Gnu Scientific Library: Reference Manual

by Gough, Brian

• ISBN13: 9780954161736
• ISBN10: 0954161734
• Edition: 2nd
• Format: Paperback
• Copyright: 2003-02-01
• Publisher: Lightning Source Inc

Summary

The GNU Scientific Library (GSL) is a free numerical library for C and C++ programmers. It provides over 1,000 routines for solving mathematical problems in science and engineering. Written by the developers of GSL this reference manual is the definitive guide to the library. The GNU Scientific Library is free software, distributed under the GNU General Public License (GPL). It is available for GNU (GNU/Linux), Unix and Microsoft Windows systems. All the money raised from the sale of this book supports the development of the GNU Scientific Library. This is the second edition of the manual, and corresponds to version 1.3 of the library.

Table of Contents

Preface	1	(2)
Introduction	3	(6)
Routines available in GSL	3	(1)
GSL is Free Software	4	(1)
Obtaining GSL	5	(1)
An Example Program	5	(1)
No Warranty	6	(1)
Reporting Bugs	6	(1)
Further Information	6	(1)
How to Help	7	(2)
Using the library	9	(8)
ANSI C Compliance	9	(1)
Compiling and Linking	9	(1)
Shared Libraries	10	(1)
Inline functions	11	(1)
Long double	11	(1)
Portability functions	12	(1)
Alternative optimized functions	12	(1)
Support for different numeric types	13	(1)
Compatibility with C++	14	(1)
Aliasing of arrays	14	(1)
Thread-safety	15	(1)
Code Reuse	15	(2)
Error Handling	17	(6)
Error Reporting	17	(1)
Error Codes	18	(1)
Error Handlers	19	(2)
Using GSL error reporting in your own functions	21	(1)
Examples	22	(1)
Mathematical Functions	23	(6)
Mathematical Constants	23	(1)
Infinities and Not-a-number	24	(1)
Elementary Functions	24	(1)
Small integer powers	25	(1)
Testing the Sign of Numbers	26	(1)
Testing for Odd and Even Numbers	26	(1)
Maximum and Minimum functions	27	(1)
Approximate Comparison of Floating Point Numbers	28	(1)
Complex Numbers	29	(10)
Complex numbers	29	(1)
Properties of complex numbers	30	(1)
Complex arithmetic operators	31	(2)
Elementary Complex Functions	33	(1)
Complex Trigonometric Functions	34	(1)
Inverse Complex Trigonometric Functions	34	(2)
Complex Hyperbolic Functions	36	(1)
Inverse Complex Hyperbolic Functions	36	(1)
References and Further Reading	37	(2)
Polynomials	39	(6)
Polynomial Evaluation	39	(1)
Divided Difference Representation of Polynomials	39	(1)
Quadratic Equations	40	(1)
Cubic Equations	41	(1)
General Polynomial Equations	41	(1)
Examples	42	(1)
References and Further Reading	43	(2)
Special Functions	45	(54)
Usage	45	(1)
The gsl_sf_result struct	46	(1)
Modes	46	(1)
Airy Functions and Derivatives	47	(2)
Airy Functions	47	(1)
Derivatives of Airy Functions	48	(1)
Zeros of Airy Functions	49	(1)
Zeros of Derivatives of Airy Functions	49	(1)
Bessel Functions	49	(11)
Regular Cylindrical Bessel Functions	50	(1)
Irregular Cylindrical Bessel Functions	50	(1)
Regular Modified Cylindrical Bessel Functions	51	(1)
Irregular Modified Cylindrical Bessel Functions	52	(2)
Regular Spherical Bessel Functions	54	(1)
Irregular Spherical Bessel Functions	55	(1)
Regular Modified Spherical Bessel Functions	56	(1)
Irregular Modified Spherical Bessel Functions	57	(1)
Regular Bessel Function - Fractional Order	58	(1)
Irregular Bessel Functions - Fractional Order	58	(1)
Regular Modified Bessel Functions - Fractional Order	58	(1)
Irregular Modified Bessel Functions - Fractional Order	59	(1)
Zeros of Regular Bessel Functions	59	(1)
Clausen Functions	60	(1)
Coulomb Functions	60	(3)
Normalized Hydrogenic Bound States	60	(1)
Coulomb Wave Functions	61	(1)
Coulomb Wave Function Normalization Constant	62	(1)
