Numerical Methods for Special Functions

Abstract

Probably, the most extended (pseudo) definition of the set of functions known as “special functions” refers to those mathematical functions which are widely used in scientific and technical applications, and of which many useful properties are known. These functions are typically used in two related contexts:
1. as a way of obtaining simple closed formulas and other analytical properties of solutions of problems from pure and applied mathematics, statistics, physics, and engineering;
2. as a way of understanding the nature of the solutions of these problems, and for obtaining numerical results from the representations of the functions.
Our book is intended to provide assistance when a researcher or a student needs to get the numbers from analytical formulas containing special functions. This book should be useful for those who need to compute a function by their own means, or for those who want to know more about the numerical methods behind the available algorithms. Our main purpose is to provide a guide of available methods for computations and when to use them. Also, because of the large variety of numerical methods that are available for computing special functions, we expect that a broader “numerical audience” will be interested in many of the topics discussed (particularly in the first part of the book). Several levels of reading are possible in this book and most of the chapters start with basic principles. Examples are given to illustrate the use of the methods, pseudoalgorithms are given to describe technical details, and published algorithms for computing a selection of functions are described as practical illustrations for the basic methods of this book.
The presentation of the topics is organized in four parts: Basic Methods, Further Tools and Methods, Related Topics and Examples, and Software. The first part (Basic Methods) describes a set of methods which, in our experience, are the most popular and important ones for computing special functions. This includes convergent and divergent series, Chebyshev expansions, linear recurrence relations, and quadrature methods. These basic chapters are mostly self-contained and start from first principles. We expect that many of the contents are appropriate for advanced numerical analysis courses (parts of the chapters are in fact based on classroom notes); however, because the main focus is on special functions, detailed examples of application are also provided.
The second part of the book (Further Tools and Methods) contains a set of important methods for computing special functions which, however, are probably not so well known as the basic methods (at least for readers who are not very familiar with special functions).

Full text

Numerical
Methods
for
Special
Functions
Amparo
Gil
Universidad
de
Cantabria
Santander, Cantabria, Spain
Javier Segura
Universidad
de
Cantabria
Santander, Cantabria, Spain
Nico M. Temme
Centrum voor Wiskunde
en
Informatica
Amsterdam,
The
Netherlands
• Society
for
Industrial and Applied Mathematics
•
Philadelphia

Contents
Preface xiii
1 Introduction 1
1 Basic Methods 13
2 Convergent and Divergent Series 15
2.1 Introduction 15
2.1.1 Power series: First steps 15
2.1.2 Further practical aspects 17
2.2 Differential equations and Frobenius series solutions 18
2.2.1 Singular points 19
2.2.2 The solution near a regular point 20
2.2.3 Power series expansions around a regular singular point . 22
2.2.4 The Liouville transformation 25
2.3 Hypergeometric series 26
2.3.1 The Gauss hypergeometric function 28
2.3.2 Other power series for the Gauss hypergeometric function 30
2.3.3 Removable singularities 33
2.4 Asymptotic expansions 34
2.4.1 Watson's lemma 36
2.4.2 Estimating the remainders of asymptotic expansions ... 38
2.4.3 Exponentially improved asymptotic expansions 39
2.4.4 Alternatives of asymptotic expansions 40
3 Chebyshev Expansions 51
3.1 Introduction 51
3.2 Basic results on interpolation 52
3.2.1 The Runge phenomenon and the Chebyshev nodes .... 54
3.3 Chebyshev polynomials: Basic properties 56
3.3.1 Properties of the Chebyshev polynomials Tn(x) 56
3.3.2 Chebyshev polynomials of the second, third, and fourth
kinds 60
vii

