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The computation of special functions
U.S.S.R. Comput. Maths. Math. Phys. Vol. 20, No. 5, pp. 3-12 Printed in Great Britain
0041-5553/80/050003-10$07.50/O 01981. Pergamon Press Ltd.
THECOMPUTATIONOFSPECIALFUNCTIONS* V. A. DITKIN, K. A. KARPOV and M. K. KERIMOV Moscow (Received
13 May 1980)
METHODS for computing special functions are reviewed, with the accent on methods used when tabulating the functions. Recently published tables are also surveyed.
1. INTRODUCTION Throughout the history of natural science, special functions have been a powerful instrument in the solution of a wide variety of important problems. The reduction of any given applied problem to the evaluation of special functions has always been, and still is, looked on as indicating a strong penetration into the essence of the problem. Almost all the familiar special functions have arisen from a wide diversity of applied problems, so that the study of their properties, and their tabulation, have engaged, not only mathematicians, but also physicists, astronomers, engineers and other specialists. This is still essentially true today. The problems involved in tabulating special functions are as old as the functions themselves, and have usually come to the fore as soon as the function has made its appearance. These problems have occupied many celebrated scholars, including those by whose names the functions are known. Notably scholars have devoted a life-time to the tabulation of certain special functions. Such tables have been, and still are, vital bricks in the edifice of science. Once compiled, the tables can retain their value long after the specific problems to which they owe their origin have been forgotten. Three periods can be distinguished in the tabulation of special functions: 1) laborious manual computation, by individual enthusiasts; 2) mechanical and electromechanical computations, by devices which vastly increased the scope for performing difficult and laborious calculations; 3) the rapid advances in electronic computers, whereby the scope for calculations became virtually unlimited. In the first period, a few tables of the most important functions were compiled, of limited accuracy, and for a limited range of variation of the argument. In the second period, the number of tabulated functions increased, along with the accuracy and the range of variation of the argument. The third period, which saw the appearance of the high-speed electronic computer, has been marked by the setting up of specialist groups dedicated to large-scale projects for the production of mathematical tables.
lZh. vjkhisl. Mat. mat. Fiz., 20, 5, 1093-1104,
1980. 3
4
V. A. Ditkin, K. A. Karpov and M. K. Kerimov
Particular mention must be made here of groups from the Royal Society (UK), the American National Bureau of Standards, the Computing Laboratories of Harvard University, and the Computing Centre of the Academy of Sciences of the USSR. The work of these groups (accompanied by many theoretical studies into computing methods) has resulted in the compilation and publication of entire libraries of mathematical tables, occupying many tens of volumes. It was in connection with this work that many of the now familiar methods for computing special functions were developed. We need only mention the widely known numerical method of recurrence relations, which was developed while tabulating Bessel functions. For instance, the tabulation of the zeros of Bessel functions, which occupied a section of the Royal Society for 18 years, led to completely new methods being devised for computing the zeros of special functions. In view of the amount of work and facilities needed to tabulate special functions, any new enterprise in this field clearly must be preceded by a very careful study of the tables already available. The number of tables published separately, or contained in publications devoted to other topics, is enormous, and there has long been a demand for the compilation of special indices of mathematical tables. The fllst attempt was headed by von Mises in 1928 [ 11, though only the first part of the work was published. The next was a very complete guide to tables of Bessel functions, compiled by Bateman and Archibald [2]. The fust index covering tables of all elementary and special functions was due to Fletcher et al., and was originally published in 1946. The second, greatly enlarged edition, appeared in 2 volumes [3] in England and the USA in 1962. The German index appeared in 195.5 [4], and the Russian, in 1956 [5]. A new, much more complete index, compiled by M. K. Kerimov, is currently being prepared for press at the Computing Centre of the Academy of Sciences of the USSR. It will include detailed information about virtually all the tables published anywhere in the world. and about algorithms and programs (whether published separately, or appearing in journal articles, books, guides etc.). The present article, timed to coincide with the 25th anniversary of the Computing Centre of the Academy of Sciences, surveys some of the methods used for computing special functions. Special attention is paid to the Soviet project for tabulating special functions. We also touch incidentally on some other recent work in the USSR and abroad. Naturally, it has not been possible to dwell in detail in a brief article on all the results obtained in computing special functions. Only the references to tables compiled at the Computing Centre are more or less complete. After the introduction (Section l), we describe computing methods in Section 2, survey algorithms and programs in Section 3, survey the tables of special functions in Section 4, and finally, refer briefly in Section 5 to work on nomography at the Computing Centre. Due to lack of space, it has not been possible to include full bibliographical details in the “References” to alI publications mentioned in the text.
The computation of spectd functions
2. Methods of computing special functions Many methods have been devised for computing series representation representations,
of the functions,
into different
etc. The latter methods are most commonly
for special functions: Different
polynomial.
rational: Chebyshev,
special polynomials
used when obtaining
of the arguments.
are used for computing
For instance, Bessel functions;
are used, and for large values? asymptotic methods or generalizations Asymptotic
computations
in different
for small values of the argument,
series
for moderate values, recurrence
formulae or expansions,
special functions
relations
In short, several different
of them may need to be used to compute
methods for computing
in recent years [6-l
(e.g. Chebyshev approximations
and Pade approximations.
methods are, by their nature. suitable for function
ranges of variation expansions
some make use of formulae to the integral
while others employ recurrence relations, etc. Special methods may be used,
such as the Lanczos tau method. or expansions polynomials),
special functions;
others apply different quadrature
a given function.
have become especially popular
11.
A lot of work has been done on obtaining
approximations
for special functions,
since
approximations are particularly suitable for machine computations. In this connection we must first mention Luke’s publications ([ 121 , and specially [ 131). m which are to be found excellent approximations for almost all the special functions; the coefficients of these approximations are evaluated to high accuracy. With the aid of such approximations, computed
the function
itself can be
to high accuracy.
