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 non-integer orders, spherical Bessel j and y
Modified Bessel I and K	✔		also for non-integer orders, Airy Ai and Bi
Coulomb Wave Functions F and G	✔		accurate even in quantum tunneling region
Reimann Zeta	✔	✔	also Dirichlet η
Dilogarithm Li₂ (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
Hypergeometric function	✔		₂F₁

Advanced functions of integers include:

greatest common divisors (GCD) and least common multiples (LCM)
factorials and double factorials
modular exponentiation
primality testing
binomial coefficients
Stirling numbers of the first and second kind
Fibonacci numbers
Bell numbers
Bernoulli numbers

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	Clebsch-Gordon 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 Analysis (Solvers)

For arbitrary user-supplied functions, Meta.Numerics supports optimization (minimization and maximization), root-finding, integration, and the solution of ordinary differential equations. All operations are supported on multi-dimensional functions as well as functions of simple real numbers.

Functionality	1-d	n-d	Notes
optimization	✔	✔	also global optimum search
root-finding	✔	✔
integration	✔	✔	advanced multi-grid Monte-Carlo integrator
ordinary differential equations	✔	✔	also conservative integrator

Statistics and Data Analysis

Data Wrangling

The library provides a framework similar to the data frame systems familiar to R and Pandas users. Arrays of strongly typed data can be imported and exported from and to CSV and JSON formatted files. Null data entries are supported via .NET's Nullable structure. Data can be filtered, ordered, and transformed in many different ways. The new views produced copy underlying data only when necessary, so manipulations of even large data sets are memory efficient. Data can be passed in column-oriented form to our or any other statistical analysis APIs.

Statistical Analysis

The library provides specialized classes for working with various kinds of data, including:

Type	Functionality
Univariate Sample	sample and population statistics, transformations, percentile/score conversions, fits to distributions, parametric and non-parametric tests
Bivariate Sample	sample and population statistics, regression (linear, polynomial, non-linear, logistic), parametric and non-parametric tests of association
Multivariate Sample	sample and population statistics, regression (linear and logistic), cluster and component analysis
Data with Error Bars	fit to line, constant, proportionality, polynomial, non-linear function, linear combination of functions
Contingency Table	sample and population statistics, parametric and non-parametric tests of association
Histograms	sample and population statistics, fits 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 statistical tests. All fits produce not just the best-fit parameter set, but also error bars, a covariance matrix, and a goodness-of-fit 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
one-sample t-test	sign test	compare a sample's mean or median to a reference value
two-sample t-test	Mann-Whitney U-test	compare the means or medians of two samples
one-way and two-way ANOVA	Kruskal-Wallis	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 χ² test	Kendall's exact test	detect associated between two categorical variables
Shapiro-Francia	Kolmogorov-Smirnov, Kuiper	compare continuous sample data to a reference distribution
Ljung-Box 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:

Beta	Cauchy	Chi Square	Exponential	Fisher's F
Frechet	Gamma	Gumbel	Kolmogorov	Kuiper
Logistic	Lognormal	Normal	Pareto	Pearson's r
Rayleigh	Student's t	Triangular	Uniform	Wald
Weibull

and discrete:

Bernoulli	Binomial	Geometric
Hypergeometric	Negative Binomial	Poisson
Uniform

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 likelihood fits to any user-supplied distribution.

Matrix Algebra

The library defines a number of matrix classes: rectangular, square, symmetric, and tri-diagonal. 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	Tri-diagonal
Arithmetic	✔	✔	✔	✔
Decomposition	✔	✔	✔	✔
Determinant		✔	✔	✔
Inverse		✔	✔	✔
Eigenvalues and Eigenvectors		✔	✔	✔

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

Extended Precision

We supply a quad precision floating point type that tracks approximately 60 decimal digits.

We supply Int128 and UInt128 types for 128-bit integer arithmetic that is faster than .NET's native BigInteger type.