General Features

• Easy to use even for the mathematically not-so-inclined.
• Great performance through optimized implementation of the best algorithms.
• Powerful enough to satisfy the most demanding power user.
• Intuitive object model. The objects in the Extreme Numerics.NET and the relationships between them match our every-day concepts.

Math Library Features

• General
  • Machine floating-point constants.
  • Common mathematical constants.
  • Extended elementary functions.
  • Algorithm support functions: iteration, tolerance, convergence tests.
• Complex numbers
  • Double-precision complex number value type.
  • Overloaded operators for all arithmetic operations.
  • Static operator functions for languages that don't support operator overloading.
  • Extension of functions in System.Math to complex argument.
  • Support for complex infinity and complex Not-a-Number (NaN).
  • Complex vector and matrix classes.
• Numerical integration and differentiation
  • Numerical differentiation.
  • Numerical integration using Simpson's rule and Romberg's method.
  • Non-adaptive Gauss-Kronrod numerical integrator.
  • Adaptive Gauss-Kronrod numerical integrator.
  • Integration over infinite intervals.
  • Optimizations for functions with singularities and/or discontinuities.
  • Six integration rules to choose from, or provide your own.
  • Integration in 2 or more dimensions.
• Curve fitting and interpolation
  • Interpolation using polynomials, cubic splines, piecewise constant and linear curves.
  • Linear least squares fit using polynomials or arbitrary functions.
  • Nonlinear least squares using predefined functions or your own.
  • Predefined nonlinear curves: exponential, rational, Gaussian, Lorentz, 4 and 5 parameter logistic.
  • Weighted least squares, with 4 predefined weight functions.
  • Scaling of curve parameters.
  • Constraints on curve parameters.
• Curves
  • Object-oriented approach to working with mathematical curves.
  • Methods for: evaluation, derivative, definite integral, tangent, roots.
  • Many basic types of curves: constants, lines, quadratics, polynomials, cubic splines, Chebyshev approximations, linear combinations of arbitrary functions.
• Solving equations
  • Real and complex roots of polynomials.
  • Roots of arbitrary functions: bisection, false positive, Dekker-Brent and Newton-Raphson methods.
  • Systems of simultaneous linear equations.
  • Systems of nonlinear equations: Powell's hybrid 'dogleg' method, Newton's method.
  • Least squares solutions.
• Optimization
  • Optimization in 1 dimension: Brent's algorithm, Golden Section search.
  • Quasi-Newton method in N dimensions: BFGS and DFP variants.
  • Conjugate gradient method in N dimensions: Fletcher-Reeves and Polak-Ribière variants.
  • Powell's conjugate gradient method.
  • Downhill Simplex method of Nelder and Mead.
  • Levenberg-Marquardt method for nonlinear least squares.
  • Line search algorithms: Moré-Thuente, quadratic, unit.
  • Linear program solver: Based on the Revised Simplex method.
  • Linear program solver: Import from MPS files.
• Signal processing
  • Real 1D and 2D Fast Fourier Transform.
  • Complex 2D Fast Fourier Transform.
  • Special code for factors 2, 3, 4, 5.
  • Real and complex convolution.
  • Managed, 32bit and 64bit native implementations.
• Special functions
  • Over 40 special functions not included in the standard .NET Framework class library.
  • Functions from combinatorics: factorial, combinations, variations, more.
  • Functions from number theory: greatest common divisor, least common multiple, decomposition into prime factors, primality testing.
  • Gamma and related functions, including incomplete and regularized gamma function, digamma function, beta function, harmonic numbers.
  • Hyperbolic and inverse hyperbolic functions for real and complex numbers.
  • Ordinary and Modified Bessel functions of the first and second kind.
  • Airy functions and their derivatives.
  • Exponential integral, sine and cosine integral, logarithmic integral.

