On generating polynomials which are orthogonal over several intervals
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- by Bernd Fischer and Gene H. Golub PDF
- Math. Comp. 56 (1991), 711-730 Request permission
Abstract:
We consider the problem of generating the recursion coefficients of orthogonal polynomials for a given weight function. The weight function is assumed to be the weighted sum of weight functions, each supported on its own interval. Some of these intervals may coincide, overlap or are contiguous. We discuss three algorithms. Two of them are based on modified moments, whereas the other is based on an explicit expression for the desired coefficients. Several examples, illustrating the numerical performance of the various methods, are presented.References
-
M. Branders, Application of Chebyshev polynomials in numerical integration, Dissertation, Catholic University of Leuven, Leuven, 1976. (Flemish)
- Walter Gautschi, On the construction of Gaussian quadrature rules from modified moments, Math. Comp. 24 (1970), 245–260. MR 285117, DOI 10.1090/S0025-5718-1970-0285117-6
- Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
- Walter Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 289–317. MR 667829, DOI 10.1137/0903018
- Walter Gautschi, How and how not to check Gaussian quadrature formulae, BIT 23 (1983), no. 2, 209–216. MR 697783, DOI 10.1007/BF02218441
- Walter Gautschi, On some orthogonal polynomials of interest in theoretical chemistry, BIT 24 (1984), no. 4, 473–483. MR 764820, DOI 10.1007/BF01934906
- Walter Gautschi, Orthogonal polynomials—constructive theory and applications, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 61–76. MR 793944, DOI 10.1016/0377-0427(85)90007-X
- Walter Gautschi, Questions of numerical condition related to polynomials, Studies in numerical analysis, MAA Stud. Math., vol. 24, Math. Assoc. America, Washington, DC, 1984, pp. 140–177. MR 925213
- Walter Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48 (1986), no. 4, 369–382. MR 834326, DOI 10.1007/BF01389645
- I. Gohberg, T. Kailath, and I. Koltracht, Efficient solution of linear systems of equations with recursive structure, Linear Algebra Appl. 80 (1986), 81–113. MR 851934, DOI 10.1016/0024-3795(86)90279-X
- Gene H. Golub and John H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl, A1–A10. MR 0245201, DOI 10.1090/S0025-5718-69-99647-1
- Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
- G. Heinig, P. Jankowski, and K. Rost, Fast inversion algorithms of Toeplitz-plus-Hankel matrices, Numer. Math. 52 (1988), no. 6, 665–682. MR 946382, DOI 10.1007/BF01395817 M. D. Kent, Chebyshev, Krylov, Lanczos: Matrix relationships and computations, Ph.D. thesis, Stanford, 1989.
- Hanoch Lev-Ari and Thomas Kailath, Triangular factorization of structured Hermitian matrices, I. Schur methods in operator theory and signal processing, Oper. Theory Adv. Appl., vol. 18, Birkhäuser, Basel, 1986, pp. 301–324. MR 902610, DOI 10.1007/978-3-0348-5483-2_{1}2
- Franz Peherstorfer, Extremalpolynome in der $L^1$- und $L^2$-Norm auf zwei disjunkten Intervallen, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 269–280 (German). MR 820530
- Youcef Saad, Iterative solution of indefinite symmetric linear systems by methods using orthogonal polynomials over two disjoint intervals, SIAM J. Numer. Anal. 20 (1983), no. 4, 784–811. MR 708457, DOI 10.1137/0720052
- R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math. 18 (1971/72), 465–478. MR 303693, DOI 10.1007/BF01406683 G. Szegö, Orthogonal polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1967.
- John C. Wheeler, Modified moments and Gaussian quadratures, Rocky Mountain J. Math. 4 (1974), 287–296. MR 334466, DOI 10.1216/RMJ-1974-4-2-287 —, Modified moments and continued fraction coefficients for the diatomic linear chain, J. Chem. Phys. 80 (1984), 472-476. H. Wilf, Mathematics for the physical sciences, Wiley, New York, 1962.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 56 (1991), 711-730
- MSC: Primary 33C45
- DOI: https://doi.org/10.1090/S0025-5718-1991-1068818-5
- MathSciNet review: 1068818