Numerical Aspects of Special Functions
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1 Numerical Aspects of Special Functions Nico M. Temme In collaboration with Amparo Gil and Javier Segura, Santander, Spain. Centrum voor Wiskunde en Informatica (CWI), Amsterdam Numerics of Special Functions, Castro Urdiales, SPAIN, September p.1/45
2 Contents The complete revision of Abramowitz & Stegun (1964) Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
3 Contents The complete revision of Abramowitz & Stegun (1964) Software for computing special functions Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
4 Contents The complete revision of Abramowitz & Stegun (1964) Software for computing special functions A book on numerics of special functions Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
5 Contents The complete revision of Abramowitz & Stegun (1964) Software for computing special functions A book on numerics of special functions Recursion relations to compute special functions Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
6 Contents The complete revision of Abramowitz & Stegun (1964) Software for computing special functions A book on numerics of special functions Recursion relations to compute special functions Parabolic cylinder functions Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
7 Contents The complete revision of Abramowitz & Stegun (1964) Software for computing special functions A book on numerics of special functions Recursion relations to compute special functions Parabolic cylinder functions Using special functions Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
8 Contents The complete revision of Abramowitz & Stegun (1964) Software for computing special functions A book on numerics of special functions Recursion relations to compute special functions Parabolic cylinder functions Using special functions Can we rely on Maple or Mathematica? Numerics of Special Functions, Castro Urdiales, SPAIN, September p.2/45
9 The complete revision of A & S (1964) Dan Lozier s progress report in OPSF (July 15, 2006): Looking Ahead to the DLMF The Digital Library of Mathematical Functions has been a much bigger job than any of the project participants expected. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.3/45
10 The complete revision of A & S (1964) Dan Lozier s progress report in OPSF (July 15, 2006): Looking Ahead to the DLMF The Digital Library of Mathematical Functions has been a much bigger job than any of the project participants expected. I am pleased to report that all of the chapters are in the final stages of editing and validating. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.3/45
11 The complete revision of A & S (1964) Dan Lozier s progress report in OPSF (July 15, 2006): Looking Ahead to the DLMF The Digital Library of Mathematical Functions has been a much bigger job than any of the project participants expected. I am pleased to report that all of the chapters are in the final stages of editing and validating. The process of selecting a publisher for the print edition has been mapped out according to NIST and US Government procurement rules. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.3/45
12 The complete revision of A & S (1964) Dan Lozier s progress report in OPSF (July 15, 2006): Looking Ahead to the DLMF The Digital Library of Mathematical Functions has been a much bigger job than any of the project participants expected. I am pleased to report that all of the chapters are in the final stages of editing and validating. The process of selecting a publisher for the print edition has been mapped out according to NIST and US Government procurement rules. The procurement will be competitive among qualified mathematics publishers. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.3/45
13 The complete revision of A & S (1964) In addition to the print edition, the DLMF will be distributed free from a public Web site at NIST. The Web address is A sample chapter on Airy functions has existed on this Web site since the beginning of the project. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.4/45
14 The complete revision of A & S (1964) In addition to the print edition, the DLMF will be distributed free from a public Web site at NIST. The Web address is A sample chapter on Airy functions has existed on this Web site since the beginning of the project. A tremendous amount of work has been done on the Web site since then to improve its "look and feel". Numerics of Special Functions, Castro Urdiales, SPAIN, September p.4/45
15 The complete revision of A & S (1964) In addition to the print edition, the DLMF will be distributed free from a public Web site at NIST. The Web address is A sample chapter on Airy functions has existed on this Web site since the beginning of the project. A tremendous amount of work has been done on the Web site since then to improve its "look and feel". A new sample chapter on the gamma function is being prepared. It will be on the Web site by the end of the summer. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.4/45
