MULTI-LANGUAGE

• ALGLIB is an open source numerical analysis library which may be used from C++, C#, FreePascal, Delphi, VBA.
• IMSL Numerical Libraries are libraries of numerical analysis functionality implemented in standard programming languages like C, Java, C# .NET, Fortran, and Python.
• The NAG Library is a collection of mathematical and statistical routines for multiple programming languages (C, C++, Fortran, Visual Basic, Java and C#) and packages (MATLAB, Excel, R, LabVIEW).
• GNU Octave is an open source high level programming language and library, including a command line interface, analogous to commercial alternatives such as Maple, MatLab, Mathematica, etc. APIs, functions and libraries can be called from many platforms, including high level engineering programs, where functions are, in many cases, seamlessly interpreted and integrated in similar fashion to MatLab. It also can be used with batch orientation.

C

• BLOPEX (Block Locally Optimal Preconditioned Eigenvalue Xolvers) is an open-source library for the scalable (parallel) solution of eigenvalue problems. Its object-oriented design allows easy portability.
• FFTW (Fastest Fourier Transform in the West) is a software library for computing Fourier and related transforms.
• GNU Scientific Library, a popular, free numerical analysis library implemented in C.
• GNU Multi-Precision Library is a library for doing arbitrary precision arithmetic.
• hypre (High Performance Preconditioners) is an open-source library of routines for scalable (parallel) solution of linear systems and preconditioning.
• IMSL Numerical Libraries are commercial cross-platform libraries containing a comprehensive set of mathematical and statistical functions that can be embedded in a users application.
• LabWindows/CVI is an ANSI C IDE that includes built-in libraries for analysis of raw measurement data, signal generation, windowing, filter functions, signal processing, linear algebra, array and complex operations, curve fitting and statistics.
• Lis is a scalable parallel library for solving systems of linear equations and standard eigenvalue problems with real sparse matrices using iterative methods.
• The NAG C Library is a collection of mathematical and statistical routines for C.
• Portable, Extensible Toolkit for Scientific Computation (PETSc), is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations.
• SLEPc Scalable Library for Eigenvalue Problem Computations is a PETSc-based open-source library for the scalable (parallel) solution of eigenvalue problems.
• SuiteSparse - A suite of sparse matrix software with specialized C implementations for sparse matrices - used in GoogleEarth, Googlemaps, Matlab itself and is now distributed with Ubuntu. The package also includes some Matlab modules.
• SuperLU — a direct sparse solver that uses LU decomposition and comes in distributed and shared memory parallel versions in addition to a sequential version.
• Trilinos, an effort to develop scalable (parallel) solver algorithms and libraries within an object-oriented software framework for the solution of large-scale, complex multi-physics engineering andscientific applications. A unique design feature of Trilinos is its focus on packages.

C++

• Armadillo is a C++ linear algebra library (matrix and vector maths), aiming towards a good balance between speed and ease of use. It employs template classes, and has optional links to BLAS and LAPACK. The syntax (API) is similar to MATLAB.
• Blaze is an open-source, high-performance C++ math library for dense and sparse arithmetic.
• Blitz++ is a high-performance vector mathematics library written in C++.
• Boost C++ libraries for numerical computation
• deal.II is a library supporting all the finite element solution of partial differential equations.
• Dlib is a modern C++ library with easy to use linear algebra and optimization tools which benefit from optimized BLAS and LAPACK libraries.
• Eigen is a vector mathematics library with performance comparable with Intel's Math Kernel Library
• FLENS is an extension C++ for matrix/vector types that are ideally suited for numerical linear algebra
• GMTL - Generic Math Template Library -  is a math library designed to be high-performance, extensible, and generic for graphics programmers.
• GNU Scientific Library (GSL) is a numerical library for C and C++ programmers. It is free software under the GNU General Public License. The library provides a wide range of mathematical routines such as random number generators, etc. There are over 1000 functions in total with an extensive test suite.
• Hermes Project: C++/Python library for rapid prototyping of space- and space-time adaptive hp-FEM solvers.
• IML++ is a C++ library for solving linear systems of equations, capable of dealing with dense, sparse, and distributed matrices.
• IT++ is a C++ library for linear algebra (matrices and vectors), signal processing and communications. Functionality similar to MATLAB and Octave.
• LAMA is a C++ template library for writing numerical solvers targeting various hardwares (e.g. GPUs through CUDA or OpenCL) on distributed memory systems, hiding the hardware specific programming from the program developer
• LAPACK++, a C++ wrapper library for LAPACK and BLAS
• LinBox is a C++ template library for doing exact computational linear algebra.
• MLPACK is an open-source library for machine learning, exploiting C++ language features to provide maximum performance and flexibility while providing a simple and consistent API
• MTL4 is a generic C++ template library providing sparse and dense BLAS functionality. MTL4 establishes an intuitive interface (similar to MATLAB) and broad applicability thanks to Generic programming.
• The NAG C++ Library is a collection of mathematical and statistical routines for C++.
• NT2 (linear algebra) -  is a C++ library providing various template matrix classes that can be used with a Matlab like syntax whose performances are close to those obtained with hand written C code
• NTL is a C++ library for number theory.
• PETSc - Portable, Extensible Toolkit for Scientific Computation
• Trilinos - is an effort to develop algorithms and enabling technologies for the solution of large-scale, complex multi-physics engineering and scientific problems. Its a collection of packages.

