al.  Numerical methods for the QCDd overlap operator. I. Sign-function and error bounds , 2002, hep-lat/0202025.

[20]  Lloyd N. Trefethen,et al.  Schwarz-Christoffel Mapping , 2002 .

[21]  V. Thomée,et al.  Time discretization of an evolution equation via Laplace transforms , 2004 .

[22]  A. Kennedy.  Approximation theory for matrices , 2004, hep-lat/0402037.

[23]  Philip I. Davies,et al.  Computing f (A)b for matrix functions f , 2005 .

[24]  M. Mori.  Discovery of the Double Exponential Transformation and Its Developments , 2005 .

[25]  Tobin A. Driscoll,et al.  Algorithm 843: Improvements to the Schwarz-Christoffel toolbox for MATLAB , 2005, TOMS.

[26]  Karl-Georg Steffens.  The history of approximation theory : from Euler to Bernstein , 2006 .

[27]  J. A. C. Weideman,et al.  Optimizing Talbot's Contours for the Inversion of the Laplace Transform , 2006, SIAM J. Numer. Anal..

[28]  Lloyd N. Trefethen,et al.  Parabolic and hyperbolic contours for computing the Bromwich integral , 2007, Math. Comput..

[29]  Lloyd N. Trefethen,et al.  Computing the Gamma Function Using Contour Integrals and Rational Approximations , 2007, SIAM J. Numer. Anal..

[30]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[31]  Nicholas J. Higham,et al.  Computing AAlpha, log(A), and Related Matrix Functions by Contour Integrals , 2008, SIAM J. Numer. Anal..