The numerous must be mentioned
approximations
the approximations.
by Nemeth for a large number of special functions contain tables of the coefficients
of
evaluated to high accuracy.
Many approximations, approximation
obtained
(see e.g. [ 141); all these publications
with tables of coefficients.
[ 15- 171 . and in many publications
Indices of approximations
are to be found in handbooks
on
by Cody and his associates (e.g. [ 18. 191 ).
have also begun to appear (e.g. [20] ).
Temme [2 1, 221, Kolbig [23] . Gautschi (241, and others, have been extremely active in the computation of special functions. Methods for the complete automation of the computer evaluation
of special functions
Bessel functions for computing functions,
have also been published (see e.g., [25, 261). The evaluation
in the real and complex domains is dealt with in [27, 281. Numerous parabolic cylinder functions
in [30,31].
are surveyed in (291. and for computing
A very detailed survey of methods of computing
of
methods Coulomb
special functions
is to
be found in [32]. The Computing Centre of the Academy of Sciences of the USSR has produced a variety of theoretical and computational work on special functions, including one [33] on checking the Riemann hypothesis about the zeros of the zeta function, on the properties of Lommel functions of two variables [34], on asymptotic formulae for Legendre functions [35-371. on asymptotic expansions for cylindrical functions of two variables [38-391, on the zeros of these functions [40], and on the evaluation of the Weierstrass elliptic function [41]. A lot of work has been done on applications of the operational calculus and integral transforms to the theory of special functions, see e.g. [42-441.
6
V. A. Ditkin, K. A. Karpov and M. K. Kerimov
3. Algorithms and programs When computers began to be used for evaluating special functions, programs written in the appropriate machine codes started to appear. Special algorithmic languages were later developed, and programs were written in these languages. Large stocks of such programs have now been built up, both here and abroad, and are always being supplemented. The stock of algorithms and programs of the Academy of Sciences of the USSR, is kept at the Computing Centre. Similar stocks are kept at many Academies of Sciences of other Soviet republics. For the USSR Academy stock, an automatic information search system operates for algorithms and programs, whereby required information can be found rapidly. The systematically filled data bank of the search system at present contains the description of over 2500@ works with algorithms, programs, and programming information. On the basis of the existing programs, so-called libraries and packages of programs are at present being developed, designed for the automatic solution of large classes of problems. Abroad, such packages have in fact appeared for computing special functions (see e.g. [46] ). In the USSR, most libraries of programs (e.g. at the Research Computing Centre of hioscow University, or at the Mathematical Institute of the Belorussia Academy of Sciences) include programs for computing special functions. Some of the programs from these libraries have already been published (see [46,47]). In addition, the following programs may be mentioned: for the sine, cosine, and Fresnel integrals [48] ; for the integral exponential function [49, SO], for the dilogarithm [5 11, the incomplete gamma function [52-541, the gamma function in the complex domain [55], the logarithm of the gamma function and the digamma function in the complex domain [56], the probability integral in the complex domain [57], Bessel functions [58-621, spherical Bessel functions [63], Bessel functions in the complex domain [64], the zeros of Bessel functions and their derivatives [65], Thomson functions [66], elliptic integrals and functions [67], Coulomb functions [68] , spheroidal functions [69-7 11, and Airy functions [72-741.
4. Tables The Soviet project for computing mathematical tables was devised and started at the Institute of Precision Mechanics and Computing Techniques (IEM and CT), and then moved to the Computing Centre (CC) in 1955. The project was originally headed by L. A. Lyusternik, then from 1953, by V. A. Ditkin. Of course, some work on the tabulation of special functions had been carried out earlier in the USSR. First we must mention the fundamental tables of modified Bessel functions [75], started originally back in 1930; groups from three institutes (including the Institute of Mechanics and the Mathematical Institute of the Academy of Sciences of the USSR) worked on them later. Similar work was done even earlier in the Ukraine, where Bessel functions of complex argument [76] were tabulated for the first time. The Soviet project consists of three series of tables: 1) “Mathematical tables” (edited by V. A. Ditkin); 2) “special tables”; 3) a series of translated foreign tables “Library of Mathematical Tables, ” issued from 1958 and edited by K. A. Karpov. All the tables of these series are as a rule fundamental (with an accuracy of at least 5-6 places). When selecting functions for tabulation, their practical and scientific importance was invariably the guiding factor. The wishes of research and production groups were always taken into account.
The computation of special finctions
We can only mention
here a few of the tables of special functions
appearing in these
series. and touch briefly on some other tables published in recent years in the USSR and abroad. The series “Mathematical
(L.A. Lyusternik,
and year of publication function
Tables”.
The series in fact started with tables of Bessel1 functions
1. Ya. Akushskii. V. A. Ditkin.
1949), prepared at the IEM and CT (the authors
are shown in parentheses).
Later, at the same Institute,
tables of the
In F(z) in the complex domain were prepared and published (A.A. Abramov,
The first tables computed
by the new BESM-I computer were of Fresnel integrals (1953).
though after these, several tables were computed
on keyboard
since work on them had started before production for using the electronic
1953).
and analytic-computing
and analytic-computing
machines,
of the BESM -1. A method was developed
machines. such that the published tables were
all free from errors. The following tables were subsequently integral exponential
function
published:
W(z) =e-’
(1954): the function
domain (K. A. Karpov. 1954); Jacobi polynomials on the initiative functions
of A. A. Dorodnitsyn:
for the asymptotic
1955): cylindrical
functions
Laplace’s equation
Jer2 da:
(L. N. Karmazini,
of Airy functions
solution of second-order
in the complex
1954); tables, developed
and special degenerate
differential
equations
of two variables (E. N. Dekanosidze,
hypergeometric
(A. D. Smirnov,
1956); tables for solving
in elliptic domains (A. 1. Vzorova. 1957); the integral logarithm
(K. A. Karpov. S. N. Razumovskii,
1956): Bessel functions
integrals (E. A. Chistova, 1958); Bessel functions (L. N. Karmazina,
integral sine and cosine (1954); the
of a real argument and their
of an imaginary
E. A. Chistova. 1958); integro-exponential
argument
function
and their integrals
(V. I. Pagurova, 1959);
Weber functions,
in three volumes (K. A. Karpov. 1. E. Kireeva. E. A. Chistova. 1959, 1964,
1968); Thomson
functions
functions generalized
and their first derivatives (L. I. Nosova (Osipova), in tvvo volumes (M. I. Zhurina.
