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Computer Algebra Systems
(Maple, Reduce, MuPAD, Magma, Axiom, Maxima, ...)
Computer algebra has been a central component of Mathematica's overall vision since its inception, and indeed Mathematica's original release in 1988 was what first brought computer algebra into the mainstream. For over 20 years, Wolfram Research has been a consistent leader in computer algebra research, implementing and inventing an unsurpassed number of new methods and algorithms—and indeed shaping the very concept of computer algebra.
While a variety of systems have been developed to provide basic computer algebra functionality, Mathematica is unique not only in delivering far greater depth and quality of coverage, but also in tightly integrating computer algebra into a unified overall framework. This integration is what has allowed Mathematica to bring computer algebra into industrial applications. And in education, it makes computer algebra an increasingly compelling educational tool, by linking it not only to the best in static visualization, but also now to a new generation of dynamic exploratory visualization and instant interactive interface creation technology. In addition, the Wolfram Demonstrations Project provides thousands of prebuilt resources for applying computer algebra in education.
In recent years, Mathematica's integration of efficient arbitrary-precision numerics, special functions, number theory, discrete mathematics, computational geometry and other areas has allowed Mathematica to drive the development of major new classes of computer algebra algorithms that could not realistically be implemented in narrowly defined computer algebra systems.
Computer Algebra System Features in Mathematica:
- Symbolic handling of arbitrary formulas
- Exact and arbitrary-precision arithmetic »
- Symbolic expansion, factoring, simplification and substitution »
- Symbolic integration, differentiation, summation, limits and series »
- Symbolic solvers for systems of equations, differential equations, and difference equations »
- Full support of elementary and special functions »
- Exact and symbolic matrix operations »
Key Advantages of Mathematica for Computer Algebra:
- Computer algebra seamlessly integrated with numerics, graphics, programming, etc.
- Integrated document interface with full 2D traditional math input »
- Real-time interactive computer algebra using GUI controls
- Professionally supported system, available on all standard computer platforms
- Unified system design, maximizing interoperability of all system functions »
- Consistent language and interface, optimized for rapid learning »
- Full automation of superfunctions such as Solve, with automatic algorithm selection »
- Many original mathematical algorithms developed specifically for Mathematica
- Consistent handling of mathematical subtleties such as function branch cuts »
- Complete handling of real, complex, integer domains for equations, inequalities, etc. »
- Full symbolic and numeric support for 250+ special functions »
- Fully integrated high-performance numerics, linear algebra, etc. »
- Arbitrary-precision numerics for all functions, with automatic precision control »
- Fully editable and interpretable 2D traditional math notation »
- World-class software quality assurance, with millions of computer algebra tests
- Extensive built-in mathematical data collections »
- Integration with MathWorld's encyclopedic website of mathematical information
- Thousands of ready-built interactive demonstrations of mathematical concepts »
Interoperability with Computer Algebra Systems:
- Import and export of hundreds of formats, including MathML and TEX »
- Web interface using webMathematica
- MathLink API for serializing and sending structured data between programs »
- Third-party support for OpenMath
Interesting Tidbits:
- Wolfram Research has the world's most active computer algebra research organization
- Mathematica has been the basis for many thousands of mathematical papers
- Algorithms for Mathematica are increasingly discovered using NKS algorithm search methodology
- Mathematica has been a key element of calculus reform for nearly 20 years
- Mathematica was the first commercially successful system with computer algebra capabilities
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