randommatrix

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 Go to latest Published: Jul 19, 2026 License: MIT Imports: 8 Imported by: 0

Documentation

Overview

Package randommatrix implements the core objects of random matrix theory (RMT) using only the Go standard library.

Random matrix theory studies the statistical behaviour of the eigenvalues of matrices whose entries are drawn at random. Despite the randomness of the individual entries, the collective spectrum obeys sharp, universal laws. This package provides seeded generators for the classical Gaussian and Wishart ensembles together with the analytic machinery used to describe their spectra.

The package is organised around a small number of themes:

  • Dense real and complex matrix types (Matrix and CMatrix) with the linear algebra needed to extract spectra: a Jacobi eigensolver for real symmetric matrices and a Hermitian eigensolver built on the real symmetric embedding.

  • Seeded generators for the Gaussian ensembles GOE (beta = 1), GUE (beta = 2) and GSE (beta = 4), for the Wishart / Laguerre ensembles, and for the Ginibre ensembles. Every generator takes an explicit int64 seed and uses math/rand, so results are reproducible and never depend on the wall clock or on a cryptographic source.

  • The limiting spectral laws: the Wigner semicircle law for Wigner matrices and the Marchenko-Pastur law for sample covariance (Wishart) matrices, with densities, cumulative distributions, supports and moments.

  • Edge fluctuations described by the Tracy-Widom distributions for beta = 1, 2, 4, provided through the accurate shifted-gamma approximation of Chiani (2014) together with tabulated moments and the soft-edge rescaling of the largest eigenvalue.

  • Bulk fluctuations described by the nearest-neighbour level-spacing distributions: the Poisson law for uncorrelated spectra and the Wigner surmise for the Gaussian ensembles, plus the level-spacing ratio statistics that avoid the need for spectral unfolding.

  • Spectral moments and elementary free probability: the Stieltjes / Cauchy transform, the R-transform, free cumulants via non-crossing partitions and free additive convolution of the semicircle law.

All randomness is seeded by the caller. Only the packages math, math/big, math/cmplx, sort, errors, fmt and strings are imported; there is no cgo and no third-party dependency.

Index

Examples

Constants

View Source 
const (
	BetaOrthogonal = 1
	BetaUnitary    = 2
	BetaSymplectic = 4
)

Beta values identify the three classical symmetry classes by their Dyson index: 1 for the orthogonal (real) class, 2 for the unitary (complex) class and 4 for the symplectic (quaternion) class.

Variables

View Source 
var ErrDimensionMismatch = errors.New("randommatrix: dimension mismatch")

ErrDimensionMismatch is returned when two matrices have incompatible shapes for the requested operation.

View Source 
var ErrNotSquare = errors.New("randommatrix: matrix is not square")

ErrNotSquare is returned when a square matrix is required but a rectangular one was supplied.

View Source 
var ErrNotSymmetric = errors.New("randommatrix: matrix is not symmetric")

ErrNotSymmetric is returned when a symmetric matrix is required.

Functions

func BinomialBig 

func BinomialBig(n, k int) *big.Int

BinomialBig returns the exact binomial coefficient C(n, k) as a big integer. It returns zero when k < 0 or k > n.

func BinomialCoefficient 

func BinomialCoefficient(n, k int) float64

BinomialCoefficient returns C(n, k) as a float64, computed exactly with big integers and then converted. It returns zero for k < 0 or k > n.

func CatalanBig 

func CatalanBig(n int) *big.Int

CatalanBig returns the n-th Catalan number C_n = C(2n, n)/(n+1) as an exact big integer for n >= 0.

func CatalanNumber 

func CatalanNumber(n int) float64

CatalanNumber returns the n-th Catalan number as a float64.

func CentralSpectralMoment 

func CentralSpectralMoment(eigs []float64, k int) float64

CentralSpectralMoment returns the k-th central moment (1/n) sum (lambda_i - mean)^k of the eigenvalue sample.

func ConditionNumberSymmetric 

func ConditionNumberSymmetric(m *Matrix) (float64, error)

ConditionNumberSymmetric returns the spectral condition number |lambda_max| / |lambda_min| of a real symmetric matrix. It returns +Inf when the smallest absolute eigenvalue is zero.

func ConsecutiveSpacingRatios 

func ConsecutiveSpacingRatios(eigs []float64) []float64

ConsecutiveSpacingRatios returns the restricted ratios r_i = min(s_i, s_{i-1}) / max(s_i, s_{i-1}) in [0, 1], which need no spectral unfolding.

func DeterminantSymmetric 

func DeterminantSymmetric(m *Matrix) (float64, error)

DeterminantSymmetric returns the determinant of a real symmetric matrix as the product of its eigenvalues.

func EigenvaluesHermitian 

func EigenvaluesHermitian(m *CMatrix) ([]float64, error)