P_:;+r(r). Airy, functions
1960); Legrendre
L. N. Karmazina.
for the asy,mptotic solution of differential
equations
1960, 1967): E (py’) ‘+ (q+
ET) y=! (L. I. Nosova, (Osipova), S. A. Tumarkin. 1961): cylindrical functions of two imaginary variables (L. S. Bark. P. I. Kuznetsov? 1962); tables and formulae for spherical functions PIIIViL{, (5) (M. 1. Zhurina, L. N. Karmazina, 1962); elliptic integrals of all three kinds, in two volumes (V. M. Belyakov et al., 1962, 1963): Legendre functions P-tl;:+ir (5) (M. I.
Zhurina,
L. N. Karmazina.
Rayleigh-Rice
1963); the incomplete
distributions
gamma function
(L. S. Bark et al., 1964); degenerate
1st and 2nd kinds, in two volumes (M. 1. Zhurina.
(V. I. Pagurova, 1963); the
hypergeometric
function
of the
L. N. Osipova, 1964. 1972); modified
Whittaker functions (B. 1. Korobochkin. Yu. A. Filippov, 1965); the logarithmic derivative of the gamma function and its derivatives in the complex domain (1965); Chebyshev polynomials (L. P. Grabar’, 1965): incomplete cy,lindrical functions (M. M. Agrest et al., 1966); roots and weighting factors of generalized Laguerre polynomials (E. N. Dekanosidze, 1966); modified Bessel functions
with imaginary index Ki,(x)
elliptic Weierstrass function equation
(T. D. Kuznetsova.
(Ts. D. Lomtatsi. Yu. N. Smirnov.
(M. 1. Zhurina,
L. N. Karmazina,
1967), characteristic
exponents
1967); the of Mathieu’s
1969); eigenvalues of Mathieu’s equations
(L. S. Bark er aZ., 1970); roots and weighting coefficients of Jacobi polynomials (E. N. Glonti, 1971); Legendre functions of imaginary argument (L. N. Karmazina, 1972); and modified Bessel functions
K,hiiB (5)
(Yu. M. Rapoport.
1979).
8
1’. A. Ditkin, K. A. Karpov and M. K. Kerimov
“Miscellaneous Tables” series. We shall confine ourselves here to mentioning the important tables of Coulomb wave functions (A. V. Lut’yanov et d, 1961), the Racah coefficients (A. F. Nikiforov et al., 1962), the tables for computing toroidal shells (L. N. Osipova. S. A. Tumarkin, 1963), the integrals of quantum chemistry (Yu. A. Kruglyak, D. R. Uitmen, 1965). and the tables of 9j-coefficients (Ya. 1. Vizbaraite et al.,1968). The entire series comprises 13 volumes. “Libran> of Mathematical Tables’lseries. This great series comprises 48 volumes (some of them are 2nd editions). The tables selected were as a rule subjected to careful and continuous checking by a specially developed method, with the result that large numbers of errors were detected and corrected; the corrections were sometimes so important that the translation (e.g. of the well known Hayashi tables) became virtually an original edition.
The most important of the general-purpose foreign tables, well known to specialists, were included in the series. Special functions are covered by 29 volumes. Since we cannot quote all the titles here, we shall just mention those which include considerable theoretical matter in addition to the tables as such: degenerate hypergeometric functions (L. J. Slater, 1968), Weber (parabolic cylinder) functions (J. C. P. Miller, 1968), Coulomb wave functions (A. R. Curtis, I969), and the incomplete beta function (K. Pearson, 1974). While few tables have been published in the USSR outside the walls of the Computing Centre, some of them are of great importance, notably, the tables of elliptic integrals (N. S. SamoiIova-Yakhontova, 1935): the collection of tables (B. I. Segal, K. A. Semendyaev, 1948; 3rd ed., 1962); the normalized associated Legendre polynomials (S. L. Belousov, 1956), Laguerre polynomials and functions (V. S. Aizenshtat et al., 1965); the Bessel functions J,(X) of integral indices n from 0 to 120 (V. N. Faddeeva, M. K. Gavurin, 1950); the function connected with the probability integral in the complex domain (V. N. Faddeeva, N. M. Terent’ev, 1954); the Airy functions (V. A. Fok, 1946. G. D. Yakovleva, 1969); spheroidal wave functions and their derivatives. in two volumes (S. P. Erashevska1.a er al., 1973, 1976); the fractional-exponential function of negative parameters and its integral (Vu. N. Rabotnov et al., 1969); the generarized Riemann zeta function and its derivative of complex argument, in two volumes (N. N. Voitovich et al., 1970); the probability integral (N. V. Smirnov, 1960); the two dimensional normal distribution (N. V. Smirnov, L. N. Bol’shev, 1962); and extensive tables of mathematical statistics (L. N. Bol’shev, N. V. Smirnov, 1965; 2nd ed., 1968). Some fundamental foreign tables published in recent years should also be mentioned: special functions [77] (2 vols.)! the incomplete gamma function [78, 791 , Kelvin functions (801, Pommel functions of two variables [81], Mathieu radial functions [82] (2 vols.), spheroidal wave functions [83] (6 vols.), and Jacobi elliptic functions [84].
