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Table of Contents
1. Introduction
1.1 Routines available in GSL
1.2 GSL is Free Software
1.3 Obtaining GSL
1.4 No Warranty
1.5 Reporting Bugs
1.6 Further Information
1.7 Conventions used in this manual
2. Using the library
2.1 An Example Program
2.2 Compiling and Linking
2.3 Shared Libraries
2.4 ANSI C Compliance
2.5 Inline functions
2.6 Long double
2.7 Portability functions
2.8 Alternative optimized functions
2.9 Support for different numeric types
2.10 Compatibility with C++
2.11 Aliasing of arrays
2.12 Thread-safety
2.13 Code Reuse
3. Error Handling
3.1 Error Reporting
3.2 Error Codes
3.3 Error Handlers
3.4 Using GSL error reporting in your own functions
3.5 Examples
4. Mathematical Functions
4.1 Mathematical Constants
4.2 Infinities and Not-a-number
4.3 Elementary Functions
4.4 Small integer powers
4.5 Testing the Sign of Numbers
4.6 Testing for Odd and Even Numbers
4.7 Maximum and Minimum functions
4.8 Approximate Comparison of Floating Point Numbers
5. Complex Numbers
5.1 Complex numbers
5.2 Properties of complex numbers
5.3 Complex arithmetic operators
5.4 Elementary Complex Functions
5.5 Complex Trigonometric Functions
5.6 Inverse Complex Trigonometric Functions
5.7 Complex Hyperbolic Functions
5.8 Inverse Complex Hyperbolic Functions
5.9 References and Further Reading
6. Polynomials
6.1 Polynomial Evaluation
6.2 Divided Difference Representation of Polynomials
6.3 Quadratic Equations
6.4 Cubic Equations
6.5 General Polynomial Equations
6.6 Examples
6.7 References and Further Reading
7. Special Functions
7.1 Usage
7.2 The gsl_sf_result struct
7.3 Modes
7.4 Airy Functions and Derivatives
7.4.1 Airy Functions
7.4.2 Derivatives of Airy Functions
7.4.3 Zeros of Airy Functions
7.4.4 Zeros of Derivatives of Airy Functions
7.5 Bessel Functions
7.5.1 Regular Cylindrical Bessel Functions
7.5.2 Irregular Cylindrical Bessel Functions
7.5.3 Regular Modified Cylindrical Bessel Functions
7.5.4 Irregular Modified Cylindrical Bessel Functions
7.5.5 Regular Spherical Bessel Functions
7.5.6 Irregular Spherical Bessel Functions
7.5.7 Regular Modified Spherical Bessel Functions
7.5.8 Irregular Modified Spherical Bessel Functions
7.5.9 Regular Bessel Function - Fractional Order
7.5.10 Irregular Bessel Functions - Fractional Order
7.5.11 Regular Modified Bessel Functions - Fractional Order
7.5.12 Irregular Modified Bessel Functions - Fractional Order
7.5.13 Zeros of Regular Bessel Functions
7.6 Clausen Functions
7.7 Coulomb Functions
7.7.1 Normalized Hydrogenic Bound States
7.7.2 Coulomb Wave Functions
7.7.3 Coulomb Wave Function Normalization Constant
7.8 Coupling Coefficients
7.8.1 3-j Symbols
7.8.2 6-j Symbols
7.8.3 9-j Symbols
7.9 Dawson Function
7.10 Debye Functions
7.11 Dilogarithm
7.11.1 Real Argument
7.11.2 Complex Argument
7.12 Elementary Operations
7.13 Elliptic Integrals
7.13.1 Definition of Legendre Forms
7.13.2 Definition of Carlson Forms
7.13.3 Legendre Form of Complete Elliptic Integrals
7.13.4 Legendre Form of Incomplete Elliptic Integrals
7.13.5 Carlson Forms
7.14 Elliptic Functions (Jacobi)
7.15 Error Functions
7.15.1 Error Function
7.15.2 Complementary Error Function
7.15.3 Log Complementary Error Function
7.15.4 Probability functions
7.16 Exponential Functions
7.16.1 Exponential Function
7.16.2 Relative Exponential Functions
7.16.3 Exponentiation With Error Estimate
7.17 Exponential Integrals
7.17.1 Exponential Integral
7.17.2 Ei(x)
7.17.3 Hyperbolic Integrals
7.17.4 Ei_3(x)
7.17.5 Trigonometric Integrals
7.17.6 Arctangent Integral
7.18 Fermi-Dirac Function
7.18.1 Complete Fermi-Dirac Integrals
7.18.2 Incomplete Fermi-Dirac Integrals
7.19 Gamma Function
7.20 Gegenbauer Functions
7.21 Hypergeometric Functions
7.22 Laguerre Functions
7.23 Lambert W Functions
7.24 Legendre Functions and Spherical Harmonics
7.24.1 Legendre Polynomials
7.24.2 Associated Legendre Polynomials and Spherical Harmonics
7.24.3 Conical Functions
7.24.4 Radial Functions for Hyperbolic Space
