Table of Contents

1. Introduction

1.1 Routines available in GSL
1.2 GSL is Free Software
1.3 Obtaining GSL
1.4 An Example Program
1.5 No Warranty
1.6 Further Information

2. Using the library

2.1 ANSI C Compliance
2.2 Compiling and Linking
2.3 Shared Libraries
2.4 Autoconf macros
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 Code Reuse

3. Error Handling

3.1 Error Reporting
3.2 Error Handlers
3.3 Using GSL error reporting in your own functions

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

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. Roots of Polynomials

6.1 Polynomial evaluation
6.2 Quadratic equations
6.3 Cubic equations
6.4 General polynomial equations
6.5 Examples
6.6 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 Hurwitz Zeta Function
7.31.3 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 Examples
9.9 References and Further Reading

10. Sorting

10.1 Sorting objects
10.2 Sorting vectors
10.3 Selecting the k-th smallest or largest elements
10.4 Computing the rank
10.5 Examples
10.6 References and Further Reading

11. BLAS Support

11.1 GSL BLAS Interface

11.1.1 Level 1
11.1.2 Level 2
11.1.3 Level 3

11.2 Examples
11.3 References and Further Reading

12. Linear Algebra

12.1 LU Decomposition
12.2 QR Decomposition
12.3 QR Decomposition with Column Pivoting
12.4 Singular Value Decomposition
12.5 Cholesky Decomposition
12.6 Tridiagonal Decomposition of Real Symmetric Matrices
12.7 Tridiagonal Decomposition of Hermitian Matrices
12.8 Bidiagonalization
12.9 Householder solver for linear systems
12.10 Tridiagonal Systems
12.11 Examples
12.12 References and Further Reading

13. Eigensystems

13.1 Real Symmetric Matrices
13.2 Complex Hermitian Matrices
13.3 Sorting Eigenvalues and Eigenvectors
13.4 Examples
13.5 References and Further Reading

14. Fast Fourier Transforms (FFTs)

14.1 Mathematical Definitions
14.2 Overview of complex data FFTs
14.3 Radix-2 FFT routines for complex data
14.4 Mixed-radix FFT routines for complex data
14.5 Overview of real data FFTs
14.6 Radix-2 FFT routines for real data
14.7 Mixed-radix FFT routines for real data
14.8 References and Further Reading

15. Numerical Integration

15.1 Introduction
15.2 QNG non-adaptive Gauss-Kronrod integration
15.3 QAG adaptive integration
15.4 QAGS adaptive integration with singularities
15.5 QAGP adaptive integration with known singular points
15.6 QAGI adaptive integration on infinite intervals
15.7 QAWC adaptive integration for Cauchy principal values
15.8 QAWS adaptive integration for singular functions
15.9 QAWO adaptive integration for oscillatory functions
15.10 QAWF adaptive integration for Fourier integrals
15.11 Error codes
15.12 Examples
15.13 References and Further Reading

16. Random Number Generation

16.1 General comments on random numbers
16.2 The Random Number Generator Interface
16.3 Random number generator initialization
16.4 Sampling from a random number generator
16.5 Auxiliary random number generator functions
16.6 Random number environment variables
16.7 Saving and restoring random number generator state
16.8 Random number generator algorithms
16.9 Unix random number generators
16.10 Numerical Recipes generators
16.11 Other random number generators
16.12 Random Number Generator Performance
16.13 References and Further Reading
16.14 Acknowledgements

17. Quasi-Random Sequences

17.1 Quasi-random number generator initialization
17.2 Sampling from a quasi-random number generator
17.3 Auxiliary quasi-random number generator functions
17.4 Saving and resorting quasi-random number generator state
17.5 Quasi-random number generator algorithms
17.6 Examples
17.7 References

18. Random Number Distributions

18.1 The Gaussian Distribution
18.2 The Gaussian Tail Distribution
18.3 The Bivariate Gaussian Distribution
18.4 The Exponential Distribution
18.5 The Laplace Distribution
18.6 The Exponential Power Distribution
18.7 The Cauchy Distribution
18.8 The Rayleigh Distribution
18.9 The Rayleigh Tail Distribution
18.10 The Landau Distribution
18.11 The Levy alpha-Stable Distributions
18.12 The Levy skew alpha-Stable Distribution
18.13 The Gamma Distribution
18.14 The Flat (Uniform) Distribution
18.15 The Lognormal Distribution
18.16 The Chi-squared Distribution
18.17 The F-distribution
18.18 The t-distribution
18.19 The Beta Distribution
18.20 The Logistic Distribution
18.21 The Pareto Distribution
18.22 The Spherical Distribution (2D & 3D)
18.23 The Weibull Distribution
18.24 The Type-1 Gumbel Distribution
18.25 The Type-2 Gumbel Distribution
18.26 General Discrete Distributions
18.27 The Poisson Distribution
18.28 The Bernoulli Distribution
18.29 The Binomial Distribution
18.30 The Negative Binomial Distribution
18.31 The Pascal Distribution
18.32 The Geometric Distribution
18.33 The Hypergeometric Distribution
18.34 The Logarithmic Distribution
18.35 Shuffling and Sampling
18.36 Examples
18.37 References and Further Reading

19. Statistics

19.1 Mean, Standard Deviation and Variance
19.2 Absolute deviation
19.3 Higher moments (skewness and kurtosis)
19.4 Autocorrelation
19.5 Covariance
19.6 Weighted Samples
19.7 Maximum and Minimum values
19.8 Median and Percentiles
19.9 Example statistical programs
19.10 References and Further Reading

