Skip to main content
SpringerLink
Log in

Jacobi matrices for measures modified by a rational factor

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1972).

    Google Scholar 

  2. T.S. Chihara,An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).

    Google Scholar 

  3. S. Elhay, G.H. Golub and J. Kautsky, Jacobi matrices for sums of weight functions, BIT 32 (1992) 143–166.

    Article  Google Scholar 

  4. S. Elhay and J. Kautsky, Generalized Kronrod-Patterson type imbedded quadratures, Aplikace Matematiky 37 (1992) 81–103.

    Google Scholar 

  5. W. Gautschi, Minimal solutions of three-term recurrence relations and orthogonal polynomials, Math. Comp. 36 (1981) 547–554.

    Google Scholar 

  6. W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comp. 3 (1982) 289–317.

    Article  Google Scholar 

  7. 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 

  8. J. Kautsky ad G.H. Golub, On the calculation of Jacobi matrices, Lin. Alg. Appl. 52/53 (1983) 439–455.

    Google Scholar 

  9. R.A. Sack and A.F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math. 18 (1971/72) 465–478.

    Article  Google Scholar 

  10. C. Szegö,Orthogonal Polynomials, AMS Colloquium Publ. 23, 4th ed. (AMS, Providence, RI, 1975).

    Google Scholar 

  11. V.B. Uvarov, The connection between systems of polynomials orthogonal with respect to different distribution functions, USSR Comp. Math. and Math. Phys. 9 (1969) 25–36.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, University of Adelaide, 5005, South Australia, Australia

    Sylvan Elhay

  2. School of Information Science and Technology, Flinders University, 5042, Bedford Park, South Australia, Australia

    Jaroslav Kautsky

Authors
  1. Sylvan Elhay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaroslav Kautsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by G.H. Golub

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02142672

Keywords

Search

Navigation