16 The complete revision of A & S (1964) In addition to the print edition, the DLMF will be distributed free from a public Web site at NIST. The Web address is A sample chapter on Airy functions has existed on this Web site since the beginning of the project. A tremendous amount of work has been done on the Web site since then to improve its "look and feel". A new sample chapter on the gamma function is being prepared. It will be on the Web site by the end of the summer. More than 50 individuals are contributing to the project as paid authors and validators. The staff at NIST consists of another dozen or so people. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.4/45
17 Software for computing special functions A complete survey of the available software: Lozier & Olver (1994); last update: December Numerics of Special Functions, Castro Urdiales, SPAIN, September p.5/45
18 Software for computing special functions A complete survey of the available software: Lozier & Olver (1994); last update: December Interactive systems based on computer algebra: Matlab, Maple, Mathematica. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.5/45
19 Software for computing special functions A complete survey of the available software: Lozier & Olver (1994); last update: December Interactive systems based on computer algebra: Matlab, Maple, Mathematica. Mathematical libraries: CALGO, SLATEC, CERN, IMSL, NAG. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.5/45
20 Software for computing special functions A complete survey of the available software: Lozier & Olver (1994); last update: December Interactive systems based on computer algebra: Matlab, Maple, Mathematica. Mathematical libraries: CALGO, SLATEC, CERN, IMSL, NAG. Books with software: Baker, Moshier, Numerical Recipes, Thompson, Wong & Guo, Zhang & Jin. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.5/45
21 A book on numerics of special functions The topics mentioned in this lecture, and several other topics, will be discussed extensively, with examples of software, in a new book with the title Numerical Methods for Special Functions. Written together with my co-authors Amparo Gil and Javier Segura (Santander, Spain). We have just submitted the first version for review. To be published by SIAM. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.6/45
22 A book on numerics of special functions As the basic methods for computing special functions we consider Power series: convergent, asymptotic Chebyshev series Recurrence relations and continued fractions Quadrature methods Numerics of Special Functions, Castro Urdiales, SPAIN, September p.7/45
23 A book on numerics of special functions Other important methods and aspects are Continued fractions Computation of the zeros of special functions Padé approximations Sequence transformations Best rational approximations Numerics of Special Functions, Castro Urdiales, SPAIN, September p.8/45
24 A book on numerics of special functions Other topics are The Euler summation formula Numerical solution of ODEs: Taylor expansion method Uniform asymptotic expansions Asymptotic inversion of distribution functions Symmetric elliptic integrals Numerical inversion of Laplace transforms Approximations of Stirling Numbers Oscillatory integrals Numerics of Special Functions, Castro Urdiales, SPAIN, September p.9/45
25 A book on numerics of special functions In the software section we discuss mainly the G-S-T Fortran codes published earlier. Routines can be downloaded from Airy and Scorer functions of complex arguments and their derivatives. Legendre functions of integer and half-odd degrees; toroidal harmonics. Modified Bessel functions of integer and half integer orders. Modified Bessel functions of purely imaginary orders. Parabolic cylinder functions. Zeros of the Bessel function. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.10/45
26 Recursion relations To compute Bessel functions Legendre functions Confluent hypergeometric functions (Kummer, Whittaker, Coulomb) Parabolic cylinder functions Gauss hypergeometric functions Incomplete gamma and beta functions Orthogonal polynomials recursion relations are important and frequently used. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.11/45
27 Recursion relations The recursion relations are of the form When there are two linearly independent solutions, such that and the computation of the minimal solution from and The solution! (forward recursion) is usually very unstable. is called a dominant solution. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.12/45
28 Recursion relations For example, the functions satisfy the recursion relation with Maple, standard 10 Digits, we see Computing that a negative number. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.13/45
29 Recursion relations The function is for a dominant solution of the recursion. It can be computed in a stable way with starting values Numerics of Special Functions, Castro Urdiales, SPAIN, September p.14/45