EigenvaluesHermitian returns the eigenvalues of a Hermitian matrix in ascending order. It uses the real symmetric embedding of the Hermitian matrix H = A + iB into [[A, -B], [B, A]], whose 2n eigenvalues are those of H each repeated twice; the duplicates are collapsed.

func EigenvaluesSymmetric 

func EigenvaluesSymmetric(m *Matrix) ([]float64, error)

EigenvaluesSymmetric returns the eigenvalues of a real symmetric matrix in ascending order.

func EmpiricalCDF 

func EmpiricalCDF(data []float64, x float64) float64

EmpiricalCDF returns the value of the empirical cumulative distribution function of the sample at x, i.e. the fraction of samples not exceeding x.

func EmpiricalCauchy 

func EmpiricalCauchy(eigs []float64, z complex128) complex128

EmpiricalCauchy returns the Cauchy transform (1/n) sum 1/(z - lambda_i), the negative of the Stieltjes transform.

func EmpiricalStieltjes 

func EmpiricalStieltjes(eigs []float64, z complex128) complex128

EmpiricalStieltjes returns the Stieltjes transform (1/n) sum 1/(lambda_i - z) of the empirical spectral distribution at complex argument z.

func FactorialBig 

func FactorialBig(n int) *big.Int

FactorialBig returns n! as an exact big integer for n >= 0 and nil for n < 0.

func FreeConvolutionMoments 

func FreeConvolutionMoments(momentsA, momentsB []float64) []float64

FreeConvolutionMoments returns the moments of the free additive convolution of two distributions given their moment sequences. Free cumulants add, so the result is obtained by adding cumulants and converting back. The shorter of the two inputs sets the number of returned moments.

func FreeConvolutionSemicircleRadius 

func FreeConvolutionSemicircleRadius(r1, r2 float64) float64

FreeConvolutionSemicircleRadius returns the radius of the semicircle law obtained as the free additive convolution of two semicircle laws of the given radii, namely sqrt(R1^2 + R2^2).

func FreeConvolutionSemicircleVariance 

func FreeConvolutionSemicircleVariance(var1, var2 float64) float64

FreeConvolutionSemicircleVariance returns the variance of the free additive convolution of two semicircle laws, which is the sum of their variances.

func FreeCumulantsFromMoments 

func FreeCumulantsFromMoments(moments []float64) []float64

FreeCumulantsFromMoments returns the free cumulants kappa_1..kappa_N implied by the moments m_1..m_N by inverting the free moment-cumulant relation.

func GOEMeanRatio 

func GOEMeanRatio() float64

GOEMeanRatio returns the surmise value 0.5307 of the mean restricted spacing ratio for the GOE.

func GOESpacingCDF 

func GOESpacingCDF(s float64) float64

GOESpacingCDF returns the GOE spacing distribution 1 - exp(-pi s^2/4).

func GOESpacingDensity 

func GOESpacingDensity(s float64) float64

GOESpacingDensity returns the GOE (beta = 1) Wigner surmise density (pi/2) s exp(-pi s^2/4).

func GSEMeanRatio 

func GSEMeanRatio() float64

GSEMeanRatio returns the surmise value 0.6744 of the mean restricted spacing ratio for the GSE.

func GSESpacingCDF 

func GSESpacingCDF(s float64) float64

GSESpacingCDF returns the GSE spacing distribution, integrated numerically.

func GSESpacingDensity 

func GSESpacingDensity(s float64) float64

GSESpacingDensity returns the GSE (beta = 4) Wigner surmise density (2^18/(3^6 pi^3)) s^4 exp(-64 s^2/(9 pi)).

func GUEMeanRatio 

func GUEMeanRatio() float64

GUEMeanRatio returns the surmise value 0.5996 of the mean restricted spacing ratio for the GUE.

func GUESpacingCDF 

func GUESpacingCDF(s float64) float64

GUESpacingCDF returns the GUE spacing distribution, integrated numerically.

func GUESpacingDensity 

func GUESpacingDensity(s float64) float64

GUESpacingDensity returns the GUE (beta = 2) Wigner surmise density (32/pi^2) s^2 exp(-4 s^2/pi).

func GaussianVector 

func GaussianVector(n int, mean, std float64, seed int64) []float64

GaussianVector returns n independent N(mean, std^2) samples.

func KolmogorovSmirnovStatistic 

func KolmogorovSmirnovStatistic(data []float64, cdf func(float64) float64) float64

KolmogorovSmirnovStatistic returns the maximum absolute difference between the empirical CDF of the sample and the reference CDF, evaluated at the sample points.

func LargestEigenvalue 

func LargestEigenvalue(m *Matrix) (float64, error)

LargestEigenvalue returns the algebraically largest eigenvalue of a real symmetric matrix.

func MarchenkoPasturAtomMass 

func MarchenkoPasturAtomMass(c float64) float64

MarchenkoPasturAtomMass returns the mass max(0, 1 - 1/c) of the atom at zero of the Marchenko-Pastur law with aspect ratio c.

