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An algorithm for Gaussian quadrature given modified moments

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Summary

An algebraic algorithm, the long quotient- modified difference (LQMD) algorithm, is described for the Gaussian quadrature of the one-dimensional product integral ∫f(x)w(x)dx when the weight function ω(x) is known through modified momentsv l =; theP l (x) are any polynomials of degreel satisfying 3-term recurrence relations with known coefficients. The algorithm serves to establish the co-diagonal matrix, the eigenvalues of which are the Gaussian abscissas. Applied to ordinary moments it requires far fewer divisions than the quotient-difference algorithm; if theP l (x) are themselves orthogonal with a kernelw 0 03F0;, there is no instability due to rounding errors. For smooth kernels ω(x) it is safe to use secondorder interpolation in determining the eigenvalues by Givens' method. The Christoffel weights can be expressed as ratios of two terms which are most easily calculated in a Sturm sequence beginning with the highest value ofl. A formula for the Christoffel weights applicable for rational versions of theQR algorithm is also derived. Convergence and the propagation of rounding errors are illustrated by several examples, and anAlgol procedure is given.

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Authors and Affiliations

  1. Department of Mathematics, University of Salford, Salford, Lancashire, England

    R. A. Sack

  2. E.M.I. Electronics and Industrial Operations Limited, E.O.P.S., Feltham, Middlesex, England

    A. F. Donovan

Authors
  1. R. A. Sack
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  2. A. F. Donovan
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Additional information

Based in part on a project report presented by A. F. Donovan in partial fulfilment of the requirements for the degree of B. Sc. (Honours), University of Salford (1969).

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Sack, R.A., Donovan, A.F. An algorithm for Gaussian quadrature given modified moments. Numer. Math. 18, 465–478 (1971). https://doi.org/10.1007/BF01406683

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  • DOI: https://doi.org/10.1007/BF01406683

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