The zeros of regular Coulomb wave functions and of their derivatives
Author:
Yasuhiko Ikebe
Journal:
Math. Comp. 29 (1975), 878887
MSC:
Primary 65D20
MathSciNet review:
0378361
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A simple and efficient numerical method for computing the zeros of regular Coulomb wave functions and of their derivatives is presented. The method is based on the characterization of the zeros of the functions and of their derivatives in terms of eigenvalues of certain compact matrix operators. A similar approach has been reported for the computation of the zeros of Bessel functions and of their derivatives [9], [14].
 [1]
Handbook of mathematical functions, with formulas, graphs, and
mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun,
Dover Publications Inc., New York, 1966. MR 0208797
(34 #8606)
 [2]
W.
J. Cody and K.
E. Hillstrom, Chebyshev approximations for the
Coulomb phase shift, Math. Comp. 24 (1970), 671–677. MR 0273785
(42 #8661), http://dx.doi.org/10.1090/S00255718197002737854
 [3]
W. J. CODY & KATHLEEN A. PACIOREK, "Remark on algorithm 292regular Coulomb wave functions," Comm. ACM, v. 13, 1970, p. 573.
 [4]
A.
R. Curtis, Coulomb wave functions, Prepared under the
direction of The Coulomb Wave Functions Panel of the Mathematical Tables
Committee. Royal Society Mathematical Tables, Vol. 11, Published for the
Royal Society at the Cambridge University Press, New York, 1964. MR 0167643
(29 #4915)
 [5]
W. GAUTSCHI, "Algorithm 292regular Coulomb wave functions," Comm. ACM, v. 9, 1966, pp. 793795.
 [6]
W. GAUTSCHI, "Remark on algorithm 292regular Coulomb wave functions," Comm. ACM, v. 12, 1969, p. 280.
 [7]
Walter
Gautschi, Computational aspects of threeterm recurrence
relations, SIAM Rev. 9 (1967), 24–82. MR 0213062
(35 #3927)
 [8]
Walter
Gautschi, An application of minimal solutions of threeterm
recurrences to Coulomb wave functions, Aequationes Math.
2 (1969), 171–176. MR 0246512
(39 #7816)
 [9]
J.
Grad and E.
Zakrajšek, Method for evaluation of zeros of Bessel
functions, J. Inst. Math. Appl. 11 (1973),
57–72. MR
0331725 (48 #10057)
 [10]
J. H. GUNN, "Algorithm 300Coulomb wave functions," Comm. ACM, v. 10, 1967, pp. 244245.
 [11]
K. S. KÖLBIG, "Certification of algorithm 300Coulomb wave functions," Comm. ACM, v. 12, 1969, pp. 279280.
 [12]
K. S. KÖLBIG, "Certification algorithm 292regular Coulomb wave functions," Comm. ACM, v. 12, 1969, pp. 278279.
 [13]
K. S. KÖLBIG, "Remarks on the computation of Coulomb wave functions," Computer Physics Comm., v. 4, 1972, pp. 214220.
 [14]
Y. IKEBE, The Zeros of Bessel Functions and of Their Derivatives, Technical Report CNA81, Center for Numerical Analysis, The University of Texas, Austin, Texas, February 1974.
 [15]
M. A. KRASNOSEL'SKIĬ, G. M. VAĬNIKKO, P. P. ZABREĬKO, Ja. B. RUTICKIĬ & V. Ja. STECENKO, Approximate Solution of Operator Equations, "Nauka", Moscow, 1969; English transl., WoltersNoordhoff, Groningen, 1972. MR 41 #4271.
 [16]
B. T. SMITH, J. M. BOYLE, B. S. GARBOW, Y. IKEBE, C. KLEMA & C. B. MOLER, Matrix Eigensystem RoutinesEISPACK Guide, SpringerVerlag, New York, 1974.
 [17]
A.
J. Strecok and J.
A. Gregory, High precision evaluation of the
irregular Coulomb wave functions, Math.
Comp. 26 (1972),
955–961; addendum, ibid. 26 (1972), no. 120, loose microfiche suppl.