7.25 Logarithm and Related Functions
7.26 Power Function
7.27 Psi (Digamma) Function
7.27.1 Digamma Function
7.27.2 Trigamma Function
7.27.3 Polygamma Function
7.28 Synchrotron Functions
7.29 Transport Functions
7.30 Trigonometric Functions
7.30.1 Circular Trigonometric Functions
7.30.2 Trigonometric Functions for Complex Arguments
7.30.3 Hyperbolic Trigonometric Functions
7.30.4 Conversion Functions
7.30.5 Restriction Functions
7.30.6 Trigonometric Functions With Error Estimates
7.31 Zeta Functions
7.31.1 Riemann Zeta Function
7.31.2 Riemann Zeta Function Minus One
7.31.3 Hurwitz Zeta Function
7.31.4 Eta Function
7.32 Examples
7.33 References and Further Reading
8. Vectors and Matrices
8.1 Data types
8.2 Blocks
8.2.1 Block allocation
8.2.2 Reading and writing blocks
8.2.3 Example programs for blocks
8.3 Vectors
8.3.1 Vector allocation
8.3.2 Accessing vector elements
8.3.3 Initializing vector elements
8.3.4 Reading and writing vectors
8.3.5 Vector views
8.3.6 Copying vectors
8.3.7 Exchanging elements
8.3.8 Vector operations
8.3.9 Finding maximum and minimum elements of vectors
8.3.10 Vector properties
8.3.11 Example programs for vectors
8.4 Matrices
8.4.1 Matrix allocation
8.4.2 Accessing matrix elements
8.4.3 Initializing matrix elements
8.4.4 Reading and writing matrices
8.4.5 Matrix views
8.4.6 Creating row and column views
8.4.7 Copying matrices
8.4.8 Copying rows and columns
8.4.9 Exchanging rows and columns
8.4.10 Matrix operations
8.4.11 Finding maximum and minimum elements of matrices
8.4.12 Matrix properties
8.4.13 Example programs for matrices
8.5 References and Further Reading
9. Permutations
9.1 The Permutation struct
9.2 Permutation allocation
9.3 Accessing permutation elements
9.4 Permutation properties
9.5 Permutation functions
9.6 Applying Permutations
9.7 Reading and writing permutations
9.8 Permutations in Cyclic Form
9.9 Examples
9.10 References and Further Reading
10. Combinations
10.1 The Combination struct
10.2 Combination allocation
10.3 Accessing combination elements
10.4 Combination properties
10.5 Combination functions
10.6 Reading and writing combinations
10.7 Examples
10.8 References and Further Reading
11. Sorting
11.1 Sorting objects
11.2 Sorting vectors
11.3 Selecting the k smallest or largest elements
11.4 Computing the rank
11.5 Examples
11.6 References and Further Reading
12. BLAS Support
12.1 GSL BLAS Interface
12.1.1 Level 1
12.1.2 Level 2
12.1.3 Level 3
12.2 Examples
12.3 References and Further Reading
13. Linear Algebra
13.1 LU Decomposition
13.2 QR Decomposition
13.3 QR Decomposition with Column Pivoting
13.4 Singular Value Decomposition
13.5 Cholesky Decomposition
13.6 Tridiagonal Decomposition of Real Symmetric Matrices
13.7 Tridiagonal Decomposition of Hermitian Matrices
13.8 Bidiagonalization
13.9 Householder Transformations
13.10 Householder solver for linear systems
13.11 Tridiagonal Systems
13.12 Examples
13.13 References and Further Reading
14. Eigensystems
14.1 Real Symmetric Matrices
14.2 Complex Hermitian Matrices
14.3 Sorting Eigenvalues and Eigenvectors
14.4 Examples
14.5 References and Further Reading
15. Fast Fourier Transforms (FFTs)
15.1 Mathematical Definitions
15.2 Overview of complex data FFTs
15.3 Radix-2 FFT routines for complex data
15.4 Mixed-radix FFT routines for complex data
15.5 Overview of real data FFTs
15.6 Radix-2 FFT routines for real data
15.7 Mixed-radix FFT routines for real data
15.8 References and Further Reading
16. Numerical Integration
16.1 Introduction
16.2 QNG non-adaptive Gauss-Kronrod integration
16.3 QAG adaptive integration
16.4 QAGS adaptive integration with singularities
16.5 QAGP adaptive integration with known singular points
16.6 QAGI adaptive integration on infinite intervals
16.7 QAWC adaptive integration for Cauchy principal values
16.8 QAWS adaptive integration for singular functions
16.9 QAWO adaptive integration for oscillatory functions
16.10 QAWF adaptive integration for Fourier integrals
16.11 Error codes
16.12 Examples
16.13 References and Further Reading
