New in 9.0: Mathematics & Algorithms

Mathematica 9 adds major extensions and integration across the system. Probability and statistics adds survival and reliability analysis as well as a unified treatment for all random processes. Graphs and networks adds network flows, social network analysis, and major performance enhancements. Control systems adds PID autotuning as well as full support for descriptor and delay systems. Numerical differential equations now includes hybrid systems and parametric systems, as well as advanced support for differential algebraic equations.

Descriptive Statistics Dependency Measures »

SpearmanRho Spearman's rank correlation for data and distributions

KendallTau Kendall's rank correlation for data and distributions

BlomqvistBeta  ▪  GoodmanKruskalGamma  ▪  HoeffdingD  ▪  PillaiTrace  ▪  WilksW

Hypothesis Tests Independence Tests »

IndependenceTest test whether two vectors are independent

CorrelationTest test whether two vectors are uncorrelated

PearsonCorrelationTest  ▪  SpearmanRankTest  ▪  KendallTauTest  ▪  HoeffdingDTest  ▪  GoodmanKruskalGammaTest  ▪  BlomqvistBetaTest  ▪  PillaiTraceTest

New Parametric Distributions

HypoexponentialDistribution hypoexponential distribution

VarianceGammaDistribution variance gamma distribution

HyperexponentialDistribution  ▪  CoxianDistribution  ▪  MeixnerDistribution  ▪  TsallisQExponentialDistribution  ▪  TsallisQGaussianDistribution  ▪  LogMultinormalDistribution

Derived Distributions »

SplicedDistribution splicing several distributions such as different body and tail

CompoundPoissonDistribution Poisson distribution with given jump size distribution

OrderDistribution generalized to different and multivariate distributions

New Data Models

EventData censored (left, right, interval) and truncated data

WeightedData weighted data

Mean descriptive statistics works for censored and weighted data

EstimatedDistribution parametric distribution estimation from censored data

EmpiricalDistribution nonparametric distributions from censored data

Survival Analysis »

SurvivalModelFit survival functions with confidence bands from censored data

CoxModelFit Cox proportional hazards model from censored data

LogRankTest hypothesis test for equal hazard rates

Reliability Modeling and Analysis »

ReliabilityDistribution reliability block diagram model for a system

FailureDistribution fault tree model for a system

StandbyDistribution standby reliability distribution (cold, hot, ...)

BirnbaumImportance measure to pinpoint subsystem to improve for reliability

StructuralImportance  ▪  BarlowProschanImportance  ▪  ImprovementImportance  ▪  RiskAchievementImportance  ▪  RiskReductionImportance  ▪  CriticalityFailureImportance  ▪  CriticalitySuccessImportance  ▪  FussellVeselyImportance

Random Processes »

RandomFunction simulate a random process

TemporalData represent one or several time series data

EstimatedProcess process estimated from time series data

SliceDistribution distribution for a time slice of a random process

StationaryDistribution  ▪  CovarianceFunction  ▪  ...

Parametric Processes »

RandomWalkProcess  ▪  PoissonProcess  ▪  WienerProcess  ▪  ...

Finite Markov Processes »

DiscreteMarkovProcess  ▪  ContinuousMarkovProcess  ▪  ...

Queueing Processes »

QueueingProcess  ▪  QueueingNetworkProcess  ▪  ...

Time Series Processes »

ARMAProcess  ▪  ARIMAProcess  ▪  ...

Stochastic Differential Equation Processes »

ItoProcess  ▪  StratonovichProcess  ▪  ...

Differential Equations with Events »

WhenEvent discrete actions in differential equations when an event occurs

NDSolve now supports differential equations with events

NDSolveValue  ▪  DiscreteVariables

Differential Equations with Parameters

ParametricNDSolveValue solve differential equations depending on parameters

ParametricNDSolve  ▪  ParametricFunction

Differential Algebraic Equations

NDSolve can now solve high-index DAEs

NDSolveValue  ▪  ParametricNDSolve  ▪  ParametricNDSolveValue

Graphs & Networks »

FindPostmanTour find a tour that traverses each edge at least once

WeaklyConnectedComponents  ▪  EdgeConnectivity  ▪  VertexConnectivity  ▪  FindGraphPartition  ▪  PlanarGraphQ  ▪  ...

Social Network Analysis »

FindGraphCommunities find communities within a graph

GlobalClusteringCoefficient compute the global clustering coefficient

FindClique  ▪  FindKClique  ▪  FindKClan  ▪  FindKPlex  ▪  FindKClub  ▪  LambdaComponents  ▪  LuccioSamiComponents  ▪  GraphReciprocity  ▪  MeanClusteringCoefficient  ▪  LocalClusteringCoefficient  ▪  DegreeCentrality  ▪  BetweennessCentrality  ▪  ...

Network Flows and Cuts »

FindMaximumFlow find the maximum flow from a source vertex to a target vertex

FindMinimumCostFlow find the minimum cost flows

FindMinimumCut find the minimum cut for a graph

OptimumFlowData  ▪  FindEdgeCut  ▪  FindVertexCut  ▪  ...

Random Graphs »

GraphPropertyDistribution automatically compute properties of random graphs

SpatialGraphDistribution higher-performance random graph generation

Control Systems

PIDTune automatic design of PID controllers

Models with Time Delays »

SystemsModelDelay represent delays in state-space and transfer-function models

TransferFunctionModel  ▪  StateSpaceModel  ▪  SystemsModelDelayApproximate  ▪  SmithDelayCompensator  ▪  ...

Models with Algebraic Constraints »

DescriptorStateSpace whether to use descriptor state-space representation

StateSpaceModel  ▪  KroneckerModelDecomposition  ▪  ...

Fast Symbolic Linear and Polynomial Algebra

LinearSolve fast methods for sparse symbolic and exact systems

Expand fast methods for polynomials with numeric coefficients

NullSpace  ▪  MatrixRank  ▪  Det  ▪  Subresultants  ▪  PolynomialExtendedGCD

New Polynomial Functions

SubresultantPolynomials  ▪  SubresultantPolynomialRemainders

New Matrix Functions

MatrixLog matrix log function

MatrixFunction matrix function for a general function

New Algebraic Numbers

CubeRoot real cube root

Surd real th root

Vector Calculus »

Grad  ▪  Div  ▪  Curl  ▪  Laplacian

Coordinate Systems

CoordinateChartData properties of coordinate systems

TransformedField transform a field between coordinate systems

CoordinateTransform  ▪  CoordinateTransformData

Symbolic Tensor Algebra »

Arrays domain of arrays with given properties

Matrices  ▪  Vectors

TensorRank  ▪  TensorDimensions  ▪  TensorSymmetry

TensorProduct  ▪  TensorContract  ▪  TensorTranspose  ▪  ...

Symmetries and Symmetrized Arrays

SymmetrizedArray compact representation of arrays with symmetry

Symmetric  ▪  Antisymmetric

SymmetrizedIndependentComponents  ▪  SymmetrizedDependentComponents