.NET FRAMEWORK LANGUAGES C#, F# AND VB.NET

• FinMath is a library for scientific and financial computation on the .NET Framework. Its functionality includes vector and matrix algebra, statistics, data analysis, advanced functions, signal processing, numerical optimization and many more. It uses Intel's Math Kernel Library and Intel Integrated Performance Primitives to do most of the low-level computations while providing simple and object oriented interface.
• ILNumerics.Net Commercial high performance, typesafe numerical array classes and functions for general math, FFT and linear algebra, aims .NET/mono, 32&64 bit, script-like syntax in C#, 2D & 3D plot controls, efficient memory management
• IMSL Numerical Libraries for .NET is a commercial library of mathematical, statistical, data mining, financial and charting classes written in C#.
• Measurement Studio is a commercial integrated suite UI controls and class libraries for use in developing test and measurement applications. The analysis class libraries provide various digital signal processing, signal filtering, signal generation, peak detection, and other general mathematical functionality.
• NMath by CenterSpace Software: Commercial numerical component libraries for the .NET platform, including signal processing (FFT) classes, a linear algebra (LAPACK & BLAS) framework, and a statistics package.
• suanshu.net by Numerical Method Inc.: is a large collection of numerical algorithms including linear algebra, (advanced) optimization, interpolation, Markov model, principal component analysis, time series analysis, hypothesis testing, regressions, statistics, ordinary and partial differential equation solvers, and suanshu.
• NLinear is a generic linear algebra toolkit in C# compatible with Silverlight.
• Math.NET Numerics aims to provide methods and algorithms for numerical computations in science, engineering and every day use. Covered topics include special functions, linear algebra, probability models, random numbers, interpolation, integral transforms and more. MIT/X11 license.
• Meta.Numerics is a library for advanced scientific computation in the .NET Framework.
• The NAG Library for .NET is a collection of mathematical and statistical routines for Microsoft .NET.
• Common Intermediate Dynamic Scientific Library (CIDSL): open source automated scientific data calculation library constituting automated functions that calculate the characteristics of a physical body with the data available .
• The Extreme Optimization Numerical Libraries for .NET are a comemercial collection of mathematical and statistical classes for Microsoft .NET.