17. Random Number Generation
17.1 General comments on random numbers
17.2 The Random Number Generator Interface
17.3 Random number generator initialization
17.4 Sampling from a random number generator
17.5 Auxiliary random number generator functions
17.6 Random number environment variables
17.7 Copying random number generator state
17.8 Reading and writing random number generator state
17.9 Random number generator algorithms
17.10 Unix random number generators
17.11 Other random number generators
17.12 Performance
17.13 Examples
17.14 References and Further Reading
17.15 Acknowledgements
18. Quasi-Random Sequences
18.1 Quasi-random number generator initialization
18.2 Sampling from a quasi-random number generator
18.3 Auxiliary quasi-random number generator functions
18.4 Saving and resorting quasi-random number generator state
18.5 Quasi-random number generator algorithms
18.6 Examples
18.7 References
19. Random Number Distributions
19.1 Introduction
19.2 The Gaussian Distribution
19.3 The Gaussian Tail Distribution
19.4 The Bivariate Gaussian Distribution
19.5 The Exponential Distribution
19.6 The Laplace Distribution
19.7 The Exponential Power Distribution
19.8 The Cauchy Distribution
19.9 The Rayleigh Distribution
19.10 The Rayleigh Tail Distribution
19.11 The Landau Distribution
19.12 The Levy alpha-Stable Distributions
19.13 The Levy skew alpha-Stable Distribution
19.14 The Gamma Distribution
19.15 The Flat (Uniform) Distribution
19.16 The Lognormal Distribution
19.17 The Chi-squared Distribution
19.18 The F-distribution
19.19 The t-distribution
19.20 The Beta Distribution
19.21 The Logistic Distribution
19.22 The Pareto Distribution
19.23 Spherical Vector Distributions
19.24 The Weibull Distribution
19.25 The Type-1 Gumbel Distribution
19.26 The Type-2 Gumbel Distribution
19.27 The Dirichlet Distribution
19.28 General Discrete Distributions
19.29 The Poisson Distribution
19.30 The Bernoulli Distribution
19.31 The Binomial Distribution
19.32 The Multinomial Distribution
19.33 The Negative Binomial Distribution
19.34 The Pascal Distribution
19.35 The Geometric Distribution
19.36 The Hypergeometric Distribution
19.37 The Logarithmic Distribution
19.38 Shuffling and Sampling
19.39 Examples
19.40 References and Further Reading
20. Statistics
20.1 Mean, Standard Deviation and Variance
20.2 Absolute deviation
20.3 Higher moments (skewness and kurtosis)
20.4 Autocorrelation
20.5 Covariance
20.6 Weighted Samples
20.7 Maximum and Minimum values
20.8 Median and Percentiles
20.9 Examples
20.10 References and Further Reading
21. Histograms
21.1 The histogram struct
21.2 Histogram allocation
21.3 Copying Histograms
21.4 Updating and accessing histogram elements
21.5 Searching histogram ranges
21.6 Histogram Statistics
21.7 Histogram Operations
21.8 Reading and writing histograms
21.9 Resampling from histograms
21.10 The histogram probability distribution struct
21.11 Example programs for histograms
21.12 Two dimensional histograms
21.13 The 2D histogram struct
21.14 2D Histogram allocation
21.15 Copying 2D Histograms
21.16 Updating and accessing 2D histogram elements
21.17 Searching 2D histogram ranges
21.18 2D Histogram Statistics
21.19 2D Histogram Operations
21.20 Reading and writing 2D histograms
21.21 Resampling from 2D histograms
21.22 Example programs for 2D histograms
22. N-tuples
22.1 The ntuple struct
22.2 Creating ntuples
22.3 Opening an existing ntuple file
22.4 Writing ntuples
22.5 Reading ntuples
22.6 Closing an ntuple file
22.7 Histogramming ntuple values
22.8 Examples
22.9 References and Further Reading
23. Monte Carlo Integration
23.1 Interface
23.2 PLAIN Monte Carlo
23.3 MISER
23.4 VEGAS
23.5 Examples
23.6 References and Further Reading
24. Simulated Annealing
24.1 Simulated Annealing algorithm
24.2 Simulated Annealing functions
24.3 Examples
24.3.1 Trivial example
24.3.2 Traveling Salesman Problem
25. Ordinary Differential Equations
25.1 Defining the ODE System
25.2 Stepping Functions
25.3 Adaptive Step-size Control