func MarchenkoPasturCDF 

func MarchenkoPasturCDF(x, c, sigma2 float64) float64

MarchenkoPasturCDF returns the cumulative distribution function of the Marchenko-Pastur law with aspect ratio c and variance sigma2 at x, including the atom at zero when c > 1. The continuous part is integrated numerically.

func MarchenkoPasturDensity 

func MarchenkoPasturDensity(x, c, sigma2 float64) float64

MarchenkoPasturDensity returns the continuous part of the Marchenko-Pastur density with aspect ratio c and variance sigma2 at x. It is zero outside the bulk support and does not include the possible atom at zero when c > 1.

func MarchenkoPasturHasAtom 

func MarchenkoPasturHasAtom(c float64) bool

MarchenkoPasturHasAtom reports whether the Marchenko-Pastur law with aspect ratio c places an atom at zero (which happens exactly when c > 1).

func MarchenkoPasturMean 

func MarchenkoPasturMean(c, sigma2 float64) float64

MarchenkoPasturMean returns the mean sigma2 of the Marchenko-Pastur law.

func MarchenkoPasturMoment 

func MarchenkoPasturMoment(k int, c, sigma2 float64) float64

MarchenkoPasturMoment returns the k-th moment of the Marchenko-Pastur law with aspect ratio c and variance sigma2. It uses the Narayana expansion m_k = sigma2^k * sum_{r=0}^{k-1} 1/(r+1) * C(k,r) * C(k-1,r) * c^r.

func MarchenkoPasturStieltjes 

func MarchenkoPasturStieltjes(z complex128, c float64) complex128

MarchenkoPasturStieltjes returns the Stieltjes transform of the standard (sigma2 = 1) Marchenko-Pastur law with ratio c at complex argument z, using the branch that behaves like -1/z at infinity.

func MarchenkoPasturSupport 

func MarchenkoPasturSupport(c, sigma2 float64) (lo, hi float64)

MarchenkoPasturSupport returns the edges lambda_minus and lambda_plus of the Marchenko-Pastur law with aspect ratio c and per-entry variance sigma2.

func MarchenkoPasturVariance 

func MarchenkoPasturVariance(c, sigma2 float64) float64

MarchenkoPasturVariance returns the variance c*sigma2^2 of the Marchenko-Pastur law.

func Max 

func Max(xs []float64) float64

Max returns the maximum of a non-empty slice; it returns NaN for the empty slice.

func Mean 

func Mean(xs []float64) float64

Mean returns the arithmetic mean of xs, or NaN for the empty slice.

func MeanConsecutiveRatio 

func MeanConsecutiveRatio(eigs []float64) float64

MeanConsecutiveRatio returns the mean of the restricted spacing ratios of the eigenvalue sample.

func MeanSpacing 

func MeanSpacing(eigs []float64) float64

MeanSpacing returns the mean nearest-neighbour spacing of the eigenvalue sample.

func Min 

func Min(xs []float64) float64

Min returns the minimum of a non-empty slice; it returns NaN for the empty slice.

func MomentsFromFreeCumulants 

func MomentsFromFreeCumulants(kappa []float64) []float64

MomentsFromFreeCumulants returns the moments m_1..m_N implied by the free cumulants kappa_1..kappa_N via the free moment-cumulant relation m_n = sum over non-crossing partitions of products of cumulants.

func NearestNeighbourSpacings 

func NearestNeighbourSpacings(eigs []float64) []float64

NearestNeighbourSpacings returns the consecutive gaps of the sorted eigenvalue sample.

func NonCrossingPartitionsCount 

func NonCrossingPartitionsCount(n int) float64

NonCrossingPartitionsCount returns the number of non-crossing partitions of an n-element set, which equals the n-th Catalan number.

func NormalizedSpacings 

func NormalizedSpacings(eigs []float64) []float64

NormalizedSpacings returns the nearest-neighbour spacings rescaled by their mean, a crude global unfolding that produces unit mean spacing.

func PoissonMeanRatio 

func PoissonMeanRatio() float64

PoissonMeanRatio returns the mean restricted spacing ratio 2 ln 2 - 1 for a Poisson spectrum.

func PoissonRatioDensity 

func PoissonRatioDensity(r float64) float64

PoissonRatioDensity returns the surmise density 1/(1+r)^2 of the unrestricted consecutive-spacing ratio for a Poisson spectrum.

func PoissonSpacingCDF 

func PoissonSpacingCDF(s float64) float64

PoissonSpacingCDF returns the cumulative distribution 1 - exp(-s) of Poisson level spacings.

func PoissonSpacingDensity 

func PoissonSpacingDensity(s float64) float64

PoissonSpacingDensity returns the density exp(-s) of nearest-neighbour level spacings for an uncorrelated (Poisson) spectrum with unit mean spacing.