Vector and Matrix Library Features

• General
  • Single, double, or quad precision real or complex components.
  • Based on standard BLAS and LAPACK routines.
  • 100% managed implementation for security, portability and small sizes.
  • Native, processor-optimized implementation for speed with large sizes based on the Intel® Math Kernel Library.
  • Native 64bit support.
• GPU computing
  • GPU computing: offload computations to the GPU.
  • Data is kept on the GPU as long as possible for optimal performance.
• Vectors
  • Dense vectors.
  • Band vectors.
  • Constant vectors.
  • Row, column and diagonal vectors.
  • Vector views.
• Vector Operations
  • Basic arithmetic operations.
  • Element-wise operations.
  • Overloaded arithmetic operators.
  • Norms, dot products.
  • Largest and smallest values.
  • Functions of vectors (sine, cosine, etc.)
• Matrices
  • General matrices.
  • Triangular matrices.
  • Real symmetric matrices and complex Hermitian matrices.
  • Band matrices.
  • Diagonal matrices.
  • Matrix views.
• Matrix Operations
  • Basic arithmetic operations.
  • Matrix-vector products.
  • Overloaded arithmetic operations.
  • Element-wise operations.
  • Row and column scaling.
  • Norms, rank, condition numbers.
  • Singular values, eigenvalues and eigenvectors.
• Matrix Decompositions
  • LU decomposition.
  • QR decomposition.
  • Cholesky decomposition.
  • Singular value decomposition.
  • Symmetric eigenvalue decomposition.
  • Non-symmetric eigenvalue decomposition.
  • Banded LU and Cholesky decomposition.
• Sparse Matrices
  • Sparse vectors.
  • Sparse matrices.
  • Matrices in Compressed Sparse Column format.
  • Sparse LU Decomposition.
  • Read matrices in Matrix Market format.
• Linear equations and least squares
  • Shared API for matrices and decompositions.
  • Determinants, inverses, numerical rank, condition numbers.
  • Solve equations with 1 or multiple right-hand sides.
  • Least squares solutions using QR or Singular Value Decomposition.
  • Moore-Penrose Pseudo-inverse.
  • Non-negative least squares (NNLS).

Statistics Library Features

• Descriptive Statistics
  • Measures of central tendency: mean, median, trimmed mean, harmonic mean, geometric mean.
  • Measures of scale: variance, standard deviation, range, interquartile range, absolute deviation from mean and median.
  • Higher moments: skewness, kurtosis.
• Probability Distributions
  • Probability density function (PDF).
  • Cumulative distribution function (CDF).
  • Percentile or inverse cumulative distribution function.
  • Moments: mean, variance, skewness and kurtosis.
  • Generate random samples from any distribution.
  • Parameter estimation for selected distributions.
• Continuous Probability Distributions
  • Beta distribution.
  • Cauchy distribution.
  • Chi-squared distribution.
  • Erlang distribution.
  • Exponential distribution.
  • F distribution.
  • Gamma distribution.
  • Generalized Pareto distribution.
  • Gumbel distribution.
  • Laplace distribution.
  • Logistic distribution.
  • Lognormal distribution.
  • Normal distribution.
  • Pareto distribution.
  • Piecewise distribution.
  • Rayleigh distribution.
  • Student t distribution.
  • Transformed beta distribution.
  • Transformed gamma distribution.
  • Triangular distribution.
  • Uniform distribution.
  • Weibull distribution.
• Discrete Probability Distributions
  • Bernoulli distribution.
  • Binomial distribution.
  • Geometric distribution.
  • Hypergeometric distribution.
  • Negative binomial distribution.
  • Poisson distribution.
  • Uniform distribution.
• Multivariate Probability Distributions
  • Multivariate normal distribution.
  • Dirichlet distribution.
• Histograms
  • One-dimensional histograms.
  • Probability distribution associated with a histogram.
• General Linear Models
  • Infrastructure for General Linear Model and Generalized Linear Model calculations.
  • Analysis of variance.
  • Regression analysis.
  • Model-specific hypothesis tests.
• Analysis of variance (ANOVA)
  • One and two-way ANOVA.
  • One-way ANOVA with repeated measures.
• Regression analysis
  • Simple, multiple, and polynomial regression.
  • Nonlinear regression.
  • Logistic regression.
  • Generalized linear models.
  • Flexible regression models.
  • Variance-covariance matrix, regression matrix.
  • Confidence intervals and significance tests for regression parameters.
• Time series analysis
  • Treat several observation variables as a unit.
  • Change frequency of time series.
  • Automatically apply predefined aggregators.
  • Advanced aggregators: volume weighted average.
• Transformations of Time Series Data
  • Lagged time series, sums, products.
  • Change, percent change, growth rate.
  • Extrapolated change, percent change, growth rate.
  • Period to date sums and differences.
  • Simple, exponential, weighted moving average.
  • Savitsky-Golay smoothing.
• Multivariate Models
  • Principal Component Analysis (PCA).
  • Hierarchical clustering.
  • K-means clustering.
• Statistical tests
  • Tests for the mean: one sample z-test, one sample t-test.
  • Paired and unpaired two-sample t test for the difference between two sample means.
  • Two Sample z-test for ratios.
  • One sample chi-squared test for variance.
  • F-test for the ratio of two variances.
  • One and two sample Kolmogorov-Smirnov test.
  • Anderson-Darling test for normality.
  • Chi-squared goodness-of-fit test.
  • Bartlett and Levene tests for homogeneity of variances.
  • McNemar and Stuart-Maxwell test.
• Random number generation
  • Compatible with the .NET Framework's System.Random.
  • Four generators, with varying quality, period and speed to suit your application.
  • Generate random samples from any distribution.
  • Fauré and Halton sequences.
  • Shufflers and randomized enumerators.