5. Nomograms
In parallel with the work on compiling tables at the Computer Centre of the Academy of Sciences, studies were directed by G. S. Khovanskii on the development of methods of nomography and the production of working nomograms, i.e. “graphical tables,” which are an indispensable aid to many engineers. Several textbooks on nomography were prepared and published, along with atlases of working nomograms. while other work on nomography in the USSR was directed
9
The computation of special functions
and coordinated. As a result, 12 collections have so far appeared (with two more in the press). Experience suggests that, in spite of the wide introduction of computing techniques, nomograms, like tables, will not lose their value and will long remain a useful instrument for research-workers and engineers. Translated
by D. E. Brown.
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OLVER, F. W. J., Unsolved problems in the asymptotic estimation of special functions, in: 7”heov and applications of special functions, Acad. Press, New York, 99- 142, 1975.
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OLVER, F. W. J., Asymptotics and special functions,
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DINGLE, R. B., Asymptotic expansions: their derivation and interpretation, Acad. Press, New York, 1973.
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11
obespechenie ES EVM), No. 3, In-t matem
48. BULIRSCH. B., Numerical calculation of the sine. cosine and Fresnel integrals, Numer. Marh., 9, 380-385, 1967. 49. HERSHEY, A. V., Computing programs for the complex exponential integrals, U.S. Naval Proving Ground, Dahlgren, Virginia. NAVORD Rep. No. 5909, NPG Rept. No. 1646, June, 1959. 50. GAUTSCHI, W., Algorithm 471, Exponential integrals, Communs. AChf, 16, 761-763,
1973.
51. GINSBERG. E. S., and ZABOROB’SKI. D.. Algorithm 490, The dilogarithm function of a real argument, Communs. AChf, 18, 200-202, 1975. 52. BHATTACHARJEE, G. P., Algorithm AS32, The incomplete gamma integral, Appl. Starist.. 19,285-287, 1970. 53. GAUTSCHI. W., Algorithm 542. incomplete gamma functions, ACM Trans. Math. Sofrware, 5,482-489, 1979. 54. FULLERTON, W., Algorithm 435. Slodified incomplete gamma function, Communs. ACM, 15, 993-995, 1972. 55. NG, E. W., A comparison of computational methods and algorithms for the complex gamma function, ACM Trans. Math. Sofrware, 1, 56-70. 1975. 56. KGLBIG, K. S.. Program for computing the logarithm of the gamma function, and the diagamma function, for complex argument, Comput. PhJ,s. Communs., 4, 221-226, 1972. 57. ANOLIK, M. V., Standard program for computing the probability integral of complex argument, in: Computing methods (hletody vychislenii), No. 6, Izd-vo LGU, Leningrad, 1970. 58. PLESHAKOV, G. li., Programs for computing Bessel functions of fractional index on the basis of power and asymptotic expansions, in: Computing methods and programming (Vychisl. metody i programmirovanie), hio. 3. Izd-vo Saratovsk. un-ta? Saratov, 1970. 59. CHISTOVA, E. A., Standard programs for the Strela.3 (Standartnye programmy dlya mashiny “Strela-3”), VTs Akad. Nauk SSSR. Moscow. 1961. 60. TROFIMOV, E. N., and STELINA. N. Yu., Srandard program for computing Bessel functions (in IP-3 system), Nl VTs MGU. Moscow. 1970. 61. ALEKSENKO, A. M., and SPOSOBNY’I. V. V., Computation of the modified Bessel function of the first kind, in: Machines for engineering computations (Mashiny dlya inzh. raschetov). Xo. 2. Kiev, 143- 146, 1970. 62. AMOS, E. D., DANIEL, S. L., and WESTON, M. K., CDC 6600 subroutines IBESS and JBESS for Bessel functionsI, and J,(x),x 2 0, Y > 0, ACM Trans. Math. Software, 3, 76-92, 1977. 63. ZELENKA, J., Algorithms for evaluation of spherical Bessel functions..rtplikace
Mat., 14, 175-177,
1969.
64. ARDILL, R. W. B.? and MORIARTY, K. J. M.. The Bessel functions JO and J1 of complex argument, Comput. PhJr. Communs.. 13, 17-24, 1977. 65. TEMME, N. M., An algorithm with ALGOL 60 program for computation of the zeros of ordinary Bessel functions and those of their derivatives, J. Comput. Phys., 32, 270-279, 1979. 66. JACKISCH, P., Algolprogramme zur Berechnung der Thomson-Funktionen, nmenau, 15, 181-188, 1969.
~‘ISS.Z. Tech. Hochschulc
67. BULIRSCH, R., Numerical calculation of elliptic integrals and elliptic functions, Numer. Math., 7, 78-90, 1965;11,353-354: 111,13,305-315, 1969. 68. BARDIN, C. et al., Coulomb functions in entire (7. p)-plane, Comput. Ph_vs.Communs., 3,73-87,
1972.
69. KING, B. J., BAIER, R. V.. and HANISH, S., A. FORTRAN computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives, Naval Res. lab., NRL Repr. No. 7012, Washington, D. C., 1970. 70. KING, B. J., and VAN BUREN, A. L., A. FORTRAN computer program for calculating the prolate and oblate angle functions of the first kind and their fist and second derivatives, Naval Res. Lab., NRL Repr. h’o. 7161, Washington. D. C.. 1970.
12
V. A. Ditkin, K. A. Karpov and M. K. Kerimol
71. VAN BUREN, A. L., BAIER, R. V., and HANISH, S., A FORTRAN computer program for calculating the
oblate spheroidal radia functions of the first and second kind and their fast derivatives, Naval Rcs. Lab., .&‘RLRepr. h’o. 6959, Washington, D. C., 1970. 72. DAGMAN, E. I., Program for computing Airy functions, in: Machines for engineering computations (Mashiny dlya inzh. raschetov), No. 6, 122-129, Kiev, 1972. 73. GUSHCHINA, N. A., Program for computing Airy functions Ai (z), Bi (2) and their derivatives for real values of-the argument, FTI Akad. Nauk SSSR, Leningrad, 1974. 74. PRINCE, P. J., Algorithm 498. Airy functions using Cbebyshev series approximations, Software, 1, 372-379, 1975.
ACM Trans. Math.