25.4 Evolution
25.5 Examples
25.6 References and Further Reading
26. Interpolation
26.1 Introduction
26.2 Interpolation Functions
26.3 Interpolation Types
26.4 Index Look-up and Acceleration
26.5 Evaluation of Interpolating Functions
26.6 Higher-level Interface
26.7 Examples
26.8 References and Further Reading
27. Numerical Differentiation
27.1 Functions
27.2 Examples
27.3 References and Further Reading
28. Chebyshev Approximations
28.1 The gsl_cheb_series struct
28.2 Creation and Calculation of Chebyshev Series
28.3 Chebyshev Series Evaluation
28.4 Derivatives and Integrals
28.5 Examples
28.6 References and Further Reading
29. Series Acceleration
29.1 Acceleration functions
29.2 Acceleration functions without error estimation
29.3 Examples
29.4 References and Further Reading
30. Wavelet Transforms
30.1 Definitions
30.2 Initialization
30.3 Transform Functions
30.3.1 Wavelet transforms in one dimension
30.3.2 Wavelet transforms in two dimension
30.4 Example
30.5 References and Further Reading
31. Discrete Hankel Transforms
31.1 Definitions
31.2 Functions
31.3 References and Further Reading
32. One dimensional Root-Finding
32.1 Overview
32.2 Caveats
32.3 Initializing the Solver
32.4 Providing the function to solve
32.5 Search Bounds and Guesses
32.6 Iteration
32.7 Search Stopping Parameters
32.8 Root Bracketing Algorithms
32.9 Root Finding Algorithms using Derivatives
32.10 Examples
32.11 References and Further Reading
33. One dimensional Minimization
33.1 Overview
33.2 Caveats
33.3 Initializing the Minimizer
33.4 Providing the function to minimize
33.5 Iteration
33.6 Stopping Parameters
33.7 Minimization Algorithms
33.8 Examples
33.9 References and Further Reading
34. Multidimensional Root-Finding
34.1 Overview
34.2 Initializing the Solver
34.3 Providing the function to solve
34.4 Iteration
34.5 Search Stopping Parameters
34.6 Algorithms using Derivatives
34.7 Algorithms without Derivatives
34.8 Examples
34.9 References and Further Reading
35. Multidimensional Minimization
35.1 Overview
35.2 Caveats
35.3 Initializing the Multidimensional Minimizer
35.4 Providing a function to minimize
35.5 Iteration
35.6 Stopping Criteria
35.7 Algorithms
35.8 Examples
35.9 References and Further Reading
36. Least-Squares Fitting
36.1 Linear regression
36.2 Linear fitting without a constant term
36.3 Multi-parameter fitting
36.4 Examples
36.5 References and Further Reading
37. Nonlinear Least-Squares Fitting
37.1 Overview
37.2 Initializing the Solver
37.3 Providing the Function to be Minimized
37.4 Iteration
37.5 Search Stopping Parameters
37.6 Minimization Algorithms using Derivatives
37.7 Minimization Algorithms without Derivatives
37.8 Computing the covariance matrix of best fit parameters
37.9 Examples
37.10 References and Further Reading
38. Physical Constants
38.1 Fundamental Constants
38.2 Astronomy and Astrophysics
38.3 Atomic and Nuclear Physics
38.4 Measurement of Time
38.5 Imperial Units
38.6 Nautical Units
38.7 Printers Units
38.8 Volume
38.9 Mass and Weight
38.10 Thermal Energy and Power
38.11 Pressure
38.12 Viscosity
38.13 Light and Illumination
38.14 Radioactivity
38.15 Force and Energy
38.16 Prefixes
38.17 Examples
38.18 References and Further Reading
39. IEEE floating-point arithmetic
39.1 Representation of floating point numbers
39.2 Setting up your IEEE environment
39.3 References and Further Reading
A. Debugging Numerical Programs
A.1 Using gdb
A.2 Examining floating point registers
A.3 Handling floating point exceptions
A.4 GCC warning options for numerical programs
A.5 References and Further Reading
B. Contributors to GSL
C. Autoconf Macros
D. GSL CBLAS Library
D.1 Level 1
D.2 Level 2
D.3 Level 3
D.4 Examples
Free Software Needs Free Documentation
GNU General Public License
Preamble
Appendix: How to Apply These Terms to Your New Programs
GNU Free Documentation License
ADDENDUM: How to use this License for your documents
Function Index
Variable Index
Type Index
Concept Index
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