20. Histograms

20.1 The histogram struct
20.2 Histogram allocation
20.3 Copying Histograms
20.4 Updating and accessing histogram elements
20.5 Searching histogram ranges
20.6 Histogram Statistics
20.7 Histogram Operations
20.8 Reading and writing histograms
20.9 Resampling from histograms
20.10 The histogram probability distribution struct
20.11 Example programs for histograms
20.12 Two dimensional histograms
20.13 The 2D histogram struct
20.14 2D Histogram allocation
20.15 Copying 2D Histograms
20.16 Updating and accessing 2D histogram elements
20.17 Searching 2D histogram ranges
20.18 2D Histogram Statistics
20.19 2D Histogram Operations
20.20 Reading and writing 2D histograms
20.21 Resampling from 2D histograms
20.22 Example programs for 2D histograms

21. N-tuples

21.1 The ntuple struct
21.2 Creating ntuples
21.3 Opening an existing ntuple file
21.4 Writing ntuples
21.5 Reading ntuples
21.6 Closing an ntuple file
21.7 Histogramming ntuple values
21.8 Example programs
21.9 References and Further Reading

22. Monte Carlo Integration

22.1 Interface
22.2 PLAIN Monte Carlo
22.3 MISER
22.4 VEGAS
22.5 Examples
22.6 References and Further Reading

23. Simulated Annealing

23.1 Simulated Annealing algorithm
23.2 Simulated Annealing functions
23.3 Examples with Simulated Annealing

23.3.1 Trivial example
23.3.2 Traveling Salesman Problem

24. Ordinary Differential Equations

24.1 Defining the ODE System
24.2 Stepping Functions
24.3 Adaptive Step-size Control
24.4 Evolution
24.5 Examples
24.6 References and Further Reading

25. Interpolation

25.1 Introduction
25.2 Interpolation Functions
25.3 Interpolation Types
25.4 Index Look-up and Acceleration
25.5 Evaluation of interpolating functions
25.6 Higher-level interface
25.7 Examples
25.8 References and Further Reading

26. Numerical Differentiation

26.1 Functions
26.2 Example
26.3 References and Further Reading

27. Chebyshev Approximations

27.1 The gsl_cheb_series struct
27.2 Creation and Calculation of Chebyshev Series
27.3 Chebyshev Series Evaluation
27.4 Derivatives and Integrals
27.5 Examples
27.6 References and Further Reading

28. Series Acceleration

28.1 Acceleration functions
28.2 Acceleration functions without error estimation
28.3 Example of accelerating a series
28.4 References and Further Reading

29. Discrete Hankel Transforms

29.1 Definitions
29.2 Functions
29.3 References and Further Reading

30. One dimensional Root-Finding

30.1 Overview
30.2 Caveats
30.3 Initializing the Solver
30.4 Providing the function to solve
30.5 Search Bounds and Guesses
30.6 Iteration
30.7 Search Stopping Parameters
30.8 Root Bracketing Algorithms
30.9 Root Finding Algorithms using Derivatives
30.10 Examples
30.11 References and Further Reading

31. One dimensional Minimization

31.1 Overview
31.2 Caveats
31.3 Initializing the Minimizer
31.4 Providing the function to minimize
31.5 Iteration
31.6 Stopping Parameters
31.7 Minimization Algorithms
31.8 Examples
31.9 References and Further Reading

32. Multidimensional Root-Finding

32.1 Overview
32.2 Initializing the Solver
32.3 Providing the function to solve
32.4 Iteration
32.5 Search Stopping Parameters
32.6 Algorithms using Derivatives
32.7 Algorithms without Derivatives
32.8 Examples
32.9 References and Further Reading

33. Multidimensional Minimization

33.1 Overview
33.2 Caveats
33.3 Initializing the Multidimensional Minimizer
33.4 Providing a function to minimize
33.5 Iteration
33.6 Stopping Criteria
33.7 Algorithms
33.8 Examples
33.9 References and Further Reading

34. Least-Squares Fitting

34.1 Linear regression
34.2 Linear fitting without a constant term
34.3 Multi-parameter fitting
34.4 Examples
34.5 References and Further Reading

35. Nonlinear Least-Squares Fitting

35.1 Overview
35.2 Initializing the Solver
35.3 Providing the Function to be Minimized
35.4 Iteration
35.5 Search Stopping Parameters
35.6 Minimization Algorithms using Derivatives
35.7 Minimization Algorithms without Derivatives
35.8 Computing the covariance matrix of best fit parameters
35.9 Examples
35.10 References and Further Reading

36. Physical Constants

36.1 Fundamental Constants
36.2 Astronomy and Astrophysics
36.3 Atomic and Nuclear Physics
36.4 Measurement of Time
36.5 Imperial Units
36.6 Nautical Units
36.7 Printers Units
36.8 Volume
36.9 Mass and Weight
36.10 Thermal Energy and Power
36.11 Pressure
36.12 Viscosity
36.13 Light and Illumination
36.14 Radioactivity
36.15 Examples
36.16 References and Further Reading

37. IEEE floating-point arithmetic

37.1 Representation of floating point numbers
37.2 Setting up your IEEE environment
37.3 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

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Preamble
Appendix: How to Apply These Terms to Your New Programs

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Function Index
Variable Index
Type Index
Concept Index