Coupling Coefficients	63	(1)
3-j Symbols	63	(1)
6-j Symbols	63	(1)
9-j Symbols	64	(1)
Dawson Function	64	(1)
Debye Functions	64	(1)
Dilogarithm	65	(1)
Real Argument	65	(1)
Complex Argument	65	(1)
Elementary Operations	66	(1)
Elliptic Integrals	66	(3)
Definition of Legendre Forms	66	(1)
Definition of Carlson Forms	67	(1)
Legendre Form of Complete Elliptic Integrals	67	(1)
Legendre Form of Incomplete Elliptic Integrals	68	(1)
Carlson Forms	69	(1)
Elliptic Functions (Jacobi)	69	(1)
Error Functions	70	(1)
Error Function	70	(1)
Complementary Error Function	70	(1)
Log Complementary Error Function	70	(1)
Probability functions	70	(1)
Exponential Functions	71	(2)
Exponential Function	71	(1)
Relative Exponential Functions	71	(1)
Exponentiation With Error Estimate	72	(1)
Exponential Integrals	73	(2)
Exponential Integral	73	(1)
Ei(x)	73	(1)
Hyperbolic Integrals	74	(1)
Ei_3(x)	74	(1)
Trigonometric Integrals	74	(1)
Arctangent Integral	74	(1)
Fermi-Dirac Function	75	(1)
Complete Fermi-Dirac Integrals	75	(1)
Incomplete Fermi-Dirac Integrals	76	(1)
Gamma Function	76	(5)
Gegenbauer Functions	81	(1)
Hypergeometric Functions	82	(2)
Laguerre Functions	84	(1)
Lambert W Functions	85	(1)
Legendre Functions and Spherical Harmonics	85	(4)
Legendre Polynomials	85	(1)
Associated Legendre Polynomials and Spherical Harmonics	86	(1)
Conical Functions	87	(1)
Radial Functions for Hyperbolic Space	88	(1)
Logarithm and Related Functions	89	(1)
Power Function	90	(1)
Psi (Digamma) Function	91	(1)
Digamma Function	91	(1)
Trigamma Function	91	(1)
Polygamma Function	91	(1)
Synchrotron Functions	92	(1)
Transport Functions	92	(1)
Trigonometric Functions	93	(2)
Circular Trigonometric Functions	93	(1)
Trigonometric Functions for Complex Arguments	93	(1)
Hyperbolic Trigonometric Functions	94	(1)
Conversion Functions	94	(1)
Restriction Functions	94	(1)
Trigonometric Functions With Error Estimates	95	(1)
Zeta Functions	95	(1)
Riemann Zeta Function	95	(1)
Hurwitz Zeta Function	96	(1)
Eta Function	96	(1)
Examples	96	(2)
References and Further Reading	98	(1)
Vectors and Matrices	99	(32)
Data types	99	(1)
Blocks	100	(2)
Block allocation	100	(1)
Reading and writing blocks	101	(1)
Example programs for blocks	102	(1)
Vectors	102	(12)
Vector allocation	103	(1)
Accessing vector elements	104	(1)
Initializing vector elements	105	(1)
Reading and writing vectors	105	(1)
Vector views	106	(3)
Copying vectors	109	(1)
Exchanging elements	109	(1)
Vector operations	110	(1)
Finding maximum and minimum elements of vectors	111	(1)
Vector properties	111	(1)
Example programs for vectors	112	(2)
Matrices	114	(15)
Matrix allocation	115	(1)
Accessing matrix elements	115	(1)
Initializing matrix elements	116	(1)
Reading and writing matrices	117	(1)
Matrix views	118	(3)
Creating row and column views	121	(1)
Copying matrices	122	(1)
Copying rows and columns	123	(1)
Exchanging rows and columns	123	(1)
Matrix operations	124	(1)
Finding maximum and minimum elements of matrices	125	(1)
Matrix properties	126	(1)
Example programs for matrices	126	(3)
References and Further Reading	129	(2)
Permutations	131	(10)
The Permutation struct	131	(1)
Permutation allocation	131	(1)
Accessing permutation elements	132	(1)
Permutation properties	133	(1)
Permutation functions	133	(1)
Applying Permutations	134	(1)
Reading and writing permutations	135	(1)
Permutations in Cyclic Form	136	(1)
Examples	137	(2)
References and Further Reading	139	(2)
Combinations	141	(6)
The Combination struct	141	(1)
Combination allocation	141	(1)
Accessing combination elements	142	(1)
Combination properties	142	(1)
Combination functions	143	(1)