30 Recursion relations The recursion for the functions simpler (inhomogeneous) recursion follows from the Use this in backward direction with false starting value. Then! which is correct in all shown digits. Backward recursion for a minimal solution with false starting values is the basis for the Miller algorithm. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.15/45
31 Parabolic cylinder functions PCFs are solutions of the differential equation Special cases: Hermite polynomials, complementary error function when. when PCFs are special cases of Kummer functions (confluent hypergeometric functions). Numerics of Special Functions, Castro Urdiales, SPAIN, September p.16/45
32 Parabolic cylinder functions Two linearly independent solutions of the differential equations are denoted by and. Another notation is. Scaling is very important to prevent underflow and overflow (and to extend the domains for computation). When the computation is not too difficult. When and, oscillations occur. Large parameters difficult to handle. and, with, are especially Numerics of Special Functions, Castro Urdiales, SPAIN, September p.17/45
33 Parabolic cylinder functions The regions in the -plane where different methods of computation for PCFs are considered. Only for ; for negative connection formulas are used f 1 3 a f f 2 A f 5 f 12 f 11 4 f 6 B f x Numerics of Special Functions, Castro Urdiales, SPAIN, September p.18/45
34 Parabolic cylinder functions Maclaurin series In region 1 (small and ) we use Maclaurin series Recursion relations for the coefficients are used. and are solutions of the differential equation, but do not constitute a numerically satisfactory pair of solutions for large values of. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.19/45
35 Parabolic cylinder functions are used in and with Numerics of Special Functions, Castro Urdiales, SPAIN, September p.20/45
36 Parabolic cylinder functions In other regions we use Quadrature Recursions with computed starting values Poincaré asymptotic expansions (in powers of Uniform asymptotic expansions in terms of elementary functions Uniform asymptotic expansions in terms of Airy functions ) Numerics of Special Functions, Castro Urdiales, SPAIN, September p.21/45
37 Parabolic cylinder functions Quadrature For quadrature we transform some standard integrals into integrals that can easily be evaluated by using the trapezoidal rule. It is well known that integrals of the type with analytic inside a strip, for some excellent starting points for the trapezoidal rule., are Numerics of Special Functions, Castro Urdiales, SPAIN, September p.22/45
38 Parabolic cylinder functions In many cases we transform finite integrals, say of the form into an integral of the form by using the error function: and the integral becomes Numerics of Special Functions, Castro Urdiales, SPAIN, September p.23/45
39 Using special functions Special functions play an important role in The solution of problems in mathematical physics, engineering, statistics, and so on. As kernels in integral transforms Numerics of Special Functions, Castro Urdiales, SPAIN, September p.24/45
40 Using special functions The non-central chi-squared distributions : The definition are for positive are the incomplete gamma functions: and where Numerics of Special Functions, Castro Urdiales, SPAIN, September p.25/45
41 Using special functions The non-central chi-squared distributions Recursion relations are: where is the modified Bessel function. Stability aspects are very important here. From these recursions homogeneous four-term recursions can be derived for and. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.26/45
42 Using special functions The non-central chi-squared distributions Integral representations are: Numerics of Special Functions, Castro Urdiales, SPAIN, September p.27/45
43 Using special functions The non-central chi-squared distributions By using an integral representation of the Bessel function: which in fact are inverse Laplace transforms. They are useful for deriving uniform asymptotic expansions. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.28/45
44 Using special functions The non-central chi-squared distributions In problems in radar communications very large values of are used, say, about 10,000. Asymptotic analysis shows a transition when passes the value. There is a fast transition from 0 to 1. The main approximant is in that case the complementary error function. See T(1993) and T(1996). Numerics of Special Functions, Castro Urdiales, SPAIN, September p.29/45
45 Using special functions A series of Bessel functions Consider the series in terms of modified Bessel functions The function is the solution of an elliptic boundary value problem inside a circle. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.30/45
