## 16 Search Results

### Fermat

Fermat is a super calculator - computer algebra system, in which the basic items being computed can be rational numbers, modular numbers, elements of finite fields, multivariable polynomials, multivariable rational functions, or multivariable polynomials modulo other polynomials. Fermat is available for Mac OS, Windows, Unix, and Linux. It is shareware. The basic “ground ring" F is the field of rational numbers. One may choose to work modulo a specified integer n, thereby changing the ground ring F from Q to Z/n. On top of this may be attached any number of unevaluated variables t_1, t_2, .. t_n., thereby creating the polynomial ring F[t_1, t_2, .. t_n] and its quotient field, the rational functions. Further, polynomials p, q, .. can be chosen to mod out with, creating the quotient ring F(t_1, t_2, ..)/[p, q, ...]. It is possible to allow Laurent polynomials. Once the computational ring is established in this way, all computations are of elements of this ring.

More information### GAP

GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. GAP is developed by international cooperation. The system, including source, is distributed freely under the terms of the GNU General Public License. You can study and easily modify or extend GAP for your special use. The current version is GAP 4, the older version GAP 3 is still available.

More information### Global Optimization Toolbox For Maple

Optimization is the science of finding solutions that satisfy complicated constraints and objectives. In engineering, constraints may arise from technical issues. In business, constraints are related to many factors, including cost, time, and staff. The objective of global optimization is to find [numerically] the absolute best solution of highly nonlinear optimization models that may have a number of locally optimal solutions. Global optimization problems can be extremely difficult. Frequently engineers and researchers are forced to settle for solutions that are “good enough” at the expense of extra time, money, and resources, because the best solution has not been found. Using the Global Optimization Toolbox, you can formulate your optimization model easily inside the powerful Maple numeric and symbolic system, and then use world-class Maple numeric solvers to return the best answer, fast! Illustrative references: 1. Pintér, J. D. Global Optimization in Action. Springer Science, 1996, 512 p., ISBN: 978-0-7923-3757-7 Winner of the 2000 INFORMS Computing Society Prize. 2. Pintér, J. D., Linder, D. and Chin, P. Global Optimization Toolbox for Maple: An introduction with illustrative applications. Optimization Methods and Software 21 (2006) (4) 565-582.

More information### HiFlow³

HiFlow³ is a multi-purpose finite element software providing powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differential equations. Based on object-oriented concepts and the full capabilities of C++ the HiFlow³ project follows a modular and generic approach for building efficient parallel numerical solvers. It provides highly capable modules dealing with the mesh setup, finite element spaces, degrees of freedom, linear algebra routines, numerical solvers, and output data for visualization. Parallelism – as the basis for high performance simulations on modern computing systems – is introduced on two levels: coarse-grained parallelism by means of distributed grids and distributed data structures, and fine-grained parallelism by means of platform-optimized linear algebra back-ends.

More information### Isabelle

Isabelle is a popular generic theorem prover developed at Cambridge University and TU Munich. Existing logics like Isabelle/HOL provide a theorem proving environment ready to use for sizable applications. Isabelle may also serve as framework for rapid prototyping of deductive systems. It comes with a large library including Isabelle/HOL (classical higher-order logic), Isabelle/HOLCF (Scott's Logic for Computable Functions with HOL), Isabelle/FOL (classical and intuitionistic first-order logic), and Isabelle/ZF (Zermelo-Fraenkel set theory on top of FOL).

More information### LiE

LiE is the name of a software package that enables mathematicians and physicists to perform computations of a Lie group theoretic nature. It focuses on the representation theory of complex semisimple (reductive) Lie groups and algebras, and on the structure of their Weyl groups and root systems. LiE does not compute directly with elements of the Lie groups and algebras themselves; it rather computes with weights, roots, characters and similar objects.

More information### LinBox

LinBox is a C++ template library for exact, high-performance linear algebra computation with dense, sparse, and structured matrices over the integers and over finite fields. LinBox has the following top-level functions: solve linear system, matrix rank, determinant, minimal polynomial, characteristic polynomial, Smith normal form and trace. A good collection of finite field and ring implementations is provided, for use with numerous black box matrix storage schemes.

More information### MuPad

MuPAD is a mathematical expert system for doing symbolic and exact algebraic computations as well as numerical calculations with almost arbitrary accuracy. For example, the number of significant digits can be chosen freely. Apart from a vast variety of mathematical libraries the system provides tools for high quality visualization of 2- and 3-dimensional objects. On Microsoft Windows, Apple Macintosh and Linux systems, MuPAD offers a flexible notebook concept for creating mathematical documents combining texts, graphics, formulas, computations and mathematical visualizations and animations. On Microsoft Windows MuPAD further supports the technologies OLE, ActiveX Automation, DCOM, RTF and HTML. Thus it offers a natural integration in Office applications like Word or PowerPoint as well as others.