func RTransformSeries 

func RTransformSeries(kappa []float64) []float64

RTransformSeries returns the coefficients of the R-transform power series R(z) = sum_{n>=1} kappa_n z^(n-1) from the free cumulants kappa_1..kappa_N. The returned slice holds the coefficient of z^0, z^1, ... .

func RatioSurmiseDensity 

func RatioSurmiseDensity(r, beta float64) float64

RatioSurmiseDensity returns the Atas et al. surmise density for the restricted spacing ratio r in [0, 1] at Dyson index beta, P(r) = (1/Z) (r + r^2)^beta / (1 + r + r^2)^(1 + 3 beta/2), with Z fixed by numerical normalisation.

func RescaleLargestEigenvalue 

func RescaleLargestEigenvalue(lambdaMax float64, n int) float64

RescaleLargestEigenvalue maps the largest eigenvalue lambdaMax of an n-by-n Gaussian ensemble to the Tracy-Widom scale via (lambdaMax - 2 sqrt(n)) * n^(1/6).

func SampleGOE 

func SampleGOE(n int, seed int64, scale float64) ([]float64, error)

SampleGOE returns the eigenvalues of GOE(n)/scale in ascending order. Passing scale = sqrt(n) places the bulk on [-2,2].

func SampleGUE 

func SampleGUE(n int, seed int64, scale float64) ([]float64, error)

SampleGUE returns the eigenvalues of GUE(n)/scale in ascending order.

func SampleWishartReal 

func SampleWishartReal(n, p int, seed int64) ([]float64, error)

SampleWishartReal returns the eigenvalues of the real sample covariance matrix (1/p) G Gᵀ in ascending order.

func SemicircleCDF 

func SemicircleCDF(x, radius float64) float64

SemicircleCDF returns the cumulative distribution function of the semicircle law of radius R evaluated at x.

func SemicircleDensity 

func SemicircleDensity(x, radius float64) float64

SemicircleDensity returns the Wigner semicircle density (2/(pi R^2)) sqrt(R^2 - x^2) at x for radius R; it is zero outside [-R, R].

func SemicircleFreeCumulants 

func SemicircleFreeCumulants(variance float64, N int) []float64

SemicircleFreeCumulants returns the free cumulants of the semicircle law of the given variance up to order N. Only the second cumulant is non-zero and equals the variance.

func SemicircleMoment 

func SemicircleMoment(k int, radius float64) float64

SemicircleMoment returns the k-th moment of the semicircle law of radius R. Odd moments vanish; the (2m)-th moment equals the m-th Catalan number times (R/2)^(2m).

Example 
// Even moments of the semicircle law of radius 2 are the Catalan numbers.
fmt.Printf("%.0f %.0f %.0f %.0f\n",
	SemicircleMoment(0, 2), SemicircleMoment(2, 2),
	SemicircleMoment(4, 2), SemicircleMoment(6, 2))

Output:
1 1 2 5

func SemicirclePeak 

func SemicirclePeak(radius float64) float64

SemicirclePeak returns the maximum value 2/(pi R) of the semicircle density, attained at x = 0.

func SemicircleRadius 

func SemicircleRadius(variance float64) float64

SemicircleRadius returns the radius R = 2*sqrt(variance) of the Wigner semicircle law whose second moment (variance) equals the given value.

func SemicircleStieltjes 

func SemicircleStieltjes(z complex128, radius float64) complex128

SemicircleStieltjes returns the Stieltjes transform m(z) of the semicircle law of radius R, the value of the integral of 1/(x-z) against the density. The branch of the square root is chosen so that m(z) ~ -1/z as z -> infinity.

func SemicircleSupport 

func SemicircleSupport(radius float64) (lo, hi float64)

SemicircleSupport returns the endpoints (-R, R) of the semicircle law of radius R.

func SemicircleVariance 

func SemicircleVariance(radius float64) float64

SemicircleVariance returns the variance R^2/4 of the semicircle law of radius R.

func SmallestEigenvalue 

func SmallestEigenvalue(m *Matrix) (float64, error)

SmallestEigenvalue returns the algebraically smallest eigenvalue of a real symmetric matrix.

func SortSpacings 

func SortSpacings(spacings []float64) []float64

SortSpacings returns the ascending-sorted spacings, convenient for building empirical spacing distributions.

func SortedCopy 

func SortedCopy(xs []float64) []float64

SortedCopy returns a sorted (ascending) copy of xs, leaving the input intact.

func SpacingRatios 

func SpacingRatios(eigs []float64) []float64

SpacingRatios returns the ratios s_{i+1}/s_i of consecutive nearest-neighbour spacings of the eigenvalue sample.

func SpectralGaps 

func SpectralGaps(eigs []float64) []float64

SpectralGaps returns the consecutive differences of the sorted eigenvalues.

func SpectralMean 

func SpectralMean(eigs []float64) float64

SpectralMean returns the mean of the eigenvalue sample.

func SpectralMoment 

func SpectralMoment(eigs []float64, k int) float64

SpectralMoment returns the k-th empirical spectral moment (1/n) sum lambda_i^k of the eigenvalue sample eigs.