75. VINOGRADOV, I. M., and CHETAEV, N. G., Tables of Bessel functions of imaginary argument (Tablitsy znachenii funktsii Besselya ot mnimogo argumenta), Izd-vo Akad. Nauk SSSR, Moscow, 1950. 76. DINNIK, A. N., Tables of Bessel functions of 0-th and I-st order or complex argument, Zh. Russk. fir-khim ob-va, Fir. otd., 55, No. l-3, 121-127, 1923. 77. UHDE, K., Spezielle Funktionen der mathematischen Physik, I, II, Bibliogr. Inst., HochschultaxhenbucherVerlag, Mannheirn: 1964. 78. HARTER, L. H., New tables of the incomplete gamma function ratio and of percentage points of the chi. square and beta distributions, U.S. Government Printing Off., Washington D.C., 1964. 79. KHAMIS, S. H., and RUDERT, W., Tables of the incomplete gamma function ratio, Justus von Liebig Verlag, Darmstadt, 1965. 80. YOUNG, A., and KIRK, A., Kelvin functions, Cambridge U.P., Cambridge, 1964. 81. HEBERMEHL, G. (Minkwitz), and SCHULZ, G., Tabellen der Lommelschen Funktionen CJl (W, z) and LT2(w, z) zueier Veranderlicher sowie abgeleiteter Funktionen, Akad. Verlag, Berlin, 1965. 82. BLANCH, G., and CLEMM, D. S., Tables relating to the radial Mathieu functions, Vol. I, Vol. II, U.S. Government Printing Off., Washington D.C., 1962, 1965. 83. HANISH, S., et al., Tables of radial spheroidal wat’e functions, Vols. l-6, Washington DC., 1970.
U.S. Government Printing Off.,
84. CURTIS, A. R., Tables of Jacobian elliptic functions n3hose arguments are rational fractions of the quarter period, H.M. Stationery Off., London. 1964.
0041-5553/80/050003-10$07.50/O 01981. Pergamon Press Ltd.
THECOMPUTATIONOFSPECIALFUNCTIONS* V. A. DITKIN, K. A. KARPOV and M. K. KERIMOV Moscow (Received
13 May 1980)
METHODS for computing special functions are reviewed, with the accent on methods used when tabulating the functions. Recently published tables are also surveyed.
1. INTRODUCTION Throughout the history of natural science, special functions have been a powerful instrument in the solution of a wide variety of important problems. The reduction of any given applied problem to the evaluation of special functions has always been, and still is, looked on as indicating a strong penetration into the essence of the problem. Almost all the familiar special functions have arisen from a wide diversity of applied problems, so that the study of their properties, and their tabulation, have engaged, not only mathematicians, but also physicists, astronomers, engineers and other specialists. This is still essentially true today. The problems involved in tabulating special functions are as old as the functions themselves, and have usually come to the fore as soon as the function has made its appearance. These problems have occupied many celebrated scholars, including those by whose names the functions are known. Notably scholars have devoted a life-time to the tabulation of certain special functions. Such tables have been, and still are, vital bricks in the edifice of science. Once compiled, the tables can retain their value long after the specific problems to which they owe their origin have been forgotten. Three periods can be distinguished in the tabulation of special functions: 1) laborious manual computation, by individual enthusiasts; 2) mechanical and electromechanical computations, by devices which vastly increased the scope for performing difficult and laborious calculations; 3) the rapid advances in electronic computers, whereby the scope for calculations became virtually unlimited. In the first period, a few tables of the most important functions were compiled, of limited accuracy, and for a limited range of variation of the argument. In the second period, the number of tabulated functions increased, along with the accuracy and the range of variation of the argument. The third period, which saw the appearance of the high-speed electronic computer, has been marked by the setting up of specialist groups dedicated to large-scale projects for the production of mathematical tables.
lZh. vjkhisl. Mat. mat. Fiz., 20, 5, 1093-1104,
1980. 3
4
V. A. Ditkin, K. A. Karpov and M. K. Kerimov
Particular mention must be made here of groups from the Royal Society (UK), the American National Bureau of Standards, the Computing Laboratories of Harvard University, and the Computing Centre of the Academy of Sciences of the USSR. The work of these groups (accompanied by many theoretical studies into computing methods) has resulted in the compilation and publication of entire libraries of mathematical tables, occupying many tens of volumes. It was in connection with this work that many of the now familiar methods for computing special functions were developed. We need only mention the widely known numerical method of recurrence relations, which was developed while tabulating Bessel functions. For instance, the tabulation of the zeros of Bessel functions, which occupied a section of the Royal Society for 18 years, led to completely new methods being devised for computing the zeros of special functions. In view of the amount of work and facilities needed to tabulate special functions, any new enterprise in this field clearly must be preceded by a very careful study of the tables already available. The number of tables published separately, or contained in publications devoted to other topics, is enormous, and there has long been a demand for the compilation of special indices of mathematical tables. The fllst attempt was headed by von Mises in 1928 [ 11, though only the first part of the work was published. The next was a very complete guide to tables of Bessel functions, compiled by Bateman and Archibald [2]. The fust index covering tables of all elementary and special functions was due to Fletcher et al., and was originally published in 1946. The second, greatly enlarged edition, appeared in 2 volumes [3] in England and the USA in 1962. The German index appeared in 195.5 [4], and the Russian, in 1956 [5]. A new, much more complete index, compiled by M. K. Kerimov, is currently being prepared for press at the Computing Centre of the Academy of Sciences of the USSR. It will include detailed information about virtually all the tables published anywhere in the world. and about algorithms and programs (whether published separately, or appearing in journal articles, books, guides etc.). The present article, timed to coincide with the 25th anniversary of the Computing Centre of the Academy of Sciences, surveys some of the methods used for computing special functions. Special attention is paid to the Soviet project for tabulating special functions. We also touch incidentally on some other recent work in the USSR and abroad. Naturally, it has not been possible to dwell in detail in a brief article on all the results obtained in computing special functions. Only the references to tables compiled at the Computing Centre are more or less complete. After the introduction (Section l), we describe computing methods in Section 2, survey algorithms and programs in Section 3, survey the tables of special functions in Section 4, and finally, refer briefly in Section 5 to work on nomography at the Computing Centre. Due to lack of space, it has not been possible to include full bibliographical details in the “References” to alI publications mentioned in the text.