Reading and writing combinations	143	(1)
Examples	144	(3)
Sorting	147	(8)
Sorting objects	147	(2)
Sorting vectors	149	(1)
Selecting the k-th smallest or largest elements	150	(1)
Computing the rank	151	(1)
Examples	152	(2)
References and Further Reading	154	(1)
BLAS Support	155	(18)
GSL BLAS Interface	157	(13)
Level 1	157	(4)
Level 2	161	(4)
Level 3	165	(5)
Examples	170	(1)
References and Further Reading	171	(2)
Linear Algebra	173	(18)
LU Decomposition	173	(3)
QR Decomposition	176	(2)
QR Decomposition with Column Pivoting	178	(2)
Singular Value Decomposition	180	(2)
Cholesky Decomposition	182	(1)
Tridiagonal Decomposition of Real Symmetric Matrices	183	(1)
Tridiagonal Decomposition of Hermitian Matrices	184	(1)
Bidiagonalization	185	(1)
Householder solver for linear systems	186	(1)
Tridiagonal Systems	186	(1)
Examples	187	(1)
References and Further Reading	188	(3)
Eigensystems	191	(6)
Real Symmetric Matrices	191	(1)
Complex Hermitian Matrices	192	(1)
Sorting Eigenvalues and Eigenvectors	193	(1)
Examples	194	(2)
References and Further Reading	196	(1)
Fast Fourier Transforms (FFTs)	197	(22)
Mathematical Definitions	197	(1)
Overview of complex data FFTs	198	(1)
Radix-2 FFT routines for complex data	199	(4)
Mixed-radix FFT routines for complex data	203	(5)
Overview of real data FFTs	208	(1)
Radix-2 FFT routines for real data	209	(1)
Mixed-radix FFT routines for real data	210	(7)
References and Further Reading	217	(2)
Numerical Integration	219	(14)
Introduction	219	(2)
QNG non-adaptive Gauss-Kronrod integration	221	(1)
QAG adaptive integration	221	(1)
QAGS adaptive integration with singularities	222	(1)
QAGP adaptive integration with known singular points	223	(1)
QAGI adaptive integration on infinite intervals	224	(1)
QAWC adaptive integration for Cauchy principal values	225	(1)
QAWS adaptive integration for singular functions	225	(2)
QAWO adaptive integration for oscillatory functions	227	(1)
QAWF adaptive integration for Fourier integrals	228	(2)
Error codes	230	(1)
Examples	230	(1)
References and Further Reading	231	(2)
Random Number Generation	233	(20)
General comments on random numbers	233	(1)
The Random Number Generator Interface	234	(1)
Random number generator initialization	234	(1)
Sampling from a random number generator	235	(1)
Auxiliary random number generator functions	236	(1)
Random number environment variables	237	(2)
Saving and restoring random number generator state	239	(1)
Random number generator algorithms	239	(4)
Unix random number generators	243	(2)
Other random number generators	245	(5)
Performance	250	(1)
Examples	250	(2)
References and Further Reading	252	(1)
Acknowledgements	252	(1)
Quasi-Random Sequences	253	(6)
Quasi-random number generator initialization	253	(1)
Sampling from a quasi-random number generator	253	(1)
Auxiliary quasi-random number generator functions	254	(1)
Saving and resorting quasi-random number generator state	254	(1)
Quasi-random number generator algorithms	255	(1)
Examples	255	(2)
References	257	(2)
Random Number Distributions	259	(46)
The Gaussian Distribution	260	(2)
The Gaussian Tail Distribution	262	(2)
The Bivariate Gaussian Distribution	264	(1)
The Exponential Distribution	265	(1)
The Laplace Distribution	266	(1)
The Exponential Power Distribution	267	(1)
The Cauchy Distribution	268	(1)
The Rayleigh Distribution	269	(1)
The Rayleigh Tail Distribution	270	(1)
The Landau Distribution	271	(1)
The Levy alpha-Stable Distributions	272	(1)
The Levy skew alpha-Stable Distribution	273	(1)
The Gamma Distribution	274	(1)
The Flat (Uniform) Distribution	275	(1)
The Lognormal Distribution	276	(1)
The Chi-squared Distribution	277	(1)
