Gnu Scientific Library Reference Manual

by Galassi, Mark; Davies, Jim; Theiler, James; Gough, Brian; Jungman, Gerard; Booth, Michael; Rossi, Fabrice

• ISBN13: 9780954161705
• ISBN10: 095416170X
• Format: Paperback
• Copyright: 2001-12-01
• Publisher: Network Theory Ltd

Summary

The GNU Scientific Library (GSL) is a free numerical library for C and C++ programmers. This reference manual is the definitive guide to the library.

Table of Contents

Preface	1	(2)
Introduction	3	(4)
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)
Further Information	6	(1)
Using the library	7	(8)
ANSI C Compliance	7	(1)
Compiling and Linking	7	(1)
Shared Libraries	8	(1)
Autoconf macros	8	(2)
Inline functions	10	(1)
Long double	10	(1)
Portability functions	11	(1)
Alternative optimized functions	12	(1)
Support for different numeric types	12	(2)
Compatibility with C++	14	(1)
Aliasing of arrays	14	(1)
Code Reuse	14	(1)
Error Handling	15	(6)
Error Reporting	15	(2)
Error Handlers	17	(2)
Using GSL error reporting in your own functions	19	(2)
Mathematical Functions	21	(6)
Mathematical Constants	21	(1)
Infinities and Not-a-number	22	(1)
Elementary Functions	22	(1)
Small integer powers	23	(1)
Testing the Sign of Numbers	24	(1)
Testing for Odd and Even Numbers	24	(1)
Maximum and Minimum functions	24	(3)
Complex Numbers	27	(10)
Complex Numbers	27	(1)
Properties of complex numbers	28	(1)
Complex arithmetic operators	29	(2)
Elementary Complex Functions	31	(1)
Complex Trigonometric Functions	32	(1)
Inverse Complex Trigonometric Functions	32	(2)
Complex Hyperbolic Functions	34	(1)
Inverse Complex Hyperbolic Functions	35	(1)
References and Further Reading	36	(1)
Polynomials	37	(6)
Polynomial Evaluation	37	(1)
Divided Difference Representation of Polynomials	37	(1)
Quadratic Equations	38	(1)
Cubic Equations	39	(1)
General Polynomial Equations	39	(1)
Examples	40	(1)
References and Further Reading	41	(2)
Special Functions	43	(54)
Usage	43	(1)
The gsl_sf_result struct	44	(1)
Modes	44	(1)
Airy Functions and Derivatives	45	(2)
Airy Functions	45	(1)
Derivatives of Airy Functions	46	(1)
Zeros of Airy Functions	47	(1)
Zeros of Derivatives of Airy Functions	47	(1)
Bessel Functions	47	(11)
Regular Cylindrical Bessel Functions	48	(1)
Irregular Cylindrical Bessel Functions	48	(1)
Regular Modified Cylindrical Bessel Functions	49	(1)
Irregular Modified Cylindrical Bessel Functions	50	(2)
Regular Spherical Bessel Functions	52	(1)
Irregular Spherical Bessel Functions	53	(1)
Regular Modified Spherical Bessel Functions	54	(1)
Irregular Modified Spherical Bessel Functions	55	(1)
Regular Bessel Function - Fractional Order	56	(1)
Irregular Bessel Function - Fractional Order	56	(1)
Regular Modified Bessel Functions-Fractional Order	56	(1)
Irregular Modified Bessel Functions-Fractional Order	57	(1)
Zeros of Regular Bessel Functions	57	(1)
Clausen Functions	58	(1)
Coulomb Functions	58	(3)
Normalized Hydrogenic Bound States	58	(1)
Coulomb Wave Functions	59	(1)
Coulomb Wave Function Normalization Constant	60	(1)
Coupling Coefficients	61	(1)
3-j Symbols	61	(1)
6-j Symbols	61	(1)
9-j Symbols	62	(1)
Dawson Function	62	(1)
Debye Functions	62	(1)
Dilogarithm	63	(1)
Real Argument	63	(1)
Complex Argument	63	(1)
Elementary Operations	64	(1)
Elliptic Integrals	64	(3)
Definition of Legendre Forms	64	(1)
Definition of Carlson Forms	65	(1)
Legendre Form of Complete Elliptic Integrals	65	(1)
Legendre Form of Incomplete Elliptic Integrals	65	(1)
