Why Gaussian quadrature in the complex plane?

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude cTA−1b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use cTx(k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/λ. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.

[1]  Claude Brezinski,et al.  Pade-Type Approximation and General Orthogonal Polynomials , 1981, The Mathematical Gazette.

[2]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .

[3]  K. Paulsen,et al.  A scalar and vector potential formulation for finite element solutions to Maxwell's equations , 1992, IEEE Antennas and Propagation Society International Symposium 1992 Digest.

[4]  C. Brezinski.  Biorthogonality and its Applications to Numerical Analysis , 1991 .

[5]  W. Chew.  Waves and Fields in Inhomogeneous Media , 1990 .

[6]  Gene H. Golub,et al.  An adaptive Chebyshev iterative method\newline for nonsymmetric linear systems based on modified moments , 1994 .

[7]  W. Gautschi.  Numerical analysis: an introduction , 1997 .

[8]  D. Arnett,et al.  Supernovae and Nucleosynthesis , 1996 .

[9]  T. Manteuffel,et al.  Conjugate gradient algorithms using multiple recursions , 1996 .

[10]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[11]  Jirí Hrebícek,et al.  Solving Problems in Scientific Computing Using Maple and MATLAB® , 2004, Springer Berlin Heidelberg.

[12]  T. Manteuffel,et al.  A taxonomy for conjugate gradient methods , 1990 .

[13]  Tobin A. Driscoll,et al.  From Potential Theory to Matrix Iterations in Six Steps , 1998, SIAM Rev..

[14]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[15]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[16]  F. Low.  Classical Field Theory , 1997 .

[17]  R. W. Freund,et al.  Gauss Quadratures Associated with the Arnoldi Process and the Lanczos Algorithm , 1993 .

[18]  Martin H. Gutknecht,et al.  Lanczos-type solvers for nonsymmetric linear systems of equations , 1997, Acta Numerica.

[19]  Keith D. Paulsen,et al.  Nodal-based finite-element modeling of Maxwell's equations , 1992 .

[20]  H. Wilf.  Mathematics for the Physical Sciences , 1976 .

[21]  George W. Kattawar,et al.  Scattering theory of waves and particles (2nd ed.) , 1983 .

[22]  R. Gordon.  Error Bounds in Spectroscopy and Nonequilibrium Statistical Mechanics , 1968 .

[23]  K D Paulsen,et al.  Finite-element solution of Maxwell's equations with Helmholtz forms. , 1994, Journal of the Optical Society of America. A, Optics, image science, and vision.

[24]  O. Axelsson.  Iterative solution methods , 1995 .

[25]  Michael T. Heath,et al.  Scientific Computing: An Introductory Survey , 1996 .

[26]  N. Dallaporta.  L. D. Landau undE. M. Lifshitz -Quantum Mechanics. Non relativistic Theory , 1958 .

[27]  Åke Björck,et al.  Numerical Methods , 2021, Markov Renewal and Piecewise Deterministic Processes.

[28]  W. Gragg.  Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle , 1993 .

[29]  G. Golub,et al.  On the Error Computation for Polynomial Based Iteration Methods , 1994 .

[30]  W. Gautschi.  Orthogonal polynomials: applications and computation , 1996, Acta Numerica.

[31]  G. Golub,et al.  Matrices, moments and quadrature II; How to compute the norm of the error in iterative methods , 1997 .

[32]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[33]  R. Newton.  Scattering theory of waves and particles , 1966 .

[34]  Roy G. Gordon,et al.  Error Bounds in Equilibrium Statistical Mechanics , 1968 .

[35]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[36]  F. Low,et al.  Classical Field Theory: Electromagnetism and Gravitation , 1997 .

[37]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[38]  H. Shapiro.  The Schwarz Function and Its Generalization to Higher Dimensions , 1992 .

[39]  E. Stiefel.  Kernel polynomial in linear algebra and their numerical applications, in : Further contributions to the determination of eigenvalues , 1958 .

[40]  K.D. Paulsen,et al.  Applications of Potentials to Finite Element Modeling of Maxwell's Equations , 1992, Digest of the Fifth Biennial IEEE Conference on Electromagnetic Field Computation.

[41]  G. Golub,et al.  Bounds for the error of linear systems of equations using the theory of moments , 1972 .

[42]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[43]  D. Smolarski.  Optimum semi-iterative methods for the solution of any linear algebraic system with a square matrix , 1982 .

[44]  G. Golub,et al.  Bounds for the error in linear systems , 1979 .

[45]  Roland W. Freund,et al.  An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices , 1993, SIAM J. Sci. Comput..

[46]  Gene H. Golub,et al.  Matrix computations , 1983 .

[47]  R. W. Freund,et al.  The Look-Ahead Lanczos Process for Large Nonsymmetric Matrices and Related Algorithms , 1993 .