viii Contents
3.4 Chebyshev interpolation 62
3.4.1 Computing the Chebyshev interpolation polynomial ... 64
3.5 Expansions in terms of Chebyshev polynomials 66
3.5.1 Convergence properties of Chebyshev expansions .... 68
3.6 Computing the coefficients of a Chebyshev expansion 69
3.6.1 Clenshaw's method for solutions of linear differential
equations with polynomial coefficients 70
3.7 Evaluation of a Chebyshev sum 75
3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum 75
3.8 Economization of power series 80
3.9 Example: Computation of Airy functions of real variable 80
3.10 Chebyshev expansions with coefficients in terms of special functions . 83
4 Linear Recurrence Relations and Associated Continued Fractions 87
4.1 Introduction 87
4.2 Condition of three-term recurrence relations 88
4.2.1 Minimal solutions 89
4.3 Perron's theorem 92
4.3.1 Scaled recurrence relations 94
4.4 Minimal solutions of TTRRs and continued fractions 95
4.5 Some notable recurrence relations 96
4.5.1 The confluent hypergeometric family 96
4.5.2 The Gauss hypergeometric family 102
4.6 Computing the minimal solution of a TTRR 105
4.6.1 Miller's algorithm when a function value is known .... 105
4.6.2 Miller's algorithm with a normalizing sum 107
4.6.3 "Anti-Miller" algorithm 110
4.7 Inhomogeneous linear difference equations 112
4.7.1 Inhomogeneous first order difference equations.
Examples 112
4.7.2 Inhomogeneous second order difference equations .... 115
4.7.3 Olver's method 116
4.8 Anomalous behavior of some second order homogeneous and first
order inhomogeneous recurrences 118
4.8.1 A canonical example: Modified Bessel function 118
4.8.2 Other examples: Hypergeometric recursions 120
4.8.3 A first order inhomogeneous equation 121
4.8.4 A warning 122
5 Quadrature Methods 123
5.1 Introduction 123
5.2 Newton-Cotes quadrature: The trapezoidal and Simpson's rule .... 124
5.2.1 The compound trapezoidal rule 126
5.2.2 The recurrent trapezoidal rule 129
5.2.3 Euler's summation formula and the trapezoidal rule ... 130
5.3 Gauss quadrature 132

Contents ix
5.3.1 Basics of the theory of orthogonal polynomials and Gauss
quadrature 133
5.3.2 The Golub-Welsch algorithm 141
5.3.3 Example: The Airy function in the complex plane .... 145
5.3.4 Further practical aspects of Gauss quadrature 146
5.4 The trapezoidal rule on R 147
5.4.1 Contour integral formulas for the truncation errors .... 148
5.4.2 Transforming the variable of integration 153
5.5 Contour integrals and the saddle point method 157
5.5.1 The saddle point method 158
5.5.2 Other integration contours 163
5.5.3 Integrating along the saddle point contours and examples 165
II Further Tools and Methods 171
6 Numerical Aspects of Continued Fractions 173
6.1 Introduction 173
6.2 Definitions and notation 173
6.3 Equivalence transformations and contractions 175
6.4 Special forms of continued fractions 178
6.4.1 Stieltjes fractions 178
6.4.2 Jacobi fractions 179
6.4.3 Relation with Pade approximants 179
6.5 Convergence of continued fractions 179
6.6 Numerical evaluation of continued fractions 181
6.6.1 Steed's algorithm 181
6.6.2 The modified Lentz algorithm 183
6.7 Special functions and continued fractions 185
6.7.1 Incomplete gamma function 186
6.7.2 Gauss hypergeometric functions 187
7 Computation of the Zeros of Special Functions 191
7.1 Introduction 191
7.2 Some classical methods 193
7.2.1 The bisection method 193
7.2.2 The fixed point method and the Newton-Raphson method 193
7.2.3 Complex zeros 197
7.3 Local strategies: Asymptotic and other approximations 197
7.3.1 Asymptotic approximations for large zeros 199
7.3.2 Other approximations 202
7.4 Global strategies I: Matrix methods . . . 205
7.4.1 The eigenvalue problem for orthogonal polynomials . . .206
7.4.2 The eigenvalue problem for minimal solutions of TTRRs 207
7.5 Global strategies II: Global fixed point methods 213
7.5.1 Zeros of Bessel functions 213

