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Numerical Matrix Systems
(Matlab, O-Matrix, Scilab, Octave, ...)
Evolving from the Fortran subroutine libraries of the 1970s, numerical matrix systems are based on the idea of representing all computations in terms of machine-precision numerical matrices. Mathematica takes a vastly more general approach, allowing it to integrate a fundamentally broader and deeper range of capabilities—yet still routinely outperform specialized numerical matrix systems even on basic matrix operations.
With its broader modern approach, Mathematica immediately allows a much higher level of computation—directly manipulating formulas and equations symbolically, validating and extending numerical precision, automatically selecting optimal algorithms, integrating active computation and interfaces into presentation-quality documents, and routinely handling data with arbitrarily general structure.
Particularly in recent years, Mathematica's unified design and unique integration of algorithms and methods from numerical, symbolic, geometric, discrete and other areas have allowed the creation of major new generations of algorithms which greatly extend the type and range of computations that can routinely be done—and make possible new levels of efficiency even in seemingly straightforward numerical tasks.
Long used in large-scale projects across many industries, Mathematica's scalable core language, support for modern web and grid deployment, dynamic interface creation capabilities and broad integration with external systems make it a uniquely productive development environment that provides a stable long-term platform for existing and future projects.
Numerical Matrix System Features in Mathematica:
- Full coverage of standard matrix and array operations »
- Full state-of-the-art support for sparse matrices »
- Rich interactive language for array and matrix manipulation
- Unified auto-vectorization of all functions and operations »
- Built-in just-in-time compilation »
- Full built-in 2D and 3D visualization »
- Built-in numerical differential equation solving »
- Import and export of hundreds of standard and specialized data formats »
- Fully compatible on all standard and emerging computer platforms »
- Support for the latest platform-optimized internal libraries
- Support for modern distributed computing environments »
- Full, professional development environment supporting industry standards »
- Large worldwide user community »
- Full online documentation with 50,000+ examples »
- Extensive library of Mathematica-based books, courses, etc.
- 20-year history of system stability and progressive algorithm development »
Key Advantages of Mathematica Compared to Numerical Matrix Systems:
- Single unified system, with all functionality consistently built in
- Automatic selection of optimal robust algorithms »
- Total extensibility based on fully general underlying language »
- Consistent built-in support for operations on arrays of any depth »
- Seamless support for exact, arbitrary-precision and symbolic matrices »
- Automatic numerical precision tracking available in any computation »
- Fully integrated symbolic capabilities for formula manipulation »
- Traditional mathematical notation for input and output of formulas »
- Immediate ability to use symbolic functions as parameters
- Built-in local and global constrained nonlinear optimization »
- Built-in graph theory, discrete math, number theory, etc. »
- Mixed symbolic-numeric algorithms for enhanced robustness and efficiency
- Fully adaptive 2D and 3D function and data visualization »
- Integrated geometric computing »
- Automated aesthetics methodology for optimal graphics presentation
- Modern multiparadigm programming language »
- Unified overall design, consistent across all functions
- Document-centered interface providing formatted computation history
- 2D and 3D interactive graphics fully integrated into documents
- Instant full-function interface creation for arbitrary computations »
- Integrated support for database access »
- Built-in access to extensive curated data sources (financial, geospatial, scientific, ...) »
Interoperability with Numerical Matrix Systems:
- Import and export of all standard numerical data formats (CSV, XLS, MAT, HDF and binary data, ...) »
- Import and export of sparse matrix formats »
- Uniform MathLink API for C/C++, Java, .NET, etc. »
- Mathematica Player for free delivery of interactive run-time Mathematica applications
- webMathematica for web deployment
Interesting Tidbits:
- Mathematica's state-of-the-art coverage of numerical matrix operations represents only 5% of its complete internal code base
- Wolfram Research has pioneered many numeric-symbolic hybrid algorithms
- A large fraction of Mathematica's numerical algorithms are based on original research at Wolfram Research
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