Carlson Forms	66	(1)
Elliptic Functions (Jacobi)	67	(1)
Error Functions	67	(2)
Error Function	67	(1)
Complementary Error Function	68	(1)
Log Complementary Error Function	68	(1)
Probability functions	68	(1)
Exponential Functions	69	(2)
Exponential Function	69	(1)
Relative Exponential Functions	69	(1)
Exponentiation With Error Estimate	70	(1)
Exponential Integrals	71	(2)
Exponential Integral	71	(1)
Ei(x)	71	(1)
Hyperbolic Integrals	72	(1)
Ei_3(x)	72	(1)
Trigonometric Integrals	72	(1)
Arctangent Integral	72	(1)
Fermi-Dirac Function	73	(2)
Complete Fermi-Dirac Integrals	73	(1)
Incomplete Fermi-Dirac Integrals	74	(1)
Gamma Function	75	(4)
Gegenbauer Functions	79	(1)
Hypergeometric Functions	80	(2)
Laguerre Functions	82	(1)
Lambert W Functions	82	(1)
Legendre Functions and Spherical Harmonics	83	(4)
Legendre Polynomials	83	(1)
Associated Legendre Polynomials and Spherical Harmonics	84	(1)
Conical Functions	85	(1)
Radial Functions for Hyperbolic Space	86	(1)
Logarithm and Related Functions	87	(1)
Power Function	88	(1)
Psi (Digamma) Function	88	(1)
Digamma Function	88	(1)
Trigamma Function	89	(1)
Polygamma Function	89	(1)
Synchrotron Functions	89	(1)
Transport Functions	90	(1)
Trigonometric Functions	90	(3)
Circular Trigonometric Functions	91	(1)
Trigonometric Functions for Complex Arguments	91	(1)
Hyperbolic Trigonometric Functions	92	(1)
Conversion Functions	92	(1)
Restriction Functions	92	(1)
Trigonometric Functions With Error Estimates	93	(1)
Zeta Functions	93	(1)
Riemann Zeta Function	93	(1)
Hurwitz Zeta Function	94	(1)
Eta Function	94	(1)
Examples	94	(1)
References and Further Reading	95	(2)
Vectors and Matrices	97	(32)
Data types	97	(1)
Blocks	98	(2)
Block allocation	98	(1)
Reading and writing blocks	99	(1)
Example programs for blocks	100	(1)
Vectors	100	(11)
Vector allocation	101	(1)
Accessing vector elements	102	(1)
Initializing vector elements	103	(1)
Reading and writing vectors	103	(1)
Vector views	104	(3)
Copying vectors	107	(1)
Exchanging elements	108	(1)
Vector operations	108	(1)
Finding maximum and minimum elements of vectors	109	(1)
Vector properties	110	(1)
Example programs for vectors	110	(1)
Matrices	111	(16)
Matrix allocation	112	(1)
Accessing matrix elements	113	(1)
Initializing matrix elements	114	(1)
Reading and writing matrices	114	(1)
Matrix views	115	(4)
Creating row and column views	119	(1)
Copying matrices	120	(1)
Copying rows and columns	121	(1)
Exchanging rows and columns	121	(1)
Matrix operations	122	(1)
Finding maximum and minimum elements of matrices	123	(1)
Matrix properties	124	(1)
Example programs for matrices	124	(3)
References and Further Reading	127	(2)
Permutations	129	(8)
The Permutation struct	129	(1)
Permutation allocation	129	(1)
Accessing permutation elements	130	(1)
Permutation properties	131	(1)
Permutation functions	131	(1)
Applying Permutations	132	(1)
Reading and writing permutations	133	(1)
Examples	134	(1)
References and Further Reading	135	(2)
Sorting	137	(8)
Sorting objects	137	(1)
Sorting vectors	138	(2)
Selecting the k-th smallest or largest elements	140	(1)
Computing the rank	141	(1)
Examples	142	(1)
References and Further Reading	143	(2)
BLAS Support	145	(18)
GSL BLAS Interface	146	(15)
Level 1	147	(4)
Level 2	151	(4)
Level 3	155	(6)
Examples	161	(1)
References and Further Reading	162	(1)
Linear Algebra	163	(18)
LU Decomposition	163	(3)
QR Decomposition	166	(3)
QR Decomposition with Column Pivoting	169	(2)