The computation of spectd functions
2. Methods of computing special functions Many methods have been devised for computing series representation representations,
of the functions,
into different
etc. The latter methods are most commonly
for special functions: Different
polynomial.
rational: Chebyshev,
special polynomials
used when obtaining
of the arguments.
are used for computing
For instance, Bessel functions;
are used, and for large values? asymptotic methods or generalizations Asymptotic
computations
in different
for small values of the argument,
series
for moderate values, recurrence
formulae or expansions,
special functions
relations
In short, several different
of them may need to be used to compute
methods for computing
in recent years [6-l
(e.g. Chebyshev approximations
and Pade approximations.
methods are, by their nature. suitable for function
ranges of variation expansions
some make use of formulae to the integral
while others employ recurrence relations, etc. Special methods may be used,
such as the Lanczos tau method. or expansions polynomials),
special functions;
others apply different quadrature
a given function.
have become especially popular
11.
A lot of work has been done on obtaining
approximations
for special functions,
since
approximations are particularly suitable for machine computations. In this connection we must first mention Luke’s publications ([ 121 , and specially [ 131). m which are to be found excellent approximations for almost all the special functions; the coefficients of these approximations are evaluated to high accuracy. With the aid of such approximations, computed
the function
itself can be
to high accuracy.
The numerous must be mentioned
approximations
the approximations.
by Nemeth for a large number of special functions contain tables of the coefficients
of
evaluated to high accuracy.
Many approximations, approximation
obtained
(see e.g. [ 141); all these publications
with tables of coefficients.
[ 15- 171 . and in many publications
Indices of approximations
are to be found in handbooks
on
by Cody and his associates (e.g. [ 18. 191 ).
have also begun to appear (e.g. [20] ).
Temme [2 1, 221, Kolbig [23] . Gautschi (241, and others, have been extremely active in the computation of special functions. Methods for the complete automation of the computer evaluation
of special functions
Bessel functions for computing functions,
have also been published (see e.g., [25, 261). The evaluation
in the real and complex domains is dealt with in [27, 281. Numerous parabolic cylinder functions
in [30,31].
are surveyed in (291. and for computing
A very detailed survey of methods of computing
of
methods Coulomb
special functions
is to
be found in [32]. The Computing Centre of the Academy of Sciences of the USSR has produced a variety of theoretical and computational work on special functions, including one [33] on checking the Riemann hypothesis about the zeros of the zeta function, on the properties of Lommel functions of two variables [34], on asymptotic formulae for Legendre functions [35-371. on asymptotic expansions for cylindrical functions of two variables [38-391, on the zeros of these functions [40], and on the evaluation of the Weierstrass elliptic function [41]. A lot of work has been done on applications of the operational calculus and integral transforms to the theory of special functions, see e.g. [42-441.
6
V. A. Ditkin, K. A. Karpov and M. K. Kerimov
3. Algorithms and programs When computers began to be used for evaluating special functions, programs written in the appropriate machine codes started to appear. Special algorithmic languages were later developed, and programs were written in these languages. Large stocks of such programs have now been built up, both here and abroad, and are always being supplemented. The stock of algorithms and programs of the Academy of Sciences of the USSR, is kept at the Computing Centre. Similar stocks are kept at many Academies of Sciences of other Soviet republics. For the USSR Academy stock, an automatic information search system operates for algorithms and programs, whereby required information can be found rapidly. The systematically filled data bank of the search system at present contains the description of over 2500@ works with algorithms, programs, and programming information. On the basis of the existing programs, so-called libraries and packages of programs are at present being developed, designed for the automatic solution of large classes of problems. Abroad, such packages have in fact appeared for computing special functions (see e.g. [46] ). In the USSR, most libraries of programs (e.g. at the Research Computing Centre of hioscow University, or at the Mathematical Institute of the Belorussia Academy of Sciences) include programs for computing special functions. Some of the programs from these libraries have already been published (see [46,47]). In addition, the following programs may be mentioned: for the sine, cosine, and Fresnel integrals [48] ; for the integral exponential function [49, SO], for the dilogarithm [5 11, the incomplete gamma function [52-541, the gamma function in the complex domain [55], the logarithm of the gamma function and the digamma function in the complex domain [56], the probability integral in the complex domain [57], Bessel functions [58-621, spherical Bessel functions [63], Bessel functions in the complex domain [64], the zeros of Bessel functions and their derivatives [65], Thomson functions [66], elliptic integrals and functions [67], Coulomb functions [68] , spheroidal functions [69-7 11, and Airy functions [72-741.
4. Tables The Soviet project for computing mathematical tables was devised and started at the Institute of Precision Mechanics and Computing Techniques (IEM and CT), and then moved to the Computing Centre (CC) in 1955. The project was originally headed by L. A. Lyusternik, then from 1953, by V. A. Ditkin. Of course, some work on the tabulation of special functions had been carried out earlier in the USSR. First we must mention the fundamental tables of modified Bessel functions [75], started originally back in 1930; groups from three institutes (including the Institute of Mechanics and the Mathematical Institute of the Academy of Sciences of the USSR) worked on them later. Similar work was done even earlier in the Ukraine, where Bessel functions of complex argument [76] were tabulated for the first time. The Soviet project consists of three series of tables: 1) “Mathematical tables” (edited by V. A. Ditkin); 2) “special tables”; 3) a series of translated foreign tables “Library of Mathematical Tables, ” issued from 1958 and edited by K. A. Karpov. All the tables of these series are as a rule fundamental (with an accuracy of at least 5-6 places). When selecting functions for tabulation, their practical and scientific importance was invariably the guiding factor. The wishes of research and production groups were always taken into account.
