Mathematica 6.0 represented a major new level in Mathematica's distinguished twenty-year history of broad cutting-edge algorithm development. Mathematica's unified architecture and unique tradition of integrating disparate algorithmic areas—together with new basic scientific and mathematical methodologies created at Wolfram Research—makes possible an ever-increasing rate of algorithm innovation in Mathematica.

New Mathematical Functions »

HankelH1  ▪  KelvinKer  ▪  SphericalBesselJ  ▪  ParabolicCylinderD  ▪  WhittakerM  ▪  SpheroidalS1  ▪  BesselJZero  ▪  AiryAiZero  ▪  ZernikeR  ▪  ZetaZero  ▪  ...

Major New Numerics Capabilities

FindMinimum, FindMaximum (modified) — now support constrained nonlinear optimization

FindFit (modified) — now supports constrained nonlinear fitting

Minimize, Maximize (modified) — now support integer programming

NIntegrate — now supports efficient high-dimensional integration

Interpolation, InterpolatingPolynomial (modified) — generalized to higher dimensions

New Matrix & Linear Algebra Functions »

Band — symbolic representation of generalized banded matrices & arrays

ArrayFlatten — generalized block matrix constructor

Normalize  ▪  Orthogonalize  ▪  Projection  ▪  KroneckerProduct  ▪  ...

LeastSquares  ▪  HermiteDecomposition  ▪  RotationMatrix  ▪  ...

New Symbolic & Algebraic Computing Functions »

Discriminant  ▪  CountRoots  ▪  SymmetricPolynomial  ▪  SymmetricReduction  ▪  ...

FullSimplify (modified) — now supports general equational theorem proving

SeriesCoefficient (modified) — now supports symbolic series

PadeApproximant  ▪  PossibleZeroQ  ▪  HornerForm  ▪  ...

Expand, Apart, ... (modified) — now automatically thread over equations

New Statistics Capabilities »

Mean, Variance, ... (modified) — now support symbolic distributions

MeanDeviation  ▪  Skewness  ▪  Kurtosis  ▪  CentralMoment  ▪  ...

NormalDistribution  ▪  PoissonDistribution  ▪  ChiSquareDistribution  ▪  ...

PDF, CDF — probability density function, cumulative distribution function

New Data Analysis Capabilities »

BinCounts  ▪  Tally  ▪  MovingAverage  ▪  MovingMedian  ▪  ...

Nearest — find nearest neighbors in multidimensional data

FindClusters — cluster analysis for numerical, Boolean, string, etc. data

EuclideanDistance  ▪  ManhattanDistance  ▪  CorrelationDistance  ▪  ...

New Computational Geometry Capabilities »

PolyhedronData — database of properties of regular polyhedra

Nearest  ▪  NearestFunction  ▪  RegionPlot  ▪  RegionPlot3D

RotationTransform  ▪  AffineTransform  ▪  TransformationMatrix  ▪  ...

New in Discrete Math

GraphPlot, GraphPlot3D, LayeredGraphPlot — lay out graphs of any size

GraphData — database of named graphs and their properties

TreePlot — display trees

FindShortestTour — solve traveling salesman problems

New in Computational Systems

TuringMachine — general 1D, 2D, etc. Turing machines

CellularAutomaton (modified) — extended to allow more general rules

New Mathematical Data Sources »

PolyhedronData  ▪  GraphData  ▪  LatticeData  ▪  KnotData  ▪  ExampleData

New in Number Theory & Integer Functions »

IntegerPartitions  ▪  PowersRepresentations  ▪  ChineseRemainder  ▪  FrobeniusSolve  ▪  PrimitiveRoot  ▪  BellB  ▪  RamanujanTau  ▪  ...

AlgebraicNumber — implicit representation of a general algebraic number

AlgebraicNumberNorm  ▪  NumberFieldDiscriminant  ▪  ToNumberField  ▪  ...

RootApproximant — find an algebraic approximation to a number

Divisible  ▪  CoprimeQ  ▪  NextPrime  ▪  RandomPrime  ▪  SquareFreeQ  ▪  ...

IntegerLength  ▪  IntegerString  ▪  BitSet  ▪  BitShiftLeft  ▪  ...

New Random Generation Functions »

RandomInteger  ▪  RandomReal  ▪  RandomChoice  ▪  RandomSample  ▪  ...

Some Functions with Major New Algorithms

NIntegrate  ▪  FindMinimum  ▪  FunctionExpand  ▪  LinearSolve  ▪  DSolve  ▪  RSolve  ▪  Sum  ▪  Integrate  ▪  Series  ▪  NDSolve  ▪  Interpolation