FORTRAN

• BLAS (Basic Linear Algebra Subprograms) is a de facto application programming interface standard for publishing libraries to perform basic linear algebra operations such as vector and matrix multiplication.
• CERNLIB is a collection of FORTRAN 77 libraries and modules.
• EISPACK is a software library for numerical computation of eigenvalues and eigenvectors of matrices, written in FORTRAN. It contains subroutines for calculating the eigenvalues of nine classes of matrices: complex general, complex Hermitian, real general, real symmetric, real symmetric banded, real symmetric tridiagonal, special real tridiagonal, generalized real, and generalized real symmetric matices.
• IMSL Numerical Libraries are cross-platform libraries containing a comprehensive set of mathematical and statistical functions that can be embedded in a users application.
• Harwell Subroutine Library is a collection of Fortran 77 and 95 codes that address core problems in numerical analysis.
• LAPACK, the Linear Algebra PACKage, is a software library for numerical computing originally written in FORTRAN 77 and now written in Fortran 90.
• LINPACK is a software library for performing numerical linear algebra on digital computers. It was written in Fortran by Jack Dongarra, Jim Bunch, Cleve Moler, and Pete Stewart, and was intended for use on supercomputers in the 1970s and early 1980s. It has been largely superseded by LAPACK, which will run more efficiently on modern architectures.
• Lis is a scalable parallel library for solving systems of linear equations and standard eigenvalue problems with real sparse matrices using iterative methods.
• MINPACK is a library of FORTRAN subroutines for the solving of systems of nonlinear equations, or the least squares minimization of the residual of a set of linear or nonlinear equations.
• The NAG Fortran Library is a collection of mathematical and statistical routines for Fortran.
• NOVAS is a software library for astrometry-related numerical computations. Both Fortran and C versions are available.
• Netlib is a repository of scientific computing software which contains a large number of separate programs and libraries including BLAS, EISPACK, LAPACK and others.
• PAW is a free data analysis package developed at CERN.
• Portable, Extensible Toolkit for Scientific Computation (PETSc), is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations.
• QUADPACK is a FORTRAN 77 library for numerical integration of one-dimensional functions
• SLATEC is a FORTRAN 77 library of over 1400 general purpose mathematical and statistical routines.
• SOFA is a collection of subroutines that implement official IAU algorithms for astronomical computations. Both Fortran and C versions are available.
• SPARSKIT is a tool package for working with sparse matrices.
• ARPACK is a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems.
• SHTOOLS is an archive of fortran 95 based software that can be used to perform (among others) spherical harmonic transforms and reconstructions, rotations of spherical harmonic coefficients, and multitaper spectral analyses on the sphere.

JAVA

• Apache Commons, is an open-source for creating reusable Java components. It has numerical packages for linear algebra and non-linear optimization.
• Colt provides a set of Open Source Libraries for High Performance Scientific and Technical Computing.
• Efficient Java Matrix Library (EJML) is an open-source linear algebra library for manipulating dense matrices.
• JAMA, a numerical linear algebra toolkit for the Java programming language. No active development has taken place since 2005, but it still one of the more popular linear algebra packages in Java.
• Jblas: Linear Algebra for Java, a linear algebra library which is an easy to use wrapper around BLAS and LAPACK.
• Parallel Colt is an open source library for scientific computing. A parallel extension of Colt.
• SCaViS, an open-source Java libraries for numerical calculations, data I/O and visualisation of scientific results. Used together with Jython.
• JMSL Numerical Libraries, is a comprehensive set of mathematical, statistical, data mining, financial and 2D/3D charting classes.
• Matrix Toolkit Java is a linear algebra library based on BLAS and LAPACK.
• OjAlgo is an open source Java library for mathematics, linear algebra and optimisation.
• exp4j is a small Java library for evaluation of mathematical expressions.
• la4j is a tiny Java library and it provides linear algebra primitives and algorithms.
• Universsal Java Matrix Package UJM is a Java library which provides implementations for sparse and dense matrices, as well as linear algebra calculations such as matrix decomposition, inverse, multiply, mean, correlation, standard deviation, etc.
• Java Matrix Library A thoughtfully designed Java library for dealing with Matrices ( Matrix Theory ). Simple, Intuitive and Flexible. The library revolves around a Matrix interface. Operations are thoughtfully distributed. Not guilty of exposing all-in-one Matrix class.
• JDistLibJava Statistical Distribution Library A Java package that provides routines for various statistical distributions. A manual translation of distributions provided by R statistical package.

PERL

• Perl Data Language gives standard Perl the ability to compactly store and speedily manipulate the large N-dimensional data arrays which are the bread and butter of scientific computing.

PYTHON

• SCaViS scientific libraries based on Jython, Java implementation of the Python language.
• matplotlib, a MATLAB-like plotting library.
• NumPy, a library that adds support for the manipulation of large, multi-dimensional arrays and matrices; it also includes a large collection of high-level mathematical functions. NumPy serves as the backbone for a number of other numerical libraries, notably SciPy.
• Sage Math is a large mathematical software application which integrates the work of nearly 100 free software projects and supports linear algebra, combinatorics, numerical mathematics, calculus, and more.
• Plotly, a web-based scientific plotting library.
• SciPy, a large library of scientific tools. Includes NumPy
• ScientificPython, a library with a different set of scientific tools