GNU Scientific Library — statistics, random numbers, probability distributions, numerical integration, root finding, interpolation, FFT, ODEs, and special functions.

Overview

GSL is the reference numerical library for C scientific computing, covering virtually every classical algorithm in applied mathematics and statistics. It is tested to machine-precision accuracy and includes extensive documentation.

Statistics

uses gsl;

var data := [2.1, 4.5, 3.8, 7.2, 1.9, 5.5, 6.0, 3.3];

{ Full stats in one call }
var s := Stats(data);
Writeln('Mean:    ' + FloatToStr(s.Mean));
Writeln('StdDev:  ' + FloatToStr(s.StdDev));
Writeln('Median:  ' + FloatToStr(s.Median));
Writeln('P25/P75: ' + FloatToStr(s.P25) + ' / ' + FloatToStr(s.P75));
Writeln('Skew:    ' + FloatToStr(s.Skewness));

{ Linear regression }
var xs := [1.0, 2.0, 3.0, 4.0, 5.0];
var ys := [2.2, 4.1, 5.9, 8.0, 9.8];
var reg := LinearRegression(xs, ys);
Writeln('y = ' + FloatToStr(reg.Beta) + 'x + ' + FloatToStr(reg.Alpha));
Writeln('R² = ' + FloatToStr(reg.R2));

Random Numbers & Distributions

var rng := NewRngAuto();  { auto-seeded MT19937 }

{ Uniform }
Writeln(RngUniform(rng));           { [0, 1) }
Writeln(RngUniformInt(rng, 100));   { [0, 100) }

{ Normal }
Writeln(RngNormal(rng, 0, 1));     { standard normal }
Writeln(RngNormal(rng, 100, 15));  { IQ distribution }

{ Poisson — e.g. arrivals per minute }
Writeln(IntToStr(RngPoisson(rng, 3.5)));

{ Generate 1000 samples from any distribution }
var samples := RngSample(rng, distribGamma, 1000, 2.0, 1.0);

{ Shuffle in place }
Shuffle(rng, samples);

FreeRng(rng);

Numerical Integration

{ Integrate sin(x) from 0 to pi — result should be 2.0 }
var r := Integrate(0, 3.14159, 1e-6, 1e-6, 1000);
Writeln(FloatToStr(r.Value));   { ~2.0 }
Writeln(FloatToStr(r.AbsErr));  { absolute error estimate }

{ Integrate e^(-x²) from 0 to +inf = sqrt(pi)/2 }
var ri := IntegrateInf(0, 1e-8, 1e-8, 500);

Root Finding & Minimization

{ Find root of cos(x) in [0, 2] — should be pi/2 ≈ 1.5708 }
var root := FindRoot(0, 2, 1e-10, 100);
Writeln(FloatToStr(root.Root));

{ Minimize x² + sin(x) in [-2, 2] }
var minr := Minimize1D(-2, 2, 1e-8, 200);
Writeln('Minimum at x = ' + FloatToStr(minr.XMin));
Writeln('f(x) = '          + FloatToStr(minr.FMin));

Interpolation

var xs := [0.0, 1.0, 2.0, 3.0, 4.0];
var ys := [0.0, 0.84, 0.91, 0.14, -0.76];  { sin values }

var spl := NewSpline(xs, ys, splineCubic);

Writeln(FloatToStr(SplineEval(spl, 1.5)));      { interpolated }
Writeln(FloatToStr(SplineEvalDeriv(spl, 1.5))); { derivative }
Writeln(FloatToStr(SplineIntegral(spl, 0, 4))); { integral 0..4 }

{ Evaluate at many points at once }
var query := [0.5, 1.0, 1.5, 2.0, 2.5];
var vals  := SplineEvalArray(spl, query);

FreeSpline(spl);

FFT

{ Power spectrum of a signal with 512 samples }
var signal  := { ... array of 512 doubles ... }
var power   := PowerSpectrum(signal);
{ power[k] = magnitude² at frequency k/N }

{ Round-trip }
var freq    := RFFT(signal);
var back    := IRFFT(freq, 512);

API Reference

Statistics

Function	Description
Stats(Data)	All stats in one pass: TGSLStats record.
Mean / Variance / StdDev / Median(Data)	Individual statistics.
Percentile(Data, P)	P-th percentile (0–100).
Skewness / Kurtosis(Data)	Distribution shape.
Covariance / Correlation(X, Y)	Pearson covariance and correlation.
SpearmanCorrelation(X, Y)	Rank-based correlation.
LinearRegression(X, Y)	Y = Alpha + Beta*X with R², covariances.
WeightedMean / WeightedVariance(Data, W)	Weighted statistics.
Sort(Data) / Sorted(Data)	In-place sort or sorted copy.

Random Numbers

Function	Description
NewRng(Type) / NewRngAuto	Create RNG (auto-seeded with current time).
SeedRng(Rng, Seed)	Set deterministic seed.
RngUniform / RngUniformInt	Uniform real [0,1) or integer [0,N).
RngNormal(Rng, Mean, Sigma)	Gaussian variate.
RngExponential / RngPoisson / RngBinomial	Common distributions.
RngGamma / RngBeta	Gamma and Beta variates.
RngSample(Rng, Dist, N, P1, P2)	Generate N samples from any distribution.
Shuffle(Rng, Data)	Fisher-Yates shuffle in place.
FreeRng(Rng)	Free RNG resources.

Distributions — CDF / PDF

Function	Description
NormalPDF / NormalCDF / NormalCDFInv	Gaussian PDF, CDF, quantile.
PoissonPMF / BinomialPMF	Discrete probability mass.
ChiSqCDF / TCdf	Chi-squared and Student-t CDFs.

Integration, Roots, Minimization

Function	Description
Integrate(A, B, AbsTol, RelTol, Limit)	Adaptive Gauss-Kronrod QAGS.
IntegrateInf(A, ...)	Semi-infinite integral [A, +∞).
GaussLegendre(A, B, Order)	Fixed-order quadrature (fast).
FindRoot(A, B, AbsTol, MaxIter)	Root finding in [A,B] via Brent.
FindRootNewton(X0, AbsTol, MaxIter)	Newton-Raphson near X0.
Minimize1D(A, B, AbsTol, MaxIter)	1D minimization via Brent.

Interpolation & FFT

Function	Description
NewSpline(X, Y, Type)	Linear, cubic, Akima, or Steffen spline.
SplineEval / EvalDeriv / Integral	Evaluate spline, derivative, or integral.
SplineEvalArray(Spline, X)	Batch evaluation.
FreeSpline(Spline)	Free spline resources.
RFFT / IRFFT(Data)	Real FFT and inverse FFT. N must be power of 2.
PowerSpectrum(Data)	|FFT|² — frequency power at each bin.

Special Functions

Function	Description
GammaFn / LogGamma(X)	Gamma function and log-Gamma.
BetaFn(A, B) / BetaInc(X, A, B)	Beta and regularized incomplete beta.
Erf / Erfc(X)	Error function and complement.
BesselJ0 / J1 / Jn(N, X)	Bessel functions of the first kind.
LegendreP(L, X)	Legendre polynomial P_l(x).
BinomCoeff(N, K)	Binomial coefficient C(N,K).
Factorial(N)	N! as Double.