Jacobi matrices for measures modified by a rational factor
- 43 Downloads
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.
KeywordsOrthogonal polynomials Jacobi matrices rational factors modifying measures Gauss quadratures
Unable to display preview. Download preview PDF.
- M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1972).Google Scholar
- T.S. Chihara,An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).Google Scholar
- S. Elhay and J. Kautsky, Generalized Kronrod-Patterson type imbedded quadratures, Aplikace Matematiky 37 (1992) 81–103.Google Scholar
- W. Gautschi, Minimal solutions of three-term recurrence relations and orthogonal polynomials, Math. Comp. 36 (1981) 547–554.Google Scholar
- J. Kautsky and S. Elhay, Gauss quadratures and Jacobi matrices for weight functions not of one sign, Math. Comp. 43 (1984) 543–550.Google Scholar
- J. Kautsky ad G.H. Golub, On the calculation of Jacobi matrices, Lin. Alg. Appl. 52/53 (1983) 439–455.Google Scholar
- C. Szegö,Orthogonal Polynomials, AMS Colloquium Publ. 23, 4th ed. (AMS, Providence, RI, 1975).Google Scholar