ISSN:
1572-9125
Keywords:
primary 65F30
;
secondary 65D10
;
65D15
;
65D30
;
65D32
;
Orthogonal polynomials
;
Jacobi matrices
;
orthogonal methods
;
Lanczos methods
;
sums of weight functions
;
updating
;
downdating
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01995114
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