Meta.Numerics library features include advanced functions,
function analysis (solvers),
statistics and data analysis,
linear algebra,
and Fourier transforms.
Advanced Functions
The library defines a large number of simple and advanced mathematical functions on
real numbers, Complex numbers, integers, and other specialized mathematical objects.
Advanced functions of real and complex numbers include:
Function 
Real 
Complex 
Notes 
Gamma 


also ln Γ, incomplete Gamma 
Psi (Digamma) 


also polygamma ψ^{(n)} 
Beta 


also incomplete Beta 
Error Function 


also erfc, erf^{1}, Faddeeva, Fresnel C and S 
Exponential Integrals 


includes Ein, Ei, E_{n}, and trigonometric integrals Ci and Si 
Bessel J and Y 


also for noninteger orders, spherical Bessel j and y 
Modified Bessel I and K 


also for noninteger orders, Airy Ai and Bi 
Coulomb Wave Functions F and G 


accurate even in quantum tunneling region 
Reimann Zeta 


also Dirichlet η 
Dilogarithm Li_{2} (Spence's Function) 


also polylogarithm Li_{n} 
Orthogonal polynomials 


Chebyshev T,
Hermite H,
Legendre P,
Laguerre L,
Zernike R 
Elliptic Integrals 


Legendre F, K, E; Carlson R_{F} and R_{D}, and R_{G} 
Elliptic Functions 


Jacobi cn, sn, and dn 
Advanced functions of integers include:
Meta.Numerics also defines various specialized mathematical objects and associated functions:
Object 
Functions 
Complex numbers 
arithmetic, basic and advanced functions 
Vectors and matrices 
arithmetic, inversion, decompositions 
Spinors 
ClebschGordon coefficients, 3j and 6j symbols 
Uncertain values 
arithmetic and basic functions with error propagation, confidence intervals 
Polynomials 
arithmetic, evaluation, composition, integration and differentiation 
Permutations 
generation, multiplication, inversion, cyclic decomposition and other properties 
Integer partitions 
generation 
Numerical Function Analysis (Solvers)
For arbitrary usersupplied functions, Meta.Numerics supports optimization (minimization and maximization), rootfinding,
integration, and the solution of ordinary differential equations. All operations are supported on multidimensional functions
as well as functions of simple real numbers.
Function Property 
onedimensional 
multidimensional 
maxima and minima 


roots 


integration 


differentiation 


ordinary differential equations 


Statistics and Data Analysis
Data Analysis
The library provides specialized classes for working with various kinds of data, including:
Data 
Functionality 
Univariate Sample 
sample and population statistics,
transformations,
percentile score conversion,
fit to distributions,
parametric and nonparametric tests

Bivariate Sample 
sample and population statistics,
regression (linear, polynomial, nonlinear, logistic),
parametric and nonparametric tests

Experimental Data with Error Bars 
fit to line, constant, proportionality, polynomial,
nonlinear function, linear combination of functions

Contingency Table 
sample and population statistics,
parametric and nonparametric tests

Histograms 
sample and population statistics,
fit to distributions,
parametric tests

Time Series 
sample and population statistics,
power spectrum,
difference and integrate,
fit to AR and MA models

For each kind of data, methods allow you to evaluate descriptive statistics, fit models, and
perform appropriate statisical tests. All fits produce not just the bestfit parameter set, but
also error bars, a covariance matrix, and a goodnessoffit test. Specialized methods make it
easy to add, remove, update, and locate data.
Statistical Tests
Some of the many statistical tests supported by the library include:
Parametric Test 
Nonparametric Alternative 
Purpose 
onesample ttest 
sign test 
compare a sample's mean or median to a reference value 
twosample ttest 
MannWhitney Utest 
compare the means or medians of two samples 
oneway and twoway ANOVA 
KruskalWallis 
compare the means or medians of three or more samples 
Pearson's r 
Spearman's rho,
Kendall's tau 
detect association between two continuous variables 
Pearson's χ^{2} test 
Kendall's exact test 
detect associated between two categorical variables 

KolmogorovSmirnov, Kuiper 
compare continuous sample data to a reference distribution 
LjungBox test 

detect autocorrelation in time series 
For all tests, we provide exact null distributions for small samples.
Distributions
Meta.Numerics defines a large number of probability distributions, both continuous:
and discrete:
For all defined distributions, you can obtain:
 Basic Descriptive Statistics: mean, median, variance, standard deviation, skewness, excess kurtosis
 Probability Mass and Probability Density Function (PDF) values
 Cumulative Distribution Function (CDF) values, integrated from the left or right
 Inverse CDF values, i.e. percentile to score conversions
 Arbitrary raw moments, central moments, and cumulants
 Random deviates
You can also fit sample data to many of the distributions and perform maximum likelyhood fits to any
usersupplied distribution.
Matrix Algebra
The library defines a number of matrix classes: rectangular, square, symmetric, and tridiagonal. Each class defines operations appropriate to that
matrix type, implemented to exploit the matrix structure for optimum performance. The following table summarizes the available operations:
Operation 
Rectangular 
Square 
Symmetric 
Tridiagonal 
Arithmetic 




Decomposition 




Determinant 




Inverse 




Eigenvalues and Eigenvectors 




Available decompositions include LU, QR, and singular value decompositions (SVD).