func SpectralMoments 

func SpectralMoments(eigs []float64, kmax int) []float64

SpectralMoments returns the empirical spectral moments of orders 1..kmax.

func SpectralQuantile 

func SpectralQuantile(eigs []float64, q float64) float64

SpectralQuantile returns the q-quantile (0 <= q <= 1) of the eigenvalue sample using linear interpolation between order statistics.

func SpectralRadius 

func SpectralRadius(m *Matrix) (float64, error)

SpectralRadius returns the largest absolute eigenvalue of a real symmetric matrix.

func SpectralVariance 

func SpectralVariance(eigs []float64) float64

SpectralVariance returns the population variance of the eigenvalue sample.

func TracyWidomApproxParams 

func TracyWidomApproxParams(beta int) (k, theta, shift float64, ok bool)

TracyWidomApproxParams returns the shifted-gamma approximation parameters (shape k, scale theta, location shift) for the Tracy-Widom distribution of Dyson index beta. The approximation, due to Chiani (2014), models TW_beta ~ theta * Gamma(k) - shift and is accurate to about 0.01 in distribution. It returns ok = false for an unsupported beta.

func TracyWidomCDF 

func TracyWidomCDF(s float64, beta int) float64

TracyWidomCDF returns the Chiani shifted-gamma approximation of the Tracy-Widom cumulative distribution function of Dyson index beta at s. It returns NaN for an unsupported beta.

func TracyWidomDensity 

func TracyWidomDensity(s float64, beta int) float64

TracyWidomDensity returns the Chiani shifted-gamma approximation of the Tracy-Widom probability density of Dyson index beta at s.

func TracyWidomEdgeCenter 

func TracyWidomEdgeCenter(n int) float64

TracyWidomEdgeCenter returns the soft-edge centring 2*sqrt(n) for the largest eigenvalue of an n-by-n Gaussian ensemble in the normalisation where the bulk fills [-2 sqrt(n), 2 sqrt(n)].

func TracyWidomEdgeScale 

func TracyWidomEdgeScale(n int) float64

TracyWidomEdgeScale returns the soft-edge scaling n^(-1/6), so that (lambda_max - 2 sqrt(n)) / scale converges to the Tracy-Widom law.

func TracyWidomExcessKurtosis 

func TracyWidomExcessKurtosis(beta int) float64

TracyWidomExcessKurtosis returns the excess kurtosis 6/k of the shifted-gamma approximation to the Tracy-Widom distribution of Dyson index beta.

func TracyWidomMean 

func TracyWidomMean(beta int) float64

TracyWidomMean returns the tabulated mean of the Tracy-Widom distribution of Dyson index beta.

func TracyWidomSkewness 

func TracyWidomSkewness(beta int) float64

TracyWidomSkewness returns the skewness 2/sqrt(k) of the shifted-gamma approximation to the Tracy-Widom distribution of Dyson index beta.

func TracyWidomStdDev 

func TracyWidomStdDev(beta int) float64

TracyWidomStdDev returns the standard deviation of the Tracy-Widom distribution of Dyson index beta.

func TracyWidomSupportedBeta 

func TracyWidomSupportedBeta(beta int) bool

TracyWidomSupportedBeta reports whether beta is one of the supported Dyson indices 1, 2 or 4.

func TracyWidomVariance 

func TracyWidomVariance(beta int) float64

TracyWidomVariance returns the variance k*theta^2 of the shifted-gamma approximation, which matches the tabulated Tracy-Widom variance.

func Unfold 

func Unfold(eigs []float64, cdf func(float64) float64) []float64

Unfold maps each eigenvalue lambda to n*cdf(lambda), where n is the number of eigenvalues and cdf is the integrated limiting spectral density. The transformed levels have unit mean spacing.

func UnfoldedSpacings 

func UnfoldedSpacings(eigs []float64, cdf func(float64) float64) []float64

UnfoldedSpacings returns the nearest-neighbour spacings of the eigenvalues after unfolding with the given integrated spectral density.

func WignerSurmise 

func WignerSurmise(s, beta float64) float64

WignerSurmise returns the Wigner surmise density for Dyson index beta at spacing s, normalised to unit total mass and unit mean spacing. The classical cases are beta = 1 (GOE), beta = 2 (GUE) and beta = 4 (GSE).

Example 
// The Wigner surmise for the GUE (beta=2) at unit spacing.
fmt.Printf("%.4f\n", WignerSurmise(1, 2))

Output:
0.9076

func WignerSurmiseCDF 

func WignerSurmiseCDF(s, beta float64) float64

WignerSurmiseCDF returns the cumulative distribution of the Wigner surmise for Dyson index beta at spacing s, integrated numerically.

func WishartTracyWidomCenter 

func WishartTracyWidomCenter(n, p int) float64

WishartTracyWidomCenter returns the Johnstone centring (sqrt(n-1) + sqrt(p))^2 for the largest eigenvalue of a real Wishart matrix built from an n-by-p Gaussian data matrix.

func WishartTracyWidomScale 

func WishartTracyWidomScale(n, p int) float64

WishartTracyWidomScale returns the Johnstone scaling (sqrt(n-1) + sqrt(p)) (1/sqrt(n-1) + 1/sqrt(p))^(1/3) for the largest eigenvalue of a real Wishart matrix, so that (lambda_max - center)/scale converges to the Tracy-Widom law with beta = 1.

