General Mathematical Libraries:

Linear Algebra

• BabelInterface Definition Language for scientific problems
• LAPACK
• BLAS
• High-Performance BLAS by Kazushige Goto
• ATLASImproved Lapack used in GAUSSIAN
• LinAl home page Good Linear algebra C++ library  with interface to LAPACK and more - not complete enough?
• UMFPACKUniveristy of Florida Library
• hypre ProjectHigh-Performance Preconditioners
• ARPACK++, a C++ template library for solving large-scale standard and generalized eigenvalue problems. Looks good but not developed any more?
• PyMat, a matrix package for the Python programming language.
• ESI, Equation  Solver Interface
• TeraScale - Parallel Mesh Object (PMO)
• JAMA, an elementary linear algebra package for Java.
• MTL, the Matrix Template Library. Dense and sparse matrices and vectors; banded, symmetric, triangular matrices; basic algorithms. C++.
• uBLAS, BLAS in C++ with expression templates.
• tvmet, a C++ library for "tiny" vectors and matrices with expression templates.
• GMM++, generic C++ template library for sparse, dense and skyline matrices, with solvers from ITL.
• MET, a C++ matrix library with expression templates, which eliminates the overhead of overloaded operators.
• SL++, the Scientific Library project. Will provide matrices, random numbers, complex, quaternions, plotting, and FFTs. C++.
• ALP, linear and polynomial algebra. Vectors, matrices, polynomials.
• SVMT: E. Robert Tisdale's proposal for a standard C++ Scalar, Vector, Matrix and Tensor Class Library (with implementation). Note: this is a proposal, not an official standard.
• GNUSSL [ftp only], the GNU Scientific Software Library. Linear algebra and arrays. C++.
• LAPACK++ (Linear Algebra PACKage in C++), solves systems of linear equations and eigenvalue problems on high performance computer architectures.
• IML++ A C++ template library for numerical iterative methods.
• MV++ Numerical Matrix/Vector Classes in C++
• SparseLib++ A library for sparse matrix computations, including the Sparse BLAS (Basic Linear Algebra Subprograms). C++.
• ISIS++, an object-oriented framework for solving sparse linear systems of equations. C++.
• The Template Numerical Toolkit (TNT) for linear algebra is a successor to the Lapack++, Sparselib++, IML++, and MV++ packages. Its goal is to integrate these ideas into a generic algorithmic library, supporting generic user-defined data types, and increasing its functionality. C++.
• LinAlg, basic linear algebra and optimization classes. C++.
• CAM C++ Class Library (Matrix, vector, and graphics classes)
• Newmat, a C++ matrix library (docs, download)
• CLHEP includes matrix classes, random number generators for the High Energy Physics (HEP) community. C++.
• BPKIT, Block Preconditioning Toolkit for iterative solution of linear systems. Callable from C++, C, or FORTRAN.

Differential Equations

• PETSc, object-oriented software for partial differential equations (programmed in C) - Seemed to be well supported, parlellized etc
• Rheoolef, finite element environment in C++.
• EXPDE, a C++ library for solving partial differential equations on semi-unstructured grids. Parallel.
• PZ, a C++ library for finite elements. 1-3 dimensions with hp-adaptivity, continuous or discontinuous, variety of matrix formats.
• MBDyn, multibody dynamics analysis in C++. Built-in parallelization via MPI/Metis.
• GETFEM++, a C++ finite element library, generic, arbitrary dimensions.
• MOUSE, a C++ library for finite volume computations on unstructured grids.
• DEAL, a C++ library for adaptive finite elements and error estimation. Supports SMPs.
• ODE++, a class library for ordinary differential equations. Explicit and linear-implicit ODE systems, IVP and BVP parameters, various solvers (Runge-Kutta, multistep, BDF). [English] [Deutsch]
• Diffpack, partial differential equations in C++. ($)
• TIDE: Classes for Ordinary Differential Equations. Provides vectors and matrices, extrapolation integrator for ODEs, sparse matrix classes, eigenvalues, shooting for two point BVPs, nonlinear solver, continuation
• BoxLib (docs), a class library supporting adaptive mesh refinement (AMR) schemes for computational fluid dynamics.
• OVERTURE, a C++ class library for solving PDEs in complicated domains. Includes adaptive mesh refinement and overlapping grids.
• FEMLIB (ftp, www) A Finite Element package [ftp only] by Michael Tiller, no longer supported)

Graph Theory

• Boost Graph Library, a general purpose, generic C++ library for graph data structures and graph algorithms.
• GTL, the Graph Template Library (C++).
• LEDA, a C++ library for graph theory and combinatorial computing.
• Graph Theory resources by Sanders
• The Stony Brook Algorithm Repository
• NAUTYGraph isomorphism library
• CLUTO A Software Package for Clustering High-Dimensional Datasets
• ParMETIS - Graph Partioning Sparse Matrix reodering library - Many libraries use it - probably good
• Eric Weisstein's World of Mathematics
• Bookshelf for Algorithms and Programming
• graph theory and its applications -- graph theory resources
• VFLIB Graph matching library - use in HARLEM

RANDOM Number generators

Finite-Element analysis

• Engineers Mall Finite Element Analysis (FEA)
• Deal IILarge package for finite element analysis

