On sensitivity of Gauss–Christoffel quadrature

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss–Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. That is perhaps one of the reasons why, to our knowledge, the sensitivity question has not been formulated and addressed in the literature, though several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates. We then discuss conditions under which the Gauss–Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems.

[1]  Walter Gautschi,et al.  THE INTERPLAY BETWEEN CLASSICAL ANALYSIS AND (NUMERICAL) LINEAR ALGEBRA — A TRIBUTE TO GENE H. GOLUB , 2002 .

[2]  G. Golub,et al.  A survey of matrix inverse eigenvalue problems , 1986 .

[3]  W. Gautschi.  On the construction of Gaussian quadrature rules from modified moments. , 1970 .

[4]  I. Dhillon.  Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem , 1998 .

[5]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[6]  H. V. D. Vorst,et al.  The rate of convergence of Conjugate Gradients , 1986 .

[7]  Z. Strakos,et al.  On the real convergence rate of the conjugate gradient method , 1991 .

[8]  Gene H. Golub,et al.  Matrix Computation and the Theory of Moments , 1995 .

[9]  Sergey V. Kuznetsov.  Perturbation bounds of the krylov bases and associated hessenberg forms , 1997 .

[10]  W. Gautschi.  A Survey of Gauss-Christoffel Quadrature Formulae , 1981 .

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

[12]  H. Keller,et al.  Analysis of Numerical Methods , 1967 .

[13]  Bruno Lang,et al.  Computing the Bidiagonal SVD Using Multiple Relatively Robust Representations , 2006, SIAM J. Matrix Anal. Appl..

[14]  Gene H. Golub,et al.  The numerically stable reconstruction of a Jacobi matrix from spectral data , 1977, Milestones in Matrix Computation.

[15]  Dirk P. Laurie,et al.  Questions Related to Gaussian Quadrature Formulas and Two-Term Recursions , 1999 .

[16]  Walter Gautschi,et al.  Numerical Analysis , 1978, Mathemagics: A Magical Journey Through Advanced Mathematics.

[17]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[18]  Z. Strakos,et al.  On error estimation in the conjugate gradient method and why it works in finite precision computations. , 2002 .

[19]  Philip J. Davis,et al.  Chapter 6 – Automatic Integration , 1984 .

[20]  Walter Gautschi,et al.  Error Bounds for Gaussian Quadrature of Analytic Functions , 1983 .

[21]  Gene H. Golub,et al.  Jacobi matrices for sums of weight functions , 1992 .

[22]  W. Gautschi.  Construction of Gauss-Christoffel quadrature formulas , 1968 .

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

[24]  Dirk Laurie,et al.  Computation of Gauss-type quadrature formulas , 2001 .

[25]  Bernhard Beckermann,et al.  How to choose modified moments , 1998 .

[26]  G. Golub,et al.  Updating and downdating of orthogonal polynomials with data fitting applications , 1991 .

[27]  Walter Gautschi,et al.  Is the recurrence relation for orthogonal polynomials always stable? , 1993 .

[28]  Paul Nevai,et al.  Distribution of zeros of orthogonal polynomials , 1979 .

[29]  I︠u︡. V. Vorobʹev.  Method of moments in applied mathematics , 1965 .

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

[31]  Shu-Fang Xu.  A stability analysis of the jacobi matrix inverse eigenvalue problem , 1993 .

[32]  W. Gautschi.  On Generating Orthogonal Polynomials , 1982 .

[33]  G. Stewart,et al.  Estimating the Largest Eigenvalue of a Positive Definite Matrix , 1979 .

[34]  W. Gautschi.  Questions of Numerical Condition Related to Polynomials , 1978 .

[35]  W. Gautschi,et al.  THE INTERPLAY BETWEEN CLASSICAL ANALYSIS AND ( NUMERICAL ) LINEAR ALGEBRA — A TRIBUTE TO GENE , 2002 .

[36]  B. Parlett,et al.  On estimating the largest eigenvalue with the Lanczos algorithm , 1982 .

[37]  H. V. D. Vorst,et al.  The convergence behavior of ritz values in the presence of close eigenvalues , 1987 .

[38]  S. Godunov,et al.  Condition number of the Krylov bases and subspaces , 1996 .

[39]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[40]  W. Gragg,et al.  The numerically stable reconstruction of Jacobi matrices from spectral data , 1984 .

[41]  V. B. Uvarov.  The connection between systems of polynomials orthogonal with respect to different distribution functions , 1969 .

[42]  B. Parlett,et al.  Relatively robust representations of symmetric tridiagonals , 2000 .

[43]  H.-J. Fischer.  On Generating Orthogonal Polynomials for Discrete Measures , 1998 .

[44]  Inderjit S. Dhillon,et al.  Orthogonal Eigenvectors and Relative Gaps , 2003, SIAM J. Matrix Anal. Appl..

[45]  A. Stroud,et al.  Gaussian quadrature formulas , 1966 .

[46]  Bruno Lang,et al.  On Symmetric Eigenproblems Induced by the Bidiagonal SVD , 2005, SIAM J. Matrix Anal. Appl..

[47]  Dirk P. Laurie.  Accurate recovery of recursion coe cients from Gaussian quadrature formulas , 1999 .

[48]  G. Meurant,et al.  The Lanczos and conjugate gradient algorithms in finite precision arithmetic , 2006, Acta Numerica.

[49]  L. M. Milne-Thomson,et al.  The Calculus Of Finite Differences , 1934 .

[50]  C. Lanczos.  Applied Analysis , 1961 .

[51]  Liliana Borcea,et al.  On the sensitivity of Lanczos recursions to the spectrum , 2005 .

[52]  B. Parlett,et al.  Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices , 2004 .

[53]  Paul N. Swarztrauber,et al.  On Computing the Points and Weights for Gauss-Legendre Quadrature , 2002, SIAM J. Sci. Comput..

[54]  James Demmel,et al.  Accurate Singular Values of Bidiagonal Matrices , 1990, SIAM J. Sci. Comput..

[55]  Gene H. Golub,et al.  On the calculation of Jacobi Matrices , 1983 .

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

[57]  Lothar Reichel,et al.  Fast QR decomposition of Vandermonde-like matrices and polynomial least squares approximation , 1991 .

[58]  Paul Van Dooren,et al.  Sensitivity analysis of the Lanczos reduction , 1999 .

[59]  Ren-Cang Li.  Relative perturbation theory. III. More bounds on eigenvalue variation , 1997 .