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Jacobi matrices for measures modified by a rational factor

Abstract

This paper describes how, given the Jacobi matrixJ for the measure dσ(t), it is possible to produce the Jacobi matrix Ĵ for the measurer(t)dσ(t) wherer(t) is a quotient of polynomials. The method uses a new factoring algorithm to generate the Jacobi matrices associated with the partial fraction decomposition ofr(t) and then applies a previously developed summing technique to merge these Jacobi matrices. The factoring method performs best just where Gautschi's minimal solution method for this problem is weakest and vice versa. This suggests a hybrid strategy which is believed to be the most powerful yet for solving this problem. The method is demonstrated on a simple example and some numerical tests illustrate its performance characteristics.

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Communicated by G.H. Golub

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Elhay, S., Kautsky, J. Jacobi matrices for measures modified by a rational factor. Numer Algor 6, 205–227 (1994). https://doi.org/10.1007/BF02142672

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Keywords

  • Orthogonal polynomials
  • Jacobi matrices
  • rational factors modifying measures
  • Gauss quadratures