Types

type CMatrix 

type CMatrix struct {
	// contains filtered or unexported fields
}

CMatrix is a dense, row-major matrix of complex128 values.

func CIdentity 

func CIdentity(n int) *CMatrix

CIdentity returns the n-by-n complex identity matrix.

func GSE 

func GSE(n int, seed int64) *CMatrix

GSE returns a sample from the Gaussian Symplectic Ensemble as a 2n-by-2n Hermitian, self-dual complex matrix built from an n-by-n array of quaternions. Each quaternion is embedded as a 2-by-2 complex block. Diagonal quaternions are real N(0,1); the four components of each off-diagonal quaternion are independent N(0,1/2). Every eigenvalue is doubly degenerate (Kramers degeneracy).

func GUE 

func GUE(n int, seed int64) *CMatrix

GUE returns a sample from the Gaussian Unitary Ensemble: a Hermitian n-by-n matrix whose diagonal entries are real N(0,1) and whose off-diagonal entries have real and imaginary parts drawn independently from N(0,1/2). The eigenvalues of GUE(n)/sqrt(n) converge to the semicircle law on [-2,2].

func GaussianCMatrix 

func GaussianCMatrix(rows, cols int, std float64, seed int64) *CMatrix

GaussianCMatrix returns a rows-by-cols complex matrix whose real and imaginary parts are independent N(0, std^2) samples.

func GinibreComplex 

func GinibreComplex(n int, seed int64) *CMatrix

GinibreComplex returns an n-by-n complex matrix whose entries have real and imaginary parts drawn independently from N(0,1/2), so that each entry has unit expected squared modulus (the complex Ginibre ensemble).

func LaguerreUnitary 

func LaguerreUnitary(n, p int, seed int64) *CMatrix

LaguerreUnitary is a synonym for WishartComplex.

func NewCMatrix 

func NewCMatrix(rows, cols int) *CMatrix

NewCMatrix returns a rows-by-cols complex matrix of zeros.

func NewCMatrixFrom 

func NewCMatrixFrom(rows, cols int, data []complex128) *CMatrix

NewCMatrixFrom builds a complex matrix from a flat row-major slice, which is copied.

func WishartComplex 

func WishartComplex(n, p int, seed int64) *CMatrix

WishartComplex returns the complex Wishart matrix G G†, where G is an n-by-p complex matrix with real and imaginary parts drawn independently from N(0,1/2). The result is an n-by-n Hermitian positive semidefinite matrix (the Laguerre unitary ensemble).

func (*CMatrix) At 

func (m *CMatrix) At(i, j int) complex128

At returns the entry at row i, column j.

func (*CMatrix) Clone 

func (m *CMatrix) Clone() *CMatrix

Clone returns an independent deep copy.

func (*CMatrix) Cols 

func (m *CMatrix) Cols() int

Cols returns the number of columns.

func (*CMatrix) ConjugateTranspose 

func (m *CMatrix) ConjugateTranspose() *CMatrix

ConjugateTranspose returns the conjugate (Hermitian) transpose m†.

func (*CMatrix) FrobeniusNorm 

func (m *CMatrix) FrobeniusNorm() float64

FrobeniusNorm returns the Frobenius norm of the complex matrix.

func (*CMatrix) Hermitianize 

func (m *CMatrix) Hermitianize() *CMatrix

Hermitianize returns (m + m†)/2, the Hermitian part of a square matrix.

func (*CMatrix) ImagPart 

func (m *CMatrix) ImagPart() *Matrix

ImagPart returns the matrix of imaginary parts.

func (*CMatrix) IsHermitian 

func (m *CMatrix) IsHermitian(tol float64) bool

IsHermitian reports whether m is square and Hermitian to within tol.

func (*CMatrix) IsSquare 

func (m *CMatrix) IsSquare() bool

IsSquare reports whether the matrix is square.

func (*CMatrix) Minus 

func (m *CMatrix) Minus(other *CMatrix) (*CMatrix, error)

Minus returns the entrywise difference m - other.

func (*CMatrix) Mul 

func (m *CMatrix) Mul(other *CMatrix) (*CMatrix, error)

Mul returns the matrix product m * other.

func (*CMatrix) Plus 

func (m *CMatrix) Plus(other *CMatrix) (*CMatrix, error)

Plus returns the entrywise sum m + other.

