F# is wellsuited to numerical and statistical programming because of its focus on data transformations and a natural translation of the underlying mathematics.
Efficient execution of numeric code is essential. F# executes as native code, normally through JIT compilation. 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 listed below. On Linux, Mono provides easy access to SIMD instructions.
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.

Fmat  a 100% F# library for manipulating ndimensional dense matrices.
Concrete matrix implementations cover single and doubleprecision floating point, 32bit integer, bool, and string types. Fmat contains four matrix factorization and linear solver algorithms, random number generators for 7 distributions, special functions and basic statistical procedures. Fmat is available on GitHub and from NuGet. License: MIT/X11
Commercial libraries

Alea.cuBase  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.

StatFactory FCore  a highperformance numerical library supporting both CPU and GPGPU computing.
The library includes multidimensional dense matrix and 2D sparse matrix support, standard linear algebra routines, and summary statistics. The library provides options to run both 100% managed code or to use optimized native libraries such as MKL.

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.