Jacobi matrices for sums of weight functions
Tóm tắt
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.
Tài liệu tham khảo
S. Elhay, G. H. Golub and J. Kautsky.Updating and downdating of orthogonal polynomials with data fitting applications. SIAM J. Matrix Anal. Appl., 12(2): 327–353, 1991.
S. Elhay and J. Kautsky.Generalized Kronrod Patterson type imbedded quadratures. Aplikace Matematiky, 1991, to appear.
B. Fisher and G. H. Golub.On generating polynomials which are orthogonal over several intervals. Technical Report NA 89-09, Computer Science Dept, Stanford University, Stanford, California, 94305, 1989.
W. Gautschi.On generating orthogonal polynomials. SIAM J. Sci. Statist. Comput., 3: 289–317, 1982.
G. H. Golub and J. Kautsky.Calculation of Gauss quadratures with multiple free and fixed knots. Numer. Math., 41: 147–163, 1983.
G. H. Golub and J. H. Welsh.Calculation of Gauss quadrature rules. Math. Comp., 23: 221–230, 1969.
J. Kautsky and S. Elhay.Calculation of the weights of interpolatory quadratures. Numer. Math., 40: 407–422, 1982.
J. Kautsky and S. Elhay.Gauss quadratures and Jacobi matrices for weight functions not of one sign. Math. Comp., 43 (168): 543–550, 1984.
J. Kautsky and G. H. Golub.On the calculation of Jacobi matrices. Linear Algebra Appl., 52/53: 439–455, 1983.
B. N. Parlett.The Symmetric Eigenvalue Problem. Englewood Cliffs, NJ: Prentice Hall, 1980.
H. Rutishauser.On Jacobi rotation patterns. Proc. Symp. in Appl. Maths., 15: 219–239, 1963.
J. R. Wilkinson and C. Reinsch.Linear Algebra.Handbook for automatic computation, volume II, pages 274–283. Springer, 1971. Contribution II/8.