The computation of special finctions
We can only mention
here a few of the tables of special functions
appearing in these
series. and touch briefly on some other tables published in recent years in the USSR and abroad. The series “Mathematical
(L.A. Lyusternik,
and year of publication function
Tables”.
The series in fact started with tables of Bessel1 functions
1. Ya. Akushskii. V. A. Ditkin.
1949), prepared at the IEM and CT (the authors
are shown in parentheses).
Later, at the same Institute,
tables of the
In F(z) in the complex domain were prepared and published (A.A. Abramov,
The first tables computed
by the new BESM-I computer were of Fresnel integrals (1953).
though after these, several tables were computed
on keyboard
since work on them had started before production for using the electronic
1953).
and analytic-computing
and analytic-computing
machines,
of the BESM -1. A method was developed
machines. such that the published tables were
all free from errors. The following tables were subsequently integral exponential
function
published:
W(z) =e-’
(1954): the function
domain (K. A. Karpov. 1954); Jacobi polynomials on the initiative functions
of A. A. Dorodnitsyn:
for the asymptotic
1955): cylindrical
functions
Laplace’s equation
Jer2 da:
(L. N. Karmazini,
of Airy functions
solution of second-order
in the complex
1954); tables, developed
and special degenerate
differential
equations
of two variables (E. N. Dekanosidze,
hypergeometric
(A. D. Smirnov,
1956); tables for solving
in elliptic domains (A. 1. Vzorova. 1957); the integral logarithm
(K. A. Karpov. S. N. Razumovskii,
1956): Bessel functions
integrals (E. A. Chistova, 1958); Bessel functions (L. N. Karmazina,
integral sine and cosine (1954); the
of a real argument and their
of an imaginary
E. A. Chistova. 1958); integro-exponential
argument
function
and their integrals
(V. I. Pagurova, 1959);
Weber functions,
in three volumes (K. A. Karpov. 1. E. Kireeva. E. A. Chistova. 1959, 1964,
1968); Thomson
functions
functions generalized
and their first derivatives (L. I. Nosova (Osipova), in tvvo volumes (M. I. Zhurina.
P_:;+r(r). Airy, functions
1960); Legrendre
L. N. Karmazina.
for the asy,mptotic solution of differential
equations
1960, 1967): E (py’) ‘+ (q+
ET) y=! (L. I. Nosova, (Osipova), S. A. Tumarkin. 1961): cylindrical functions of two imaginary variables (L. S. Bark. P. I. Kuznetsov? 1962); tables and formulae for spherical functions PIIIViL{, (5) (M. 1. Zhurina, L. N. Karmazina, 1962); elliptic integrals of all three kinds, in two volumes (V. M. Belyakov et al., 1962, 1963): Legendre functions P-tl;:+ir (5) (M. I.
Zhurina,
L. N. Karmazina.
Rayleigh-Rice
1963); the incomplete
distributions
gamma function
(L. S. Bark et al., 1964); degenerate
1st and 2nd kinds, in two volumes (M. 1. Zhurina.
(V. I. Pagurova, 1963); the
hypergeometric
function
of the
L. N. Osipova, 1964. 1972); modified
Whittaker functions (B. 1. Korobochkin. Yu. A. Filippov, 1965); the logarithmic derivative of the gamma function and its derivatives in the complex domain (1965); Chebyshev polynomials (L. P. Grabar’, 1965): incomplete cy,lindrical functions (M. M. Agrest et al., 1966); roots and weighting factors of generalized Laguerre polynomials (E. N. Dekanosidze, 1966); modified Bessel functions
with imaginary index Ki,(x)
elliptic Weierstrass function equation
(T. D. Kuznetsova.
(Ts. D. Lomtatsi. Yu. N. Smirnov.
(M. 1. Zhurina,
L. N. Karmazina,
1967), characteristic
exponents
1967); the of Mathieu’s
1969); eigenvalues of Mathieu’s equations
(L. S. Bark er aZ., 1970); roots and weighting coefficients of Jacobi polynomials (E. N. Glonti, 1971); Legendre functions of imaginary argument (L. N. Karmazina, 1972); and modified Bessel functions
K,hiiB (5)
(Yu. M. Rapoport.
1979).
8
1’. A. Ditkin, K. A. Karpov and M. K. Kerimov
“Miscellaneous Tables” series. We shall confine ourselves here to mentioning the important tables of Coulomb wave functions (A. V. Lut’yanov et d, 1961), the Racah coefficients (A. F. Nikiforov et al., 1962), the tables for computing toroidal shells (L. N. Osipova. S. A. Tumarkin, 1963), the integrals of quantum chemistry (Yu. A. Kruglyak, D. R. Uitmen, 1965). and the tables of 9j-coefficients (Ya. 1. Vizbaraite et al.,1968). The entire series comprises 13 volumes. “Libran> of Mathematical Tables’lseries. This great series comprises 48 volumes (some of them are 2nd editions). The tables selected were as a rule subjected to careful and continuous checking by a specially developed method, with the result that large numbers of errors were detected and corrected; the corrections were sometimes so important that the translation (e.g. of the well known Hayashi tables) became virtually an original edition.
