Oberwolfach References on Mathematical Software

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The primary goal of 3D-XplorMath is to allow users with little or no programming experience to see, with minimal effort, concrete visual representations of many different categories of mathematical objects and processes. To accomplish this, objects from each category are described internally by well-designed, parameterized data structures, and for each category a variety of rendering methods is provided to permit visualization of objects of the category in ways that are appropriate for various purposes. Each of the hundreds of built-in objects known to the program is assigned carefully chosen defaults so that, when the object is selected from a menu, the program can immediately construct a standard example of the object and render it in an optimized view. The user may then use various menus and dialogs to alter the parameters describing the shape and coloration of the object, change the viewpoint from which it is seen, select different rendering methods, etc. Moreover, as its name suggests, the program can display objects such as surfaces, space curves and polyhedra using various stereo techniques. In addition to the many built-in objects known to the program, a user can create "user-defined" objects by entering formulas using standard mathematical notation. Visualizations created by the program can be saved in jpeg and other graphic formats and the data defining 3D objects can be exported to other 3D programs (e.g., Bryce or POV-Ray) in formats such as .obj and .inc. Both built-in and user-defined objects can depend on parameters, and the program can create morphing animations by moving along a path in the parameter space, and these animations can then be saved as QuickTime movies. Each of the built-in objects has associated to it a so-called ATO (About This Object) file that provides documentation for the object. An early and more developed version of the program, written in Object Pascal, runs under the Macintosh Operating System and a Java-based cross-platform version is now also available.

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Axiom is a general purpose Computer Algebra system. It is useful for research and development of mathematical algorithms. It defines a strongly typed, mathematically correct type hierarchy. It has a programming language and a built-in compiler.

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Empirical gramians can be computed for linear and nonlinear control systems for purposes of model order reduction or system identification. Model reduction using empirical gramians can be applied to the state space, to the parameter space or to both through combined reduction. The emgr framework is a compact open source toolbox for (empirical) gramian-based model reduction and compatible with OCTAVE and MATLAB.

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LAPACK is written in Fortran77 and provides routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. The associated matrix factorizations (LU, Cholesky, QR, SVD, Schur, generalized Schur) are also provided, as are related computations such as reordering of the Schur factorizations and estimating condition numbers. Dense and banded matrices are handled, but not general sparse matrices. In all areas, similar functionality is provided for real and complex matrices, in both single and double precision.

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SuperLU is a general purpose library for the direct solution of large, sparse, nonsymmetric systems of linear equations on high performance machines. The library is written in C and is callable from either C or Fortran. The library routines will perform an LU decomposition with partial pivoting and triangular system solves through forward and back substitution. The LU factorization routines can handle non-square matrices but the triangular solves are performed only for square matrices. The matrix columns may be preordered (before factorization) either through library or user supplied routines. This preordering for sparsity is completely separate from the factorization. Working precision iterative refinement subroutines are provided for improved backward stability. Routines are also provided to equilibrate the system, estimate the condition number, calculate the relative backward error, and estimate error bounds for the refined solutions.

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SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries.

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