Singular Value Decomposition	171	(1)
Cholesky Decomposition	172	(1)
Tridiagonal Decomposition of Real Symmetric Matrices	173	(1)
Tridiagonal Decomposition of Hermitian Matrices	174	(1)
Bidiagonalization	175	(1)
Householder solver for linear systems	176	(1)
Tridiagonal Systems	176	(1)
Examples	177	(2)
References and Further Reading	179	(2)
Eigensystems	181	(6)
Real Symmetric Matrices	181	(1)
Complex Hermitian Matrices	182	(1)
Sorting Eigenvalues and Eigenvectors	183	(1)
Examples	184	(2)
References and Further Reading	186	(1)
Fast Fourier Transforms (FFTs)	187	(22)
Mathematical Definitions	187	(1)
Overview of complex data FFTs	188	(1)
Radix-2 FFT routines for complex data	189	(4)
Mixed-radix FFT routines for complex data	193	(4)
Overview of real data FFTs	197	(2)
Radix-2 FFT routines for real data	199	(1)
Mixed-radix FFT routines for real data	200	(6)
References and Further Reading	206	(3)
Numerical Integration	209	(14)
Introduction	209	(2)
QNG non-adaptive Gauss-Kronrod integration	211	(1)
QAG adaptive integration	211	(1)
QAGS adaptive integration with singularities	212	(1)
QAGP adaptive integration with known singular points	213	(1)
QAGI adaptive integration on infinite intervals	214	(1)
QAWC adaptive integration for Cauchy principal values	215	(1)
QAWS adaptive integration for singular functions	215	(2)
QAWO adaptive integration for oscillatory functions	217	(1)
QAWF adaptive integration for Fourier integrals	218	(2)
Error codes	220	(1)
Examples	220	(1)
References and Further Reading	221	(2)
Random Number Generation	223	(22)
General comments on random numbers	223	(1)
The Random Number Generator Interface	224	(1)
Random number generator initialization	224	(1)
Sampling from a random number generator	225	(1)
Auxiliary random number generator functions	226	(2)
Random number environment variables	228	(1)
Saving and restoring random number generator state	229	(1)
Random number generator algorithms	230	(4)
Unix random number generators	234	(1)
Numerical Recipes generators	235	(1)
Other random number generators	236	(5)
Random Number Generator Performance	241	(1)
Examples	242	(1)
References and Further Reading	243	(1)
Acknowledgements	244	(1)
Quasi-Random Sequences	245	(4)
Quasi-random number generator initialization	245	(1)
Sampling from a quasi-random number generator	245	(1)
Auxiliary quasi-random number generator functions	246	(1)
Saving and resorting quasi-random number generator state	246	(1)
Quasi-random number generator algorithms	247	(1)
Examples	247	(1)
References	248	(1)
Random Number Distributions	249	(44)
The Gaussian Distribution	250	(2)
The Gaussian Tail Distribution	252	(2)
The Bivariate Gaussian Distribution	254	(1)
The Exponential Distribution	255	(1)
The Laplace Distribution	256	(1)
The Exponential Power Distribution	257	(1)
The Cauchy Distribution	258	(1)
The Rayleigh Distribution	259	(1)
The Rayleigh Tail Distribution	260	(1)
The Landan Distribution	261	(1)
The Levy alpha-Stable Distributions	262	(1)
The Levy skew alpha-Stable Distribution	263	(1)
The Gamma Distribution	264	(1)
The Flat (Uniform) Distribution	265	(1)
The Lognormal Distribution	266	(1)
The Chi-squared Distribution	267	(1)
The F-distribution	268	(1)
The t-distribution	269	(1)
The Beta Distribution	270	(1)
The Logistic Distribution	271	(1)
The Pareto Distribution	272	(1)
The Spherical Distribution (2D & 3D)	273	(2)
The Weibull Distribution	275	(1)
The Type-1 Gumbel Distribution	276	(1)