The F-distribution	278	(1)
The t-distribution	279	(1)
The Beta Distribution	280	(1)
The Logistic Distribution	281	(1)
The Pareto Distribution	282	(1)
The Spherical Distribution (2D & 3D)	283	(2)
The Weibull Distribution	285	(1)
The Type-1 Gumbel Distribution	286	(1)
The Type-2 Gumbel Distribution	287	(1)
The Dirichlet Distribution	288	(1)
General Discrete Distributions	289	(2)
The Poisson Distribution	291	(1)
The Bernoulli Distribution	292	(1)
The Binomial Distribution	293	(1)
The Multinomial Distribution	294	(1)
The Negative Binomial Distribution	295	(1)
The Pascal Distribution	296	(1)
The Geometric Distribution	297	(1)
The Hypergeometric Distribution	298	(1)
The Logarithmic Distribution	299	(1)
Shuffling and Sampling	300	(1)
Examples	301	(3)
References and Further Reading	304	(1)
Statistics	305	(12)
Mean, Standard Deviation and Variance	305	(2)
Absolute deviation	307	(1)
Higher moments (skewness and kurtosis)	307	(1)
Autocorrelation	308	(1)
Covariance	309	(1)
Weighted Samples	309	(3)
Maximum and Minimum values	312	(1)
Median and Percentiles	313	(1)
Example statistical programs	314	(2)
References and Further Reading	316	(1)
Histograms	317	(24)
The histogram struct	317	(1)
Histogram allocation	318	(1)
Copying Histograms	319	(1)
Updating and accessing histogram elements	320	(1)
Searching histogram ranges	321	(1)
Histogram Statistics	321	(2)
Histogram Operations	323	(1)
Reading and writing histograms	324	(1)
Resampling from histograms	325	(1)
The histogram probability distribution struct	325	(2)
Example programs for histograms	327	(2)
Two dimensional histograms	329	(1)
The 2D histogram struct	329	(1)
2D Histogram allocation	330	(1)
Copying 2D Histograms	331	(1)
Updating and accessing 2D histogram elements	331	(2)
Searching 2D histogram ranges	333	(1)
2D Histogram Statistics	333	(2)
2D Histogram Operations	335	(1)
Reading and writing 2D histograms	336	(1)
Resampling from 2D histograms	337	(2)
Example programs for 2D histograms	339	(2)
N-tuples	341	(8)
The ntuple struct	341	(1)
Creating ntuples	341	(1)
Opening an existing ntuple file	342	(1)
Writing ntuples	342	(1)
Reading ntuples	342	(1)
Closing an ntuple file	342	(1)
Histogramming ntuple values	343	(1)
Example programs	344	(3)
References and Further Reading	347	(2)
Monte Carlo Integration	349	(14)
Interface	349	(2)
PLAIN Monte Carlo	351	(1)
MISER	352	(2)
VEGAS	354	(4)
Examples	358	(4)
References and Further Reading	362	(1)
Simulated Annealing	363	(12)
Simulated Annealing algorithm	363	(1)
Simulated Annealing functions	364	(2)
Examples with Simulated Annealing	366	(9)
Trivial example	366	(4)
Traveling Salesman Problem	370	(5)
Ordinary Differential Equations	375	(12)
Defining the ODE System	375	(1)
Stepping Functions	376	(2)
Adaptive Step-size Control	378	(2)
Evolution	380	(1)
Examples	381	(4)
References and Further Reading	385	(2)
Interpolation	387	(8)
Introduction	387	(1)
Interpolation Functions	387	(1)
Interpolation Types	388	(1)
Index Look-up and Acceleration	389	(1)
Evaluation of Interpolating Functions	389	(1)
Higher-level Interface	390	(1)
Examples	391	(2)
References and Further Reading	393	(2)
Numerical Differentiation	395	(4)
Functions	395	(1)
Examples	396	(1)
References and Further Reading	397	(2)
Chebyshev Approximations	399	(4)
The gsl_cheb_series struct	399	(1)
Creation and Calculation of Chebyshev Series	399	(1)
Chebyshev Series Evaluation	400	(1)
Derivatives and Integrals	400	(1)
Examples	401	(1)
References and Further Reading	402	(1)
Series Acceleration	403	(6)
Acceleration functions	403	(1)
Acceleration functions without error estimation	404	(1)
Example of accelerating a series	405	(2)