46 Using special functions The equation reads A series of Bessel functions with boundary condition on the boundary of the circle coordinates. We use polar Numerics of Special Functions, Castro Urdiales, SPAIN, September p.31/45
47 Using special functions A series of Bessel functions This is a so-called singular perturbation problem, small parameter. is a The solution has a boundary layer at the upper part of the circle. We recall the solution, we write, Numerics of Special Functions, Castro Urdiales, SPAIN, September p.32/45
48 Using special functions A series of Bessel functions y r θ x Boundary layer inside the circle along the upper boundary and near the points. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.33/45
49 Using special functions A series of Bessel functions It is difficult to compute when with particular in the neighborhoods of the points, in. It is not difficult to compute the modified Bessel functions. But when is large, they become exponentially large, and the convergence of the Fourier series starts with high values. In particular in the boundary layer large quantities are canceling before convergence starts. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.34/45
50 Using special functions A series of Bessel functions A perturbation analysis starts with the expansion in which the terms satisfy and and all should vanish at the lower part of the unit circle. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.35/45
51 Using special functions A series of Bessel functions Integrating the first few relations for the gives where. Observe that these and that they do not satisfy the boundary condition on the upper part of the unit circle. become singular at the points Numerics of Special Functions, Castro Urdiales, SPAIN, September p.36/45
52 Using special functions A series of Bessel functions Simple model problems in singular perturbations with known exact solutions are of interest because Numerical codes for more general problems can be tested on these problems. They may give new challenges in (uniform) asymptotic analysis of integrals or series. They may provide asymptotic approximations that are difficult to obtain from perturbation analysis. The series expansions of numerical computations. is a challenge for Numerics of Special Functions, Castro Urdiales, SPAIN, September p.37/45
53 Can we rely on Maple and Mathematica? Consider Maple 9.5, Digits = 10, for, gives Numerics of Special Functions, Castro Urdiales, SPAIN, September p.38/45
54 Can we rely on Maple and Mathematica? Consider, gives Maple 9.5, Digits = 10, for With Digits = 40, the answer is Numerics of Special Functions, Castro Urdiales, SPAIN, September p.38/45
55 Can we rely on Maple and Mathematica? Consider, gives Maple 9.5, Digits = 10, for With Digits = 40, the answer is So, the first answer seems to be correct in all shown digits. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.38/45
56 Can we rely on Maple and Mathematica? Take another integral, which is almost the same: Maple 9.5, Digits=10, for, gives Numerics of Special Functions, Castro Urdiales, SPAIN, September p.39/45
57 Can we rely on Maple and Mathematica? Take another integral, which is almost the same: Maple 9.5, Digits=10, for, gives With Digits = 40, the answer is Numerics of Special Functions, Castro Urdiales, SPAIN, September p.39/45
58 Can we rely on Maple and Mathematica? Take another integral, which is almost the same:, gives Maple 9.5, Digits=10, for With Digits = 40, the answer is we and for The correct answer is have Numerics of Special Functions, Castro Urdiales, SPAIN, September p.39/45
59 Can we rely on Maple and Mathematica? The message is: one should have some feeling about the computed result. Otherwise a completely incorrect answer can be accepted. Mathematica is more reliable here, and says: "NIntegrate failed to converge to prescribed accuracy after 7 recursive bisections in near ". Numerics of Special Functions, Castro Urdiales, SPAIN, September p.40/45
60 Can we rely on Maple and Mathematica? By the way, Maple 7 could do the following integral and the funny answer was, after some simplification, where is the error function. In Maple 9.5 the answer is Numerics of Special Functions, Castro Urdiales, SPAIN, September p.41/45
61 Can we rely on Maple and Mathematica? Consider Numerical quadrature gives Mathematica 4.1 gives for G-function: in terms of the Meijer. Mathematica evaluates:. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.42/45
62 Can we rely on Maple and Mathematica? Ask Mathematica to evaluate : This gives. So, we have three numerical results: Observe that. is correct. Numerics of Special Functions, Castro Urdiales, SPAIN, September p.43/45
63 Can we rely on Maple and Mathematica? Maple 9.5: same as Mathematica. This is a wrong answer. Next, Maple 9.5, with, giving correct answer., which is the Numerics of Special Functions, Castro Urdiales, SPAIN, September p.44/45
64 Finally,... Thank You Numerics of Special Functions, Castro Urdiales, SPAIN, September p.45/45
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