The Type-2 Gumbel Distribution	277	(1)
General Discrete Distributions	278	(2)
The Poisson Distribution	280	(1)
The Bernoulli Distribution	281	(1)
The Binomial Distribution	282	(1)
The Negative Binomial Distribution	283	(1)
The Pascal Distribution	284	(1)
The Geometric Distribution	285	(1)
The Hypergeometric Distribution	286	(1)
The Logarithmic Distribution	287	(1)
Shuffling and Sampling	288	(1)
Examples	289	(2)
References and Further Reading	291	(2)
Statistics	293	(14)
Mean, Standard Deviation and Variance	293	(2)
Absolute deviation	295	(1)
Higher moments (skewness and kurtosis)	295	(2)
Autocorrelation	297	(1)
Covariance	297	(1)
Weighted Samples	298	(3)
Maximum and Minimum values	301	(1)
Median and Percentiles	302	(1)
Example statistical programs	303	(2)
References and Further Reading	305	(2)
Histograms	307	(24)
The histogram struct	307	(1)
Histogram allocation	308	(1)
Copying Histograms	309	(1)
Updating and accessing histogram elements	310	(1)
Searching histogram ranges	311	(1)
Histogram Statistics	312	(1)
Histogram Operations	313	(1)
Reading and writing histograms	314	(1)
Resampling from histograms	315	(1)
The histogram probability distribution struct	315	(2)
Example programs for histograms	317	(2)
Two dimensional histograms	319	(1)
The 2D histogram struct	319	(1)
2D Histogram allocation	320	(1)
Copying 2D Histograms	321	(1)
Updating and accessing 2D histogram elements	321	(2)
Searching 2D histogram ranges	323	(1)
2D Histogram Statistics	324	(1)
2D Histogram Operations	324	(1)
Reading and writing 2D histograms	325	(2)
Resampling from 2D histograms	327	(2)
Example programs for 2D histograms	329	(2)
N-tuples	331	(8)
The ntuple struct	331	(1)
Creating ntuples	331	(1)
Opening an existing ntuple file	332	(1)
Writing ntuples	332	(1)
Reading ntuples	332	(1)
Closing an ntuple file	332	(1)
Histogramming ntuple values	333	(1)
Example programs	334	(3)
References and Further Reading	337	(2)
Monte Carlo Integration	339	(14)
Interface	339	(2)
PLAIN Monte Carlo	341	(1)
MISER	342	(3)
VEGAS	345	(4)
Examples	349	(3)
References and Further Reading	352	(1)
Simulated Annealing	353	(10)
Simulated Annealing algorithm	353	(1)
Simulated Annealing functions	354	(2)
Examples with Simulated Annealing	356	(7)
Trivial example	356	(3)
Traveling Salesman Problem	359	(4)
Ordinary Differential Equations	363	(10)
Defining the ODE System	363	(1)
Stepping Functions	364	(2)
Adaptive Step-size Control	366	(2)
Evolution	368	(1)
Examples	369	(3)
References and Further Reading	372	(1)
Interpolation	373	(8)
Introduction	373	(1)
Interpolation Functions	373	(1)
Interpolation Types	374	(1)
Index Look-up and Acceleration	375	(1)
Evaluation of interpolating functions	376	(1)
Higher-level interface	377	(1)
Examples	378	(1)
References and Further Reading	379	(2)
Numerical Differentiation	381	(4)
Functions	381	(1)
Example	382	(1)
References and Further Reading	383	(2)
Chebyshev Approximations	385	(4)
The gsl_cheb_series struct	385	(1)
Creation and Calculation of Chebyshev Series	385	(1)
Chebyshev Series Evaluation	386	(1)
Derivatives and Integrals	386	(1)
Examples	387	(1)
References and Further Reading	388	(1)
Series Acceleration	389	(6)
Acceleration functions	389	(1)
Acceleration functions without error estimation	390	(1)
Example of accelerating a series	391	(2)
References and Further Reading	393	(2)
Discrete Hankel Transforms	395	(2)
Definitions	395	(1)
Functions	396	(1)
References and Further Reading	396	(1)
