Meta.Numerics library features include complex numbers and other mathematical objects, special functions, numerical calculus, statistics and data analysis, linear algebra, and Fourier transforms.

Mathematical Objects

Meta.Numerics defines several specialized mathematical objects:

• Complex numbers
• Matrices
• Spinors
• Uncertain values

For each type, appropriate arithmetic operations are defined and associated classes implement appropriate operations.

Advanced Functions

The library defines a large number of advanced mathematical function on real numbers, Complex numbers, and integers. The real and complex advanced functions are summarized in the following table.

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, En, 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)
Orthogonal polynomials			Chebyshev T, Hermite H, Legendre P, Laguerre L, Zernike R
Elliptic Integrals			Legendre F, K, E; Carlson RF and RD

Functions of integers include factorials, double factorials, binomial coefficients, partitions, primality testing, greatest common denominators (GCD) and least common multiples (LCM).

Spinor functions include 3j symbols (Clebsch-Gordon coefficients) and 6j symbols.

Numerical Analysis

For arbitrary user-supplied functions, Meta.Numerics supports root-finding, optimization, and integration and differentiation. Some operations are supported on functions on R^N as well as functions on R.

Function Property	one-dimensional	multi-dimensional
maxima and minima
roots
integration
differentiation

Statistics and Data Analysis

Data Collections

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

• Univariate Samples
• Bivariate Samples
• Multivariate Samples
• Experimental Data with Error Bars
• Contingency Tables

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 best-fit parameter set, but also error bars, a covariance matrix, and a goodness-of-fit test. Specialized methods make it easy to load data for analysis from a database or spreadsheet.

Statistical Tests

Some of the many statistical tests supported by the library include:

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
• 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 and central moments
• Random deviates

You can also fit sample data to many of the distributions.

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).