References and Further Reading	407	(2)
Discrete Hankel Transforms	409	(2)
Definitions	409	(1)
Functions	410	(1)
References and Further Reading	410	(1)
One dimensional Root-Finding	411	(18)
Overview	411	(1)
Caveats	412	(1)
Initializing the Solver	412	(2)
Providing the function to solve	414	(2)
Search Bounds and Guesses	416	(1)
Iteration	417	(1)
Search Stopping Parameters	418	(1)
Root Bracketing Algorithms	419	(2)
Root Finding Algorithms using Derivatives	421	(2)
Examples	423	(5)
References and Further Reading	428	(1)
One dimensional Minimization	429	(10)
Overview	429	(1)
Caveats	430	(1)
Initializing the Minimizer	431	(1)
Providing the function to minimize	432	(1)
Iteration	432	(1)
Stopping Parameters	433	(1)
Minimization Algorithms	434	(1)
Examples	435	(2)
References and Further Reading	437	(2)
Multidimensional Root-Finding	439	(18)
Overview	439	(1)
Initializing the Solver	440	(2)
Providing the function to solve	442	(3)
Iteration	445	(1)
Search Stopping Parameters	446	(1)
Algorithms using Derivatives	447	(2)
Algorithms without Derivatives	449	(1)
Examples	450	(6)
References and Further Reading	456	(1)
Multidimensional Minimization	457	(16)
Overview	457	(1)
Caveats	458	(1)
Initializing the Multidimensional Minimizer	458	(2)
Providing a function to minimize	460	(2)
Iteration	462	(1)
Stopping Criteria	463	(1)
Algorithms	464	(2)
Examples	466	(5)
References and Further Reading	471	(2)
Least-Squares Fitting	473	(12)
Linear regression	473	(1)
Linear fitting without a constant term	474	(1)
Multi-parameter fitting	475	(2)
Examples	477	(6)
References and Further Reading	483	(2)
Nonlinear Least-Squares Fitting	485	(14)
Overview	485	(1)
Initializing the Solver	486	(1)
Providing the Function to be Minimized	487	(1)
Iteration	488	(1)
Search Stopping Parameters	489	(1)
Minimization Algorithms using Derivatives	490	(1)
Minimization Algorithms without Derivatives	491	(1)
Computing the covariance matrix of best fit parameters	491	(1)
Examples	492	(5)
References and Further Reading	497	(2)
Physical Constants	499	(10)
Fundamental Constants	499	(1)
Astronomy and Astrophysics	500	(1)
Atomic and Nuclear Physics	500	(1)
Measurement of Time	501	(1)
Imperial Units	502	(1)
Nautical Units	502	(1)
Printers Units	502	(1)
Volume	503	(1)
Mass and Weight	503	(1)
Thermal Energy and Power	504	(1)
Pressure	504	(1)
Viscosity	505	(1)
Light and Illumination	505	(1)
Radioactivity	505	(1)
Force and Energy	506	(1)
Prefixes	506	(1)
Examples	507	(1)
References and Further Reading	508	(1)
IEEE floating-point arithmetic	509	(8)
Representation of floating point numbers	509	(2)
Setting up your IEEE environment	511	(4)
References and Further Reading	515	(2)
Appendix A Debugging Numerical Programs	517	(6)
A.1 Using gdb	517	(2)
A.2 Examining floating point registers	519	(1)
A.3 Handling floating point exceptions	519	(1)
A.4 GCC warning options for numerical programs	520	(2)
A.5 References and Further Reading	522	(1)
Appendix B Contributors to GSL	523	(2)
Appendix C Autoconf Macros	525	(4)
C.1 Using autoconf to link with GSL	525	(1)
C.2 Using autoconf to test for inline	526	(1)
C.3 Using autoconf to substitute portability functions	527	(2)
Appendix D GSL CBLAS Library	529	(20)
D.1 Level 1	529	(3)
D.2 Level 2	532	(9)
D.3 Level 3	541	(5)
D.4 Examples	546	(3)
GNU General Public License	549	(10)
Preamble	549	(1)
Terms and Conditions for Copying, Distribution and Modification	550	(6)
Appendix: How to Apply These Terms to Your New Programs	556	(3)
GNU Free Documentation License	559	(8)
Addendum: How to use this License for your documents	566	(1)
Function Index	567	(20)
Variable Index	587	(2)
Type Index	589	(2)
Concept Index	591