New in Version 5.1
• New highly-enhanced algorithms for high-precision LinearSolve.
• Internal vectorization of high-precision vector operations.
• New high-performance methods for MatrixExp.
• Support for sparse singular value decomposition.
• Support for HessenbergDecomposition.
• Typeset notation for Transpose and Conjugate.
• Numerical integration of discontinuous piecewise functions.
• Numerical integration over implicitly defined regions.
• Support for event detection in NDSolve.
• Additional convenience functions Boole, Clip and Rescale.
• Package for cluster analysis and dendrograms.
• Package for interactive exploration of differential equation systems.
• General vector derivatives, including gradient, Hessian and Jacobian.
• Piecewise construct for representing general piecewise functions.
• Simplification with piecewise and nested piecewise functions.
• Reduction of piecewise equations and inequalities, including quantifiers.
• Limit, Series and D support for general piecewise functions.
• Indefinite and definite integration of general piecewise functions.
• Support for solving piecewise ordinary differential equations.
• Symbolic multiple integration over regions defined by inequalities.
• Enhanced support for solving Abel and other differential equations.
• Support for linear differential equations with non-rational coefficients.
• Nonlinear partial differential equation solutions based on complete integrals.
• Support for equations with multiple moduli in Reduce.
• Additional methods for solving Diophantine equations.
Language and core system
• Full support for optimized string pattern matching.
• Integrated string and expression pattern language.
• General support for complement patterns with Except.
• String patterns integrated into all string operations.
• Generalized StringCases for string analysis.
• New functions StringSplit, StringCount and StringReplaceList.
• RegularExpression construct for compact string pattern notation.
• English-language dictionary package.
• Support for generalized Tuples and Subsets.
• Expression filtering function Pick.
• Package for benchmarking of computer systems.
Data handling and visualization
• ArrayPlot for flexible large-scale array visualization.
• Package for fully-automated network and tree layout in 2D and 3D.
• Highly optimized import and export of binary data.
• Import and export of XLS spreadsheet files.
• Support for HDF5, MAT (v5), DIF, and PCX.
• Export of AVI movie files.
• Import from http and ftp URLs.
• Automated encoding and decoding of .gz files.
• Integrated TeX import and parsing in notebooks.
• Symbolic names for common colors such as Red and Black.
• DatabaseLink for universal cross-platform database connectivity.
• Bundled drivers for most common database systems.
• Integrated language interface for database discovery, query and updating.
• Graphical interface for database connection and exploration.
• Bundled SQL engine for creating custom databases.
• Integrated GUIKit for building standalone user interfaces.
• Platform-independent Mathematica language GUI specification.
• Over 100 types of controls and widgets.
• Automatic layout for complex dialog boxes.
• System for creating sequential wizard interfaces.
• Large library of sample GUI applications.
• Transparent access to web services from within Mathematica.
• Support for SOAP and WSDL.
• Packages for search and lookup on Wolfram Research and other sites.