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Jacobi matrices for sums of weight functions

  • Part II Numerical Mathematics
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Abstract

Orthogonal polynomials are conveniently represented by the tridiagonal Jacobi matrix of coefficients of the recurrence relation which they satisfy. LetJ 1 andJ 2 be finite Jacobi matrices for the weight functionsw 1 andw 2, resp. Is it possible to determine a Jacobi matrix\(\tilde J\), corresponding to the weight functions\(\tilde w\)=w 1+w 2 using onlyJ 1 andJ 2 and if so, what can be said about its dimension? Thus, it is important to clarify the connection between a finite Jacobi matrix and its corresponding weight function(s). This leads to the need for stable numerical processes that evaluate such matrices.

Three newO(n 2) methods are derived that “merge” Jacobi matrices directly without using any information about the corresponding weight functions. The first can be implemented using any of the updating techniques developed earlier by the authors. The second new method, based on rotations, is the most stable. The third new method is closely related to the modified Chebyshev algorithm and, although it is the most economical of the three, suffers from instability for certain kinds of data.

The concepts and the methods are illustrated by small numerical examples, the algorithms are outlined and the results of numerical tests are reported.

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Authors and Affiliations

  1. Computer Science Department, University of Adelaide, 5000, Adelaide, South Australia

    Sylvan Elhay

  2. Computer Science Department, Stanford University, 94305, Stanford, California

    Gene H. Golub

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

    Jaroslav Kautsky

Authors
  1. Sylvan Elhay
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  2. Gene H. Golub
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  3. Jaroslav Kautsky
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Elhay, S., Golub, G.H. & Kautsky, J. Jacobi matrices for sums of weight functions. BIT 32, 143–166 (1992). https://doi.org/10.1007/BF01995114

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  • DOI: https://doi.org/10.1007/BF01995114

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