A fast algorithm for the calculation of the roots of special functions

SIAM Journal on Scientific Computing

Volumn 29, Issue 4, 2007, Pages 1420-1438

  Glaser, Andreas ^a   Xiangtao, Liu ^a   Rokhlin, Vladimir ^a  

^a YALE UNIVERSITY   (United States)

Author keywords

Gaussian quadratures; Roots; Special functions; Sturm Liouville

Indexed keywords

ORTHOGONAL FUNCTIONS; POLYNOMIALS; WAVE FUNCTIONS;

CLASSICAL ORTHOGONAL POLYNOMIALS; FAST ALGORITHMS; GAUSSIAN QUADRATURES; HERMITE; LEGENDRE; ROOT; SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS; SPECIAL FUNCTIONS; STURM-LIOUVILLE; TAYLOR SERIES METHODS;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 40249111875     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/06067016X     Document Type: Article

References (15)

1. M. ABRAMOVITZ AND I. A. STEGUN, EDS., Handbook of Mathematical Functions, Dover, Mineola, NY, 1972.
2. J. S. BALL, Automatic computation of zeros of Bessel functions and other special functions, SIAM J. Sci. Comput., 21 (1999), pp. 1458-1464.
3. G. BIRKHOFF AND G.-C. ROTA, Ordinary Differential Equations, Ginn, Boston, 1962, Chap. 10.5.
4. G. DAHLQUIST AND A. BJÖRK, Numerical Methods, Dover, Mineola, NY, 2003.
5. W. GAUTSCHI, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, London, 2004.
6. G. H. GOLUB AND J. H. WELSCH, Calculation of Gauss quadrature rules, Math. Comp., 23 (1969), pp. 221-230.
7. M. GU AND S. EISENSTAT, A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 172-191.
8. I. RYZHIK AND I. GRADSTEIN, Table of Integrals, Series, and Products, Academic Press, NY, 1980.
9. Y. IKEBE, Y. KIKUCHI, AND I. FUJISHIRO, Computing zeros and orders of bessel functions, J. Comput. Appl. Math., 38 (1991), pp. 169-184.
10. V. ROKHLIN AND H. XIAO, Approximation formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit, Applied and Computational Harmonic Analysis, 22 (2007), pp. 105-123.
11. P. N. SWARZTRAUBER, On computing the points and weights for Gauss-Legendre quadrature, SIAM J. Sci. Comput., 24 (2002), pp. 945-954.
12. G. SZEGÖ, Orthogonal Polynomials, 4th ed., AMS, Providence, RI, 1978.
13. M. TYGERT, Recurrence Relations and Fast Algorithms, arXiv:cs/0609081, 2006. Technical report YALEU/DCS/TR-1343, Yale University, New Haven, CT, 2006.
14. H. XIAO, V. ROKHLIN, AND N. YARVIN, Prolate spheroidal wave functions, quadrature and interpolation, Inverse Problems, 17 (2001), pp. 805-838.
15. E. YAKIMIW, Accurate.computation of weights in classical Gauss-Christoffel quadrature rules, J. Comput. Phy., 129 (1996), pp. 406-430.