The most important of the general-purpose foreign tables, well known to specialists, were included in the series. Special functions are covered by 29 volumes. Since we cannot quote all the titles here, we shall just mention those which include considerable theoretical matter in addition to the tables as such: degenerate hypergeometric functions (L. J. Slater, 1968), Weber (parabolic cylinder) functions (J. C. P. Miller, 1968), Coulomb wave functions (A. R. Curtis, I969), and the incomplete beta function (K. Pearson, 1974). While few tables have been published in the USSR outside the walls of the Computing Centre, some of them are of great importance, notably, the tables of elliptic integrals (N. S. SamoiIova-Yakhontova, 1935): the collection of tables (B. I. Segal, K. A. Semendyaev, 1948; 3rd ed., 1962); the normalized associated Legendre polynomials (S. L. Belousov, 1956), Laguerre polynomials and functions (V. S. Aizenshtat et al., 1965); the Bessel functions J,(X) of integral indices n from 0 to 120 (V. N. Faddeeva, M. K. Gavurin, 1950); the function connected with the probability integral in the complex domain (V. N. Faddeeva, N. M. Terent’ev, 1954); the Airy functions (V. A. Fok, 1946. G. D. Yakovleva, 1969); spheroidal wave functions and their derivatives. in two volumes (S. P. Erashevska1.a er al., 1973, 1976); the fractional-exponential function of negative parameters and its integral (Vu. N. Rabotnov et al., 1969); the generarized Riemann zeta function and its derivative of complex argument, in two volumes (N. N. Voitovich et al., 1970); the probability integral (N. V. Smirnov, 1960); the two dimensional normal distribution (N. V. Smirnov, L. N. Bol’shev, 1962); and extensive tables of mathematical statistics (L. N. Bol’shev, N. V. Smirnov, 1965; 2nd ed., 1968). Some fundamental foreign tables published in recent years should also be mentioned: special functions [77] (2 vols.)! the incomplete gamma function [78, 791 , Kelvin functions (801, Pommel functions of two variables [81], Mathieu radial functions [82] (2 vols.), spheroidal wave functions [83] (6 vols.), and Jacobi elliptic functions [84].
5. Nomograms
In parallel with the work on compiling tables at the Computer Centre of the Academy of Sciences, studies were directed by G. S. Khovanskii on the development of methods of nomography and the production of working nomograms, i.e. “graphical tables,” which are an indispensable aid to many engineers. Several textbooks on nomography were prepared and published, along with atlases of working nomograms. while other work on nomography in the USSR was directed
9
The computation of special functions
and coordinated. As a result, 12 collections have so far appeared (with two more in the press). Experience suggests that, in spite of the wide introduction of computing techniques, nomograms, like tables, will not lose their value and will long remain a useful instrument for research-workers and engineers. Translated
by D. E. Brown.
REFERENCES 1.
VON MISES, R., Verzeichnis berechneter Funktionentafeln, elliptische Funktionen, Berlin, VDJ Verlag, 1928.
Erster Teil, Bessebche, Kugel- und
2.
BATEMAN, H., and ARCHIBALD, R. C., A guide to tables of Bessel functions, Math. Tables and Other Aids Comput., 1, No. 7, 205-308, 1944.
3.
FLETCHER, A., et al., An index of mathematical tables, 2nd ed., Vols. 1, 2, London, Scient. Comput. Service Ltd.; Reading, Mass., Addison-Wesley, 1962.
4.
SCH%lE,
5.
LEBEDEV, A. V., and FEDOROVA, R. M., Guide to mathematical tables, lzdvo Akad. Nauk SSSR, Moscow, 1965; BURUNOVA, N. M., Supplement No. 1, Izd-vo Akad. Nauk SSSR, Moscow, 1959.
6.
OLVER, F. W. J., Unsolved problems in the asymptotic estimation of special functions, in: 7”heov and applications of special functions, Acad. Press, New York, 99- 142, 1975.
7.
OLVER, F. W. J., Asymptotics and special functions,
8.
DINGLE, R. B., Asymptotic expansions: their derivation and interpretation, Acad. Press, New York, 1973.
9.
RUEKSTYN’SH, E. YA., Asymptotic expunsions of integrals, Vol. 1, (Asimptoticheskie integralov), Zinatne, Riga, 1974; Vol. 2, 1977.
K., Index mathematischer Tafelwerke und Tabellen, 2. AufL, Munich, R. Oldenburg, 1966.
Acad. Press, New York, 1974.
razlozheniya
10. TIKHONOV, A. N., Asymptotic behaviour of integrals containing Bessel functions, Dokl. Akad. Nauk SSSR, 12.5, No. 5,982-985, 1959. 11. TIKHONOV, A. N., SAMARSKII, A. A., and ARSEN’EV, A. A., A method for making asymptotic estimates of integrals, Zh. vj%hist.Mat. mat. FL, 12, No. 4, 1005-1012, 1972. 12. LUKE, Y. L., The special functions and their approximations, Vol. 1,2, Acad. Press, New York, 1969. 13. LUKE, Y. L., Algorithms for the computation
of mathematical functions, Acad. Press. New York, 1977.
14. NEMETH, G., Chebyshev expansions for Fresnel integrals, Numer. Math., 7, No. 4, 310-312,
1965.
15. HASTINGS, C., HAYWARD, J. T., and WONG, J. P., Approximations for digital computers, Princeton Univ. Press. Princeton, 1955. 16. HART, J. F., et al., Computer approximations, Wiley, New York, 1968. 17. ABRAMOWITZ, M., and STEGUN, I. A., Handbook of mathematical functions with formulas, graphs and mathematical tables, Nat. Bur. Standards, Washington, 1964. 18. CODY, W. J., Rational Chebyshev approximations 1969.
for the error function,Math.
19. CODY, W. J., and HILSTROM, K. E., Chebyshev approximations Cornput., 24,671-677,197O.
Cornput.. 23, 631-637,
for the Coulomb phase shift, Math
20. PRASAD, B., and NARASIMHAN, V. L., An index of approximations of functions, Computer Centre, Univ. California, San Diego/La Jolla, UCSD-6442, Feb., 1964. 21. TEMME, N. M., On the numerical evaluation of the modified Bessel function of the third kind, J. Comput. Phys, 19, 324-337, 1975. 22. TEMME, N. M., On the numerical evaluation of the ordinary Bessel function of the second kind, J. Comput. Phys, 21, 343-350,1976.
10
I’. A. Ditkin,
K. A. Karpov and M. K. Kerimov
23. KGLBIC, K. S., Complex zeros of an incomplete Riemann zeta function and of the Incomplete gamma function, Math. Comput, 24.679-696, 1970. 24.
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