B1–B10. MR
0314239 (47 #2791), http://dx.doi.org/10.1090/S00255718197203142398
 [18]
J.
H. Wilkinson, The algebraic eigenvalue problem, Clarendon
Press, Oxford, 1965. MR 0184422
(32 #1894)
 [1]
 M. ABRAMOWITZ & I. A. STEGUN (Editors), Handbook of Mathematical Functions, With Formulas, Graphs and Mathematical Tables, reprint, Dover, New York, 1966. MR 34 #8606. MR 0208797 (34:8606)
 [2]
 W. J. CODY & K. HILLSTROM, "Chebyshev approximations for the Coulomb phase shift," Math. Comp., v. 24, 1970, pp. 671677. MR 42 #8661. MR 0273785 (42:8661)
 [3]
 W. J. CODY & KATHLEEN A. PACIOREK, "Remark on algorithm 292regular Coulomb wave functions," Comm. ACM, v. 13, 1970, p. 573.
 [4]
 A. R. CURTIS, Coulomb Wave Functions, Royal Soc. Math. Tables, vol. 11, Cambridge Univ. Press, New York, 1964. MR 29 #4915. MR 0167643 (29:4915)
 [5]
 W. GAUTSCHI, "Algorithm 292regular Coulomb wave functions," Comm. ACM, v. 9, 1966, pp. 793795.
 [6]
 W. GAUTSCHI, "Remark on algorithm 292regular Coulomb wave functions," Comm. ACM, v. 12, 1969, p. 280.
 [7]
 W. GAUTSCHI, "Computational aspects of threeterm recurrence relations," SIAM Rev., v. 9, 1967, pp. 2482. MR 35 #3927. MR 0213062 (35:3927)
 [8]
 W. GAUTSCHI, "An application of minimal solutions of threeterm recurrences to Coulomb wave functions," Aequationes Math., v. 2, 1969, pp. 171176. MR 39 #7816. MR 0246512 (39:7816)
 [9]
 J. GRAD & E. ZAKRAJSĚK, "Method for evaluation of zeros of Bessel functions," J. Inst. Math. Appl., v. 11, 1973, pp. 5772. MR 0331725 (48:10057)
 [10]
 J. H. GUNN, "Algorithm 300Coulomb wave functions," Comm. ACM, v. 10, 1967, pp. 244245.
 [11]
 K. S. KÖLBIG, "Certification of algorithm 300Coulomb wave functions," Comm. ACM, v. 12, 1969, pp. 279280.
 [12]
 K. S. KÖLBIG, "Certification algorithm 292regular Coulomb wave functions," Comm. ACM, v. 12, 1969, pp. 278279.
 [13]
 K. S. KÖLBIG, "Remarks on the computation of Coulomb wave functions," Computer Physics Comm., v. 4, 1972, pp. 214220.
 [14]
 Y. IKEBE, The Zeros of Bessel Functions and of Their Derivatives, Technical Report CNA81, Center for Numerical Analysis, The University of Texas, Austin, Texas, February 1974.
 [15]
 M. A. KRASNOSEL'SKIĬ, G. M. VAĬNIKKO, P. P. ZABREĬKO, Ja. B. RUTICKIĬ & V. Ja. STECENKO, Approximate Solution of Operator Equations, "Nauka", Moscow, 1969; English transl., WoltersNoordhoff, Groningen, 1972. MR 41 #4271.
 [16]
 B. T. SMITH, J. M. BOYLE, B. S. GARBOW, Y. IKEBE, C. KLEMA & C. B. MOLER, Matrix Eigensystem RoutinesEISPACK Guide, SpringerVerlag, New York, 1974.
 [17]
 A. J. STRECOK & J. A. GREGORY, "High precision evaluation of the irregular Coulomb wave functions," Math. Comp., v. 26, 1972, pp. 955961. MR 47 #2791. MR 0314239 (47:2791)
 [18]
 J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. MR 0184422 (32:1894)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65D20
Retrieve articles in all journals
with MSC:
65D20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197503783615
PII:
S 00255718(1975)03783615
Article copyright:
© Copyright 1975 American Mathematical Society