Arrays and Images

• POOMA II framework for scientific computing on sequential and parallel computers. Provides parallel arrays; fields, meshes, particles to come in version 2.1 (June 1999). C++.
• The Blitz++ class library: Array and Vector classes which rival Fortran's performance. C++.
• The P++ Library: Serial and Parallel Array Classes. C++. [BROKEN: AUTHOR CONTACTED]
• The AIPS++ Array and Image Classes (Astronomical Information Processing System). C++.
• PETE, an expression templates library -- add expression templates to your own array class.
• NumPy, Numerical extensions to Python.
• VIGRA, generic computer vision/image processing library.
• CPPIMA A C++ image processing library
• LIMP, Large Image Manipulation Program
• valarray<Troy> [ftp only], approximation of the valarray<T> class described in Ch. 26 of the ANSI/ISO C++ Standard. Uses expression templates for efficient evaluation.
• Image Understanding Environment (IUE), a DARPA project. C++.
• WAILI, a wavelet transform library in C++.

Neural Networks, genetic algorithms, machine learning, data mining

• PDP++, a neural-network simulation system written in C++
• EO -- Evolutionary Computation Framework
• CONICAL, C++ classes for building Neural Networks
• GALib, a C++ library for genetic algorithms
• Xelopes data mining library (Java, C++, C#)
• MLC++, Machine learning algorithms and data mining.

Calculations on JAVA

Java Numerics at NIST

Plotting programs and libraries

xmgr4

High-Energy Physics and Quantum Chemistry

• QC++, quantum chemistry software in C++, supporting MNDO, AM1 and PM3 models.
• FTensor, C++ class library for tensors.
• GluCat, Clifford algebra template library.
• Computational Thermodynamics Library
• Tech-X has made available C++ libraries related to particle accelerator design.

Multiprecision, arbitrary precision data types

• NTL, arbitrary length integers, vectors/matrices/polynomials over integers and over finite fields.
• EXTNUM, version of double which has the same number of mantissa bits as IEEE 754, but extends the exponent to have range from 10^^-646456993 to 10^^646456992.
• CLN, an extensive number library. Arbitrary precision integer, float, rational, polynomials, complex, modular integers, transcendental functions, assembly language kernels for some CPUs.
• MPFUN++, a multiple precision floating point computation package in C++.
• LiDIA, A library for computational number theory. Provides a collection of highly optimized implementations of various multiprecision data types and time-intensive algorithms.
• Apfloat, a C++ High Performance Arbitrary Precision Arithmetic Package
• hfloat, An arbitrary precision package, optimized for very large (> 1000) (decimal) digit numbers.
• fPoint, a C++ class which helps convert floating-point arithmetic to fixed-point arithmetic by recommending range and precision requirements.
• doubledouble, a quad-precision (approximately 30 decimal place) floating point arithmetic class.

Automatic differentiation and interval arithmetic

• FADBAD-TADIFF, a C++ package for automatic differentiation using any arithmetic (double or interval) by operator overloading.
• PROFIL/BIAS [English] [Deutsch], a C++ interval arithmetic class library.

Visualization

• The Visualization Toolkit (vtk), an extensive, free C++ library for scientific visualization.
• VisAD, a Java library for interactive and collaborative visualization and analysis of numerical data
• Java 2D Graph package. Includes contour plots, animation.
• Ptplot, a 2D data plotter in Java

Language interoperability/scripting

• CPPF77, a utility for interfacing C++ and Fortran 77 programs.
• Paul Dubois's code for interfacing Python and C++
• SWIG, generates Perl, Python, Tcl, Java, Eiffel and Guile wrappers for C++ libraries.
• SILOON (Scripting Interface Languages for Object-Oriented Numerics), toolkits and run-time support for building scripting interfaces to existing numerical codes in C, C++, and Fortran. Generates script bindings for Perl, Tcl, and Python.
• Matwrap, a tool which generates C++ wrapper code for matrix-oriented scripting languages such as Matlab 5, Octave, and tela.

Fourie Transform and others

• FFTPACK++, a C++ wrapper for FFTPACK complex routines using LAPACK++ Matrix and Vector classes.
• The FXT library of transforms. FFTs, Hartley, Number theoretic, Walsh, others coded in C++.

Optimization

• COOOL, an object-oriented optimization library
• OptSolve++, a C++ optimization library from Tech-X. (commercial)
• StarFLIP, optimization library for combinatorial problems with fuzzy constraints (C++)
• LM (Levenberg-Marquardt) implementation in Java for nonlinear least squares problems.

Miscellaneous

• GOOSE, GNU Object-Oriented Statistics Environment (C++).
• Borneo, a dialect of the Java language designed to have true support for the IEEE 754 floating point standard.
• SDTS++, a library for manipulating SDTS datasets (geographical information systems)
• Newran, C++ library for generating streams of random numbers.
• Multivariate polynomial interpolation library in C++
• MPI-2 C++ bindings (message passing library for parallel computers)
• CNCL, Communication networks simulation/analysis library (C++).
• OOMF, Object Oriented MicroMagnetic computing Framework at ITL/NIST
• CPPF77, a utility for interfacing C++ and Fortran 77 programs.
• The Standard Template Library (STL)
• The Bench++ Benchmark Suite
• EFLIB: Extended Function Library for Object-Pascal. Generalized arithmetics, equation solvers, more.

Tools

• TAU (Tuning and Analysis Utilities), explicit instrumentation of C++ libraries for profiling and tracing. For both serial and parallel codes.