Contents
7.5.2 The general case : 219
7.6 Asymptotic methods: Further examples 224
7.6.1 Airy functions 224
7.6.2 Scorer functions 227
7.6.3 The error functions 229
7.6.4 The parabolic cylinder function 233
7.6.5 Bessel functions 233
7.6.6 Orthogonal polynomials 234
Uniform Asymptotic Expansions 237
8.1 Asymptotic expansions for the incomplete gamma functions 237
8.2 Uniform asymptotic expansions 239
8.3 Uniform asymptotic expansions for the incomplete gamma functions . 240
8.3.1 The uniform expansion 242
8.3.2 Expansions for the coefficients 244
8.3.3 Numerical algorithm for small values of
JJ
245
8.3.4 A simpler uniform expansion 247
8.4 Airy-type expansions for Bessel functions 249
8.4.1 The Airy-type asymptotic expansions 250
8.4.2 Representations of as{S),bs(S),cs(S),ds(O 253
8.4.3 Properties of the functions Av, Bv, Cv, Dv 254
8.4.4 Expansions for av(f)A(f).cs-(0>^(f) 256
8.4.5 Evaluation of the functions
Av{t;),
BP(£) by iteration . . .258
8.5 Airy-type asymptotic expansions obtained from integrals 263
8.5.1 Airy-type asymptotic expansions 264
8.5.2 How to compute the coefficients an, fin 267
8.5.3 Application to parabolic cylinder functions 270
Other Methods 275
9.1 Introduction 275
9.2 Pade approximations 276
9.2.1 Pade approximants and continued fractions 278
9.2.2 How to compute the Pade approximants 278
9.2.3 Pade approximants to the exponential function 280
9.2.4 Analytic forms of Pade approximations 283
9.3 Sequence transformations 286
9.3.1 The principles of sequence transformations 286
9.3.2 Examples of sequence transformations 287
9.3.3 The transformation of power series 288
9.3.4 Numerical examples 288
9.4 Best rational approximations . . . . l. 290
9.5 Numerical solution of ordinary differential equations: Taylor
expansion method 291
9.5.1 Taylor-series method: Initial value problems . 292
9.5.2 Taylor-series method: Boundary value problem 293
9.6 Other quadrature methods 294

Contents xi
9.6.1 Romberg quadrature 294
9.6.2 Fejer and Clenshaw-Curtis quadratures 296
9.6.3 Other Gaussian quadratures 298
9.6.4 Oscillatory integrals 301
III Related Topics and Examples 307
10 Inversion of Cumulative Distribution Functions 309
10.1 Introduction 309
10.2 Asymptotic inversion of the complementary error function 309
10.3 Asymptotic inversion of incomplete gamma functions 312
10.3.1 The asymptotic inversion method 312
10.3.2 Determination of the coefficients £, 314
10.3.3 Expansions of the coefficients £, 316
10.3.4 Numerical examples 316
10.4 Generalizations 317
10.5 Asymptotic inversion of the incomplete beta function 318
10.5.1 The nearly symmetric case 319
10.5.2 The general error function case 322
10.5.3 The incomplete gamma function case 324
10.5.4 Numerical aspects 326
10.6 High order Newton-like methods 327
11 Further Examples 331
11.1 Introduction 331
11.2 The Euler summation formula 331
11.3 Approximations of Stirling numbers 336
11.3.1 Definitions 337
11.3.2 Asymptotics for Stirling numbers of the second kind . . . 338
11.3.3 Stirling numbers of the first kind 343
11.4 Symmetric elliptic integrals 344
11.4.1 The standard forms in terms of symmetric integrals . . . 345
11.4.2 An algorithm 346
11.4.3 Other elliptic integrals 347
11.5 Numerical inversion of Laplace transforms 347
11.5.1 Complex Gauss quadrature 348
11.5.2 Deforming the contour 349
11.5.3 Using Pade approximations 352
IV Software 353
12 Associated Algorithms 355
12.1 Introduction 355
12.1.1 Errors and stability: Basic terminology 356

xii Contents
12.1.2 Design and testing of software for computing functions:
General philosophy 357
12.1.3 Scaling the functions 358
12.2 Airy and Scorer functions of complex arguments 359
12.2.1 Purpose 359
12.2.2 Algorithms 359
12.3 Associated Legendre functions of integer and half-integer degrees . . .363
12.3.1 Purpose 363
12.3.2 Algorithms 364
12.4 Bessel functions 369
12.4.1 Modified Bessel functions of integer and half-integer
orders 370
12.4.2 Modified Bessel functions of purely imaginary orders . .372
12.5 Parabolic cylinder functions 377
12.5.1 Purpose 377
12.5.2 Algorithm 378
12.6 Zeros of Bessel functions 385
12.6.1 Purpose 385
12.6.2 Algorithm 385
List of Algorithms 387
Bibliography 389
Index 405