Guide  Math and Statistics Programming with F#
F# is wellsuited to numerical and statistical programming because of its focus on data transformations and the use of functional programming to help give natural translation of the underlying mathematics into executable form.
Efficient execution of numeric code is essential, and F# executes as native code, normally through JIT compilation. Vectoried execution of F#authored code is available through RyuJIT on Windows and SIMD primitives on Linux. F# can also utilize highly optimized, native code libraries such as the Intel Math Kernel Library (MKL) through packages like Math.NET Numerics and other libraries, and can be used to call directly into Fortran dynamiclink libraries.
This guide includes resources related to math and statistics programming with F#. To contribute to this guide, log on to GitHub, edit this page and send a pull request.
Note that the resources listed below are provided only for educational purposes related to the F# programming language. The F# Software Foundation does not endorse or recommend any commercial products, processes, or services. Therefore, mention of commercial products, processes, or services should not be construed as an endorsement or recommendation.
Resources for Math and Statistics
Opensource libraries

Math.NET Numerics  provides a large collection of algorithms needed in science and engineering, including linear algebra, special functions, statistics, probability models, interpolation and FFTs.
In addition to the core .NET package, Numerics specifically supports F# 3.0 with idiomatic extension modules and maintains mathematical data structures like BigRational that originated in the F# PowerPack. If a performance boost is needed, the managedcode provider backing its linear algebra routines and decompositions can be exchanged with wrappers for optimized native implementations such as Intel MKL. Supports Mono and .NET 4.0 on Linux, Mac and Windows. The portable version also SL5 and .NET for Windows Store apps.
License: MIT/X11

ILNumerics  an open or closedsource library offering high performance numerical algorithms as well as charting and plotting capabilities.
The library is based on efficient, generalpurpose array classes implementing vectors, matrices, and ndimensional arrays. Provided algorithms include standard linear algebra transforms, a highperformance Fast Fourier Transform (FFT) library, and a collection of sorting and machine learning algorithms. Plotting is based on OpenGL and supports both 2D and 3D plots. ILNumerics supports .NET 4.0 as well as Mono (recommend 2.10 or above).
License: GPLv3 or commercial (paid) license.

DiffSharp  An automatic differentiation (AD) library for exact and efficient calculation of derivatives. Also includes symbolic and numerical differentiation.
AD allows exact and efficient calculation of derivatives, by systematically invoking the chain rule of calculus at the elementary operator level during program execution. AD is different from numerical differentiation, which is prone to truncation and roundoff errors, and symbolic differentiation, which suffers from expression swell and cannot handle algorithmic control flow.
Using the DiffSharp library, derivative calculations (gradients, Hessians, Jacobians, directional derivatives, and matrixfree Hessian and Jacobianvector products) can be incorporated with minimal change into existing algorithms.
License: GPLv3

FsAlg  A lightweight linear algebra library that supports generic types.
The library provides generic Vector and Matrix types that support most of the commonly used linear algebra operations, including matrix–vector operations, matrix inverse, determinants, eigenvalues, LU and QR decompositions. Its intended use is to enable writing generic linear algebra code with custom numeric types. It can also be used as a lightweight library for prototyping and scripting with primitive floating point types.
License: BSD
Commercial libraries

Alea GPU  a framework for developing GPUaccelerated algorithms in F# on .NET and Mono.
Utilizing F# quotations and the LLVM compiler it is able to compile GPU kernels onthefly and schedule them on one or more nVidia GPU’s. Advanced GPU features such as textures and shared memory are supported. Available from Quantalea.

Extreme Optimization Numerical Libraries for .NET  a set of three libraries focused on vector and matrix processing, linear algebra methods, and statistics functions.
The library includes a large selection of standard algorithms from matrix factorization, function optimization, numerical integration, Kmeans clustering, and PCA (principal component analysis). Options are provided to run
using pure managed code for portability or to utilize highly tuned native code for additional performance. Extreme Optimization supports .NET 3.5 and 4.0 (2.0 version available) and execution on Mono. 
Microsoft Solver Foundation (MSF)  a .NET package for designing and optimizing mathematical models.
MSF provides builtin solvers for linear and quadraticprogramming, as well as nonlinear models based on NelderMead or quasiNewtonian algorithms. Models can be built using the Optimization Modeling Language (OML) or using C# or F# and other .NET languages. MSF version 3.1 is available in a free Express Edition or via an MSDN subscription.

NMath, NMath Stats  a suite providing core math and statistics functions.
NMath provides sparse and densematrix manipulations, FFT algorithms, and numeric algorithms such as curvefitting, integration, and differentiation. NMath Stats is built on NMath and provides statistics functions such multiple linear regression, hypothesis testing, and nonnegative matrix factorization. NMath and NMath Stats support .NET 4.5 and are available from CenterSpace Software.

F# for Numerics  a collection of numeric algorithms including matrix operations, optimization and interpolation functions, 1D and 2D FFTs, and pseudorandom number generation.
The library uses the standard F# PowerPack Matrix for compatibility. F# for Numerics supports .NET. The library is available from Flying Frog Consultancy.

F# for Visualization  a 2D and 3D vector graphics library with a native F# interface.
The package provides interactive plotting from within Visual Studio and support for generating animations. F# for Visualization supports .NET. The library is available from Flying Frog Consultancy.