MATHEMATICA OVERVIEW

Advanced Numerical Integration in Mathematica

NIntegrate IntroductionNIntegrate Introduction

OverviewOverview

Symbolic Preprocessing

Quadrature Rules

Oscillatory Integration

Automatic Singularity Handling

Special Strategies

Design

Strategies, Rules, and PreprocessorsStrategies, Rules, and Preprocessors

User Extensibility

NIntegrate Integration StrategiesNIntegrate Integration Strategies

Introduction

Adaptive Strategies

Global Adaptive StrategyGlobal Adaptive Strategy

MinRecursion and MaxRecursion

"MaxErrorIncreases"

Example Implementation of a Global Adaptive Strategy

Local Adaptive StrategyLocal Adaptive Strategy

MinRecursion and MaxRecursion

"InitialEstimateRelaxation"

"Partitioning"

Reuse of Integrand Values

Example Implementation of a Local Adaptive Strategy

"GlobalAdaptive" versus "LocalAdaptive""GlobalAdaptive" versus "LocalAdaptive"

"GlobalAdaptive" Is Generally Better than "LocalAdaptive"

Singularity HandlingSingularity Handling

User-Specified SingularitiesUser-Specified Singularities

Point Singularities

Curve, Surface, and Hypersurface Singularities

Example Implementation of Curve, Surface, and Hypersurface Singularity Handling

"SingularityHandler" and "SingularityDepth"

Use of the IMT Variable Transformation

IMT Transformation by Example

Use of Double Exponential Quadrature

IMT versus "DoubleExponential" versus No Singularity Handling for One-Dimensional Integrals

IMT Multidimensional Singularity Handling

Duffy's Coordinates for Multidimensional Singularity Handling

Duffy's Coordinates Strategy

Duffy's Coordinates Generalization and Example Implementation

Ignoring the Singularity

Automatic Singularity Handling Automatic Singularity Handling

One-Dimensional