Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A fast algorithm for rational interpolation via orthogonal polynomials


Authors: Ömer Eğecioğlu and Çetin K. Koç
Journal: Math. Comp. 53 (1989), 249-264
MSC: Primary 65D05; Secondary 33A65, 41A05
DOI: https://doi.org/10.1090/S0025-5718-1989-0972369-4
MathSciNet review: 972369
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new algorithm for rational interpolation is proposed. Given the data set, the algorithm generates a set of orthogonal polynomials by the classical three-term recurrence relation and then uses Newton interpolation to find the numerator and the denominator polynomials of the rational interpolating function. The number of arithmetic operations of the algorithm to find a particular rational interpolant is $ O({N^2})$, where $ N + 1$ is the number of data points. A variant of this algorithm that avoids Newton interpolation can be used to construct all rational interpolants using only $ O({N^2})$ arithmetic operations.


References [Enhancements On Off] (What's this?)

  • [1] P. J. Davis, Interpolation and Approximation, Dover, New York, 1975. MR 0380189 (52:1089)
  • [2] Ö. Egecioglu & Ç. K. Koç, An $ O(N\log N)$ Parallel Algorithm for Rational Interpolation, Technical Report No. TRCS88-2, Department of Computer Science, University of California, Santa Barbara, August 1988.
  • [3] G. E. Forsythe, "Generation and use of orthogonal polynomials for data fitting with a digital computer," J. Soc. Indust. Appl. Math., v. 5, 1957, pp. 74-78. MR 0092208 (19:1079e)
  • [4] P. R. Graves-Morris & T. R. Hopkins, "Reliable rational interpolation," Numer. Math., v. 36, 1981, pp. 111-128. MR 611488 (82f:65008)
  • [5] F. B. Hildebrand, An Introduction to Numerical Analysis, McGraw-Hill, New York, 1956. MR 0075670 (17:788d)
  • [6] T. Muir, The Theory of Determinants, vol. 2, Dover, New York, 1960. MR 0114826 (22:5644)
  • [7] A. Ralston & P. Rabinowitz, A First Course in Numerical Analysis, McGraw-Hill, New York, 1978. MR 0494814 (58:13599)
  • [8] J. Rissanen, "Solution of linear equations with Hankel and Toeplitz matrices," Numer. Math., v. 22, 1974, pp. 361-366. MR 0351057 (50:3548)
  • [9] T. J. Rivlin, An Introduction to the Approximation of Functions, Dover, New York, 1969. MR 0249885 (40:3126)
  • [10] C. Schneider & W. Werner, "Some new aspects of rational interpolation," Math. Comp., v. 47, 1986, pp. 285-299. MR 842136 (87k:65012)
  • [11] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, R. I., 1959.
  • [12] W. F. Trench, "An algorithm for the inversion of finite Hankel matrices," J. Soc. Indust. Appl. Math., v. 13, 1965, pp. 1102-1107. MR 0189232 (32:6659)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D05, 33A65, 41A05

Retrieve articles in all journals with MSC: 65D05, 33A65, 41A05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1989-0972369-4
Keywords: Rational interpolation, orthogonal polynomials, Hankel matrices
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society