One dimensional Root-Finding	397	(18)
Overview	397	(1)
Caveats	398	(1)
Initializing the Solver	399	(1)
Providing the function to solve	400	(3)
Search Bounds and Guesses	403	(1)
Iteration	403	(1)
Search Stopping Parameters	404	(1)
Root Bracketing Algorithms	405	(2)
Root Finding Algorithms using Derivatives	407	(2)
Examples	409	(5)
References and Further Reading	414	(1)
One dimensional Minimization	415	(10)
Overview	415	(1)
Caveats	416	(1)
Initializing the Minimizer	417	(1)
Providing the function to minimize	418	(1)
Iteration	418	(1)
Stopping Parameters	419	(1)
Minimization Algorithms	420	(1)
Examples	421	(2)
References and Further Reading	423	(2)
Multidimensional Root-Finding	425	(18)
Overview	425	(1)
Initializing the Solver	426	(2)
Providing the function to solve	428	(3)
Iteration	431	(1)
Search Stopping Parameters	432	(1)
Algorithms using Derivatives	433	(2)
Algorithms without Derivatives	435	(1)
Examples	436	(5)
References and Further Reading	441	(2)
Multidimensional Minimization	443	(10)
Overview	443	(1)
Caveats	444	(1)
Initializing the Multidimensional Minimizer	444	(1)
Providing a function to minimize	445	(2)
Iteration	447	(1)
Stopping Criteria	448	(1)
Algorithms	449	(1)
Examples	450	(2)
References and Further Reading	452	(1)
Least-Squares Fitting	453	(12)
Linear regression	453	(1)
Linear fitting without a constant term	454	(1)
Multi-parameter fitting	455	(2)
Examples	457	(6)
References and Further Reading	463	(2)
Nonlinear Least-Squares Fitting	465	(14)
Overview	465	(1)
Initializing the Solver	466	(1)
Providing the Function to be Minimized	467	(1)
Iteration	468	(1)
Search Stopping Parameters	469	(1)
Minimization Algorithms using Derivatives	470	(1)
Minimization Algorithms without Derivatives	471	(1)
Computing the covariance matrix of best fit parameters	472	(1)
Examples	472	(5)
References and Further Reading	477	(2)
Physical Constants	479	(8)
Fundamental Constants	479	(1)
Astronomy and Astrophysics	480	(1)
Atomic and Nuclear Physics	480	(1)
Measurement of Time	481	(1)
Imperial Units	481	(1)
Nautical Units	482	(1)
Printers Units	482	(1)
Volume	482	(1)
Mass and Weight	483	(1)
Thermal Energy and Power	483	(1)
Pressure	484	(1)
Viscosity	484	(1)
Light and Illumination	484	(1)
Radioactivity	485	(1)
Examples	485	(1)
References and Further Reading	486	(1)
IEEE floating-point arithmetic	487	(8)
Representation of floating point numbers	487	(2)
Setting up your IEEE environment	489	(4)
References and Further Reading	493	(2)
Appendix A Debugging Numerical Programs	495	(6)
A.1 Using gdb	495	(2)
A.2 Examining floating point registers	497	(1)
A.3 Handling floating point exceptions	497	(1)
A.4 GCC warning options for numerical programs	498	(2)
A.5 References and Further Reading	500	(1)
Appendix B Contributors to GSL	501	(2)
Appendix C Autoconf Macros	503	(2)
Appendix D GSL CBLAS Library	505	(20)
D.1 Level 1	505	(3)
D.2 Level 2	508	(9)
D.3 Level 3	517	(5)
D.4 Examples	522	(3)
GNU General Public License	525	(10)
Preamble	525	(1)
Terms and Conditions for Copying, Distribution and Modification	526	(6)
Appendix: How to Apply These Terms to Your New Programs	532	(3)
GNU Free Documentation License	535	(10)
Addendum: How to use this License for your documents	543	(2)
Other Scientific Packages for the GNU System	545	(4)
GNU Octave	545	(1)
GNU R	545	(2)
GNU Multiprecision Library	547	(1)
GNU Linear Programming Kit	547	(2)
Function Index	549	(18)
Variable Index	567	(2)
Type Index	569	(2)
Concept Index	571