func (*CMatrix) RealPart 

func (m *CMatrix) RealPart() *Matrix

RealPart returns the matrix of real parts.

func (*CMatrix) Rows 

func (m *CMatrix) Rows() int

Rows returns the number of rows.

func (*CMatrix) Scale 

func (m *CMatrix) Scale(s complex128) *CMatrix

Scale multiplies every entry by the complex scalar s.

func (*CMatrix) Set 

func (m *CMatrix) Set(i, j int, v complex128)

Set assigns v to the entry at row i, column j.

func (*CMatrix) Trace 

func (m *CMatrix) Trace() complex128

Trace returns the sum of the diagonal entries.

type EigenSym 

type EigenSym struct {
	Values  []float64
	Vectors *Matrix
}

EigenSym holds the eigen-decomposition of a real symmetric matrix. Values holds the eigenvalues in ascending order and column j of Vectors is the unit eigenvector belonging to Values[j].

func EigSymmetric 

func EigSymmetric(m *Matrix) (*EigenSym, error)

EigSymmetric computes the full eigen-decomposition of a real symmetric matrix using the cyclic Jacobi algorithm. It returns an error if the matrix is not square. The matrix is symmetrised internally, so the strictly lower triangle is ignored.

type Histogram 

type Histogram struct {
	Edges  []float64
	Counts []int
	Total  int
}

Histogram is a fixed-width histogram of a real sample. Edges has NumBins+1 entries; Counts[i] counts samples in [Edges[i], Edges[i+1]) except the last bin, which is closed on the right.

func EmpiricalSpectralDensity 

func EmpiricalSpectralDensity(eigs []float64, bins int) *Histogram

EmpiricalSpectralDensity returns a histogram of the eigenvalue sample using the given number of bins over the observed spectral range. It is a synonym for NewHistogramAuto tailored to spectra.

func NewHistogram 

func NewHistogram(data []float64, bins int, lo, hi float64) *Histogram

NewHistogram bins the data into the given number of equal-width bins spanning [lo, hi]. Values outside the range are ignored. It panics if bins < 1 or hi <= lo.

func NewHistogramAuto 

func NewHistogramAuto(data []float64, bins int) *Histogram

NewHistogramAuto bins the data into the given number of bins spanning its observed range [min, max]. It panics if the data is empty or degenerate.

func (*Histogram) BinCenters 

func (h *Histogram) BinCenters() []float64

BinCenters returns the midpoint of each bin.

func (*Histogram) BinWidth 

func (h *Histogram) BinWidth() float64

BinWidth returns the common width of the bins.

func (*Histogram) Density 

func (h *Histogram) Density() []float64

Density returns the empirical probability density: Counts[i] divided by (Total * BinWidth). The returned values integrate to one over the range when no samples were discarded.

func (*Histogram) ModeBin 

func (h *Histogram) ModeBin() int

ModeBin returns the index of the bin with the most samples.

func (*Histogram) NumBins 

func (h *Histogram) NumBins() int

NumBins returns the number of bins.

func (*Histogram) Probabilities 

func (h *Histogram) Probabilities() []float64

Probabilities returns the fraction of samples in each bin.

type Matrix 

type Matrix struct {
	// contains filtered or unexported fields
}

Matrix is a dense, row-major matrix of float64 values.

func Diag 

func Diag(d []float64) *Matrix

Diag returns a square diagonal matrix whose diagonal is d.

func GOE 

func GOE(n int, seed int64) *Matrix

GOE returns a sample from the Gaussian Orthogonal Ensemble: a real symmetric n-by-n matrix whose off-diagonal entries are independent N(0,1) and whose diagonal entries are independent N(0,2). With this normalisation the eigenvalues of GOE(n)/sqrt(n) converge to the semicircle law on [-2,2].

Example 
// GOE matrices are real symmetric, and generation is fully seeded.
m := GOE(3, 42)
fmt.Println(m.IsSymmetric(1e-12))

Output:
true

func GaussianMatrix 

func GaussianMatrix(rows, cols int, mean, std float64, seed int64) *Matrix

GaussianMatrix returns a rows-by-cols matrix of independent N(mean, std^2) entries.

func GinibreReal 

func GinibreReal(n int, seed int64) *Matrix

GinibreReal returns an n-by-n matrix of independent standard normal real entries (the real Ginibre ensemble).

func Identity 

func Identity(n int) *Matrix

Identity returns the n-by-n identity matrix.

func LaguerreOrthogonal 

func LaguerreOrthogonal(n, p int, seed int64) *Matrix

LaguerreOrthogonal is a synonym for WishartReal, emphasising its role as the beta = 1 Laguerre ensemble.

func NewMatrix 

func NewMatrix(rows, cols int) *Matrix

NewMatrix returns a rows-by-cols matrix whose entries are all zero. It panics if either dimension is negative.

func NewMatrixFrom 

func NewMatrixFrom(rows, cols int, data []float64) *Matrix

NewMatrixFrom builds a rows-by-cols matrix from a flat, row-major slice. The slice is copied. It panics if len(data) != rows*cols.

func NewMatrixFromRows 

func NewMatrixFromRows(rows [][]float64) *Matrix

NewMatrixFromRows builds a matrix from a slice of equal-length rows.

func SampleCovarianceReal 

func SampleCovarianceReal(n, p int, seed int64) *Matrix

SampleCovarianceReal returns (1/p) G Gᵀ where G is n-by-p standard normal. With ratio c = n/p its eigenvalues follow the Marchenko-Pastur law with variance one and ratio c, supported on [(1-sqrt(c))^2, (1+sqrt(c))^2] when c <= 1.

func WignerMatrix 

func WignerMatrix(n int, offStd, diagStd float64, seed int64) *Matrix

WignerMatrix returns a real symmetric n-by-n Wigner matrix with independent off-diagonal entries of standard deviation offStd and diagonal entries of standard deviation diagStd.

func WishartReal 

func WishartReal(n, p int, seed int64) *Matrix

WishartReal returns the real Wishart matrix G Gᵀ, where G is an n-by-p matrix of independent standard normal entries. The result is an n-by-n symmetric positive semidefinite matrix (the Laguerre orthogonal ensemble).

func Zeros 

func Zeros(rows, cols int) *Matrix

Zeros returns a rows-by-cols zero matrix; it is an alias for NewMatrix.

func (*Matrix) Add 

func (m *Matrix) Add(i, j int, v float64)

Add increments the entry at row i, column j by v.

func (*Matrix) At 

func (m *Matrix) At(i, j int) float64

At returns the entry at row i, column j.

func (*Matrix) Clone 

func (m *Matrix) Clone() *Matrix

Clone returns an independent deep copy of the matrix.

func (*Matrix) Col 

func (m *Matrix) Col(j int) []float64

Col returns a copy of column j.

func (*Matrix) Cols 

func (m *Matrix) Cols() int

Cols returns the number of columns.

func (*Matrix) Data 

func (m *Matrix) Data() []float64

Data returns a copy of the underlying row-major slice.

func (*Matrix) Diagonal 

func (m *Matrix) Diagonal() []float64

Diagonal returns a copy of the main diagonal.

func (*Matrix) Equals 

func (m *Matrix) Equals(other *Matrix, tol float64) bool

Equals reports whether every entry of m and other agree to within tol.

func (*Matrix) FrobeniusNorm 

func (m *Matrix) FrobeniusNorm() float64

FrobeniusNorm returns the Frobenius norm, sqrt of the sum of squared entries.

func (*Matrix) IsSquare 

func (m *Matrix) IsSquare() bool

IsSquare reports whether the matrix is square.

func (*Matrix) IsSymmetric 

func (m *Matrix) IsSymmetric(tol float64) bool

IsSymmetric reports whether m is square and symmetric to within tol.

func (*Matrix) MaxAbs 

func (m *Matrix) MaxAbs() float64

MaxAbs returns the largest absolute value of any entry.

func (*Matrix) Minus 

func (m *Matrix) Minus(other *Matrix) (*Matrix, error)

Minus returns the entrywise difference m - other.

func (*Matrix) Mul 

func (m *Matrix) Mul(other *Matrix) (*Matrix, error)

Mul returns the matrix product m * other.

func (*Matrix) MulVec 

func (m *Matrix) MulVec(v []float64) ([]float64, error)

MulVec returns the matrix-vector product m * v.

func (*Matrix) Plus 

func (m *Matrix) Plus(other *Matrix) (*Matrix, error)

Plus returns the entrywise sum m + other. It returns an error on shape mismatch.

func (*Matrix) Row 

func (m *Matrix) Row(i int) []float64

Row returns a copy of row i.

func (*Matrix) Rows 

func (m *Matrix) Rows() int

Rows returns the number of rows.

func (*Matrix) Scale 

func (m *Matrix) Scale(s float64) *Matrix

Scale returns a new matrix equal to m with every entry multiplied by s.

func (*Matrix) Set 

func (m *Matrix) Set(i, j int, v float64)

Set assigns v to the entry at row i, column j.

func (*Matrix) String 

func (m *Matrix) String() string

String renders the matrix with entries formatted to three decimals.

func (*Matrix) SubMatrix 

func (m *Matrix) SubMatrix(r0, r1, c0, c1 int) *Matrix

SubMatrix returns the r0:r1, c0:c1 block (half-open ranges).

func (*Matrix) Symmetrize 

func (m *Matrix) Symmetrize() *Matrix

Symmetrize returns (m + mᵀ)/2, the symmetric part of a square matrix.

func (*Matrix) Trace 

func (m *Matrix) Trace() float64

Trace returns the sum of the diagonal entries. It panics for a non-square matrix.

func (*Matrix) Transpose 

func (m *Matrix) Transpose() *Matrix

Transpose returns the transpose of the matrix.

Source Files

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