More information### Normaliz

Normaliz is a tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones. Its input data can be specified in terms of a system of generators or vertices or a system of linear homogeneous Diophantine equations, inequalities and congruences or a binomial ideal. Normaliz computes the dual cone of a rational cone (in other words, given generators, Normaliz computes the defining hyperplanes, and vice versa), a placing (or lexicographic) triangulation of a vector configuration (resulting in a triangulation of the cone generated by it), the Hilbert basis of a rational cone, the lattice points of a lattice polytope, the normalization of an affine monoid, the Hilbert (or Ehrhart) series and the Hilbert (or Ehrhart) (quasi) polynomial under a Z-grading (for example, for rational polytopes), NEW: generalized (or weighted) Ehrhart series and Lebesgue integrals of polynomials over rational polytopes via NmzIntegrate, a description of the cone and lattice under consideration by a system of inequalities, equations and congruences

More information### PolyBoRi

The core of PolyBoRi is a C++ library, which provides high-level data types for Boolean polynomials and monomials, exponent vectors, as well as for the underlying polynomial rings and subsets of the powerset of the Boolean variables. As a unique approach, binary decision diagrams are used as internal storage type for polynomial structures. On top of this C++-library we provide a Python interface. This allows parsing of complex polynomial systems, as well as sophisticated and extendable strategies for Gröbner base computation. PolyBoRi features a powerful reference implementation for Gröbner basis computation.

More information### polymake

polymake is an object-oriented system for experimental discrete mathematics. The typical working cycle of a polymake user starts with the construction of an object of interest, auch as a convex polytope, a finite simplicial complex, a graph, etc. It is then possible to ask the system for some of the object's properties or for some form of visualization. Further steps might include more elaborate constructions based on previously defined objects. Each class of polymake objects comes with a set of rules which describe how a new property of an object can be derived from previously known ones. It is a key feature that the user can extend or modify the set of rules, add further properties or even new classes of objects (with entirely new rule bases). The functions provided include: several convex hull algorithms, face lattices of convex polytopes, Voronoi diagrams and Delaunay decompositions (in arbitrary dimensions), simplicial homology (with integer coefficients), simplicial cup and cap products, intersection forms of triangulated 4-manifolds. Several forms of (interactive) visualization via interfaces to Geomview, JavaView and other programs.

More information### rbMIT

The rbMIT © MIT software package implements in Matlab® all the general reduced basis algorithms. The rbMIT © MIT software package is intended to serve both (as Matlab® source) "Developers" — numerical analysts and computational tool-builders — who wish to further develop the methodology, and (as Matlab® "executables") "Users" — computational engineers and educators — who wish to rapidly apply the methodology to new applications. The rbMIT software package was awarded with the Springer Computational Science and Engineering Prize in 2009.

More information### RCWA

RCWA is a package for the computer algebra system GAP. It provides implementations of algorithms and methods for computing in certain infinite permutation groups. The class of groups which RCWA in principle can deal with includes the finite groups, the free groups of finite rank, the free products of finitely many finite groups, certain infinite simple groups, certain divisible torsion groups and groups of many further types. It is closed under taking direct products and under taking wreath products with finite groups and with the infinite cyclic group (Z,+).

More information### SINGULAR

SINGULAR is a Computer Algebra system for polynomial computations in commutative algebra, algebraic geometry, and singularity theory. SINGULAR's main computational objects are ideals and modules over a large variety of baserings. The baserings are polynomial rings over a field (e.g., finite fields, the rationals, floats, algebraic extensions, transcendental extensions), or localizations thereof, or quotient rings with respect to an ideal. SINGULAR features fast and general implementations for computing Groebner and standard bases, including e.g. Buchberger's algorithm and Mora's Tangent Cone algorithm. Furthermore, it provides polynomial factorizations, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, and many more related functionalities. Based on an easy-to-use interactive shell and a C-like programming language, SINGULAR's internal functionality is augmented and user-extendible by libraries written in the SINGULAR programming language. A general and efficient implementation of communication links allows SINGULAR to make its functionality available to other programs.

More information### Sums over integral points of a polygon

Maple program for computing the sum of values of a polynomial function over the set of integral points of a polygon and the corresponding weighted Ehrhart quasi-polynomial.

More information### SYNAPS

SYNAPS (Symbolic and Numeric APplicationS) is a library developed in C++. The aim of this open source project is to provide a coherent and efficient library for symbolic and numeric computation. It implements data-structures and classes for the manipulation of basic objects, such as (dense, sparse, structured) vectors, matrices, univariate and multivariate polynomials. It also provides fundamental methods such as algebraic number manipulation tools, different types of univariate and multivariate polynomial root solvers, resultant computations, ...

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