Symbolic and numerical computation on Bessel functions of complex argument and large magnitude

Journal of Computational and Applied Mathematics

Volumn 75, Issue 1, 1996, Pages 99-118

  Zhang, Jun ^a  

^a GEORGE WASHINGTON UNIVERSITY   (United States)

Author keywords

Automated method; Bessel functions; Chebyshev series; Symbolic Faber polynomials

Indexed keywords

CHEBYSHEV APPROXIMATION; COMPUTATIONAL METHODS; DIFFERENTIAL EQUATIONS; FUNCTION EVALUATION; POLYNOMIALS;

BESSEL FUNCTIONS; FABER POLYNOMIALS; HANKEL FUNCTIONS;

NUMERICAL METHODS;

EID: 0030287552     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(96)00063-5     Document Type: Article

References (37)

1. D.E. Amos, Algorithm 644 - A portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Soft. 12 (1986) 265-273.
2. J.A. Belward and J. Zhang, Chebyshev approximations for the Bessel function Yn(z) with complex argument, Department of Mathematics, the University of Queensland, Australia, preprint, 1992.
3. C.K. Chui, J. Stöckler and J.D. Ward, A Faber series approach to cardinal interpolation, Math. Comput. 58 (1992) 255-273.
4. C.W. Clenshaw, Chebyshev Series for Mathematical Functions, NPL Mathematical Tables, Vol. 5, (HMSO London, 1962).
5. W.J. Cody, Algorithm 715 - A portable Fortran package of special function routines and test drivers, ACM Trans. Math. Soft. 19 (1993) 22-32.
6. J.P. Coleman, The Lanczos tau-method, J. Inst. Math. Appl. 17 (1976) 85-97.
7. J.P. Coleman, Polynomial approximations in the complex plane, J. Comput. Appl. Math. 18 (1987) 193-211.
8. J.P. Coleman, Complex polynomial approximations by the Lanczos τ-method: Dawson's integral, J. Comput. Appl. Math. 20 (1987) 137-151.
9. n(z) in the complex plane, Math. Comput. 40 (1983) 343-366.
10. J.P. Coleman and R.A. Smith, Faber polynomials for circular sectors, Math. Comput. 49 (1987) 231-241.
11. J.P. Coleman and R.A. Smith, Supplement to Faber polynomials for circular sectors, Math. Comput. 49 (1987) S1-S4.
12. R.M. Corless, D.J. Jeffrey and H. Rasmussen, Numerical evaluation of airy functions with complex arguments, J. Comput. Phys. 99 (1992) 106-114.
13. J.H. Curtiss, Faber polynomials and the Faber series, Amer. Math. Monthly 78 (1971) 577-596.
14. M. Eiermann, On semiiterative methods generated by Faber polynomials, Numer. Math. 56 (1989) 139-156.
15. S.W. Ellacott, Computation of Faber series with application to numerical polynomial approximation in the complex plane, Math. Comput. 40 (1983) 575-587.
16. S.W. Ellacott, On the Faber transformation and efficient numerical rational approximation, SIAM J. Numer. Anal. 20 (1983) 989-1000.
17. S.W. Ellacott and E.B. Saff, Computing with the Faber transform, in: P.R. Graves-Morris, E. Saff and R.S. Varga, Eds., Rational Approximation and Interpolation, Lecture Notes in Math., Vol. 1105 (Springer, Berlin, 1984) 412-418.
18. S.W. Ellacott and E.B. Saff, On Clenshaw's method and a generalization to Faber series, Numer. Math. 52 (1988) 499-509.
19. G.H. Elliott, The construction of Chebyshev approximations in the complex plane, Ph.D. Thesis, University of London, 1978.
20. L. Fox, Chebyshev methods for ordinary differential equations, Computer J. 4 (1962) 318-331.
21. L. Fox and I.B. Parker, Chebyshev Polynomials in Numerical Analysis (Oxford University Press, London, 1968).
22. K. Gatermann, C. Hoffmann and G. Opfer, Supplement to explicit Faber polynomials on circular sectors, Math. Comput. 58 (1992) S1-S6.
23. K. Gatermann, C. Hoffmann and G. Opfer, Explicit Faber polynomials on circular sectors. Math. Comput. 58 (1992) 241-253.
24. C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J. Math, and Phys. 17 (1938) 123-199.
25. C. Lanczos, Applied Analysis (Pitman, London, 1957).
26. D.W. Lozier and F.W.J. Olver, Numerical evaluation of special functions, in: W. Gautschi, Ed., Mathematics of Computation, 1943-1993: A Half-Century of Computational Mathematics, Proc. of Symposia in Appl. Math., Vol. 48, (Amer. Math. Soc., Providence, RI., 1993) 79-125.
27. Y.L. Luke, Mathematical Functions and their Approximations (Academic Press, New York, 1969).
28. Y.L. Luke, Algorithms for the Computation of Mathematical Functions (Academic Press, New York, 1977).
29. W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (3rd ed, Springer, New York, 1966).
30. S. Namasivayam and E.L. Ortiz, A hierarchy of truncation error estimates for the numerical solution of a system of ordinary differential equations with techniques based on the Tau Method, in: K. Strehmel, Ed., Numerical Treatment of Differential Equations (Teubner, Leipzig, 1988) 113-121.
31. S. Namasivayam and E.L. Ortiz, Error analysis of the tau method: dependence of the approximation error on the choice of perturbation term, Comput. Math. Appl. 25 (1993) 89-104.
32. E.L. Ortiz, The tau method, SIAM J. Numer. Anal. 6 (1969) 480-491.
33. G. Stark and R.S. Varga, A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations, Numer. Math. 64 (1993) 213-240.
34. J.A.C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal. 31 (1994) 1497-1518.
35. 1(z), Comput. Math. Appl. 30 (1995) 15-20.
36. 1(z), Comput. Math. Appl. 30 (1996) 63-70.
37. 0(z), Computer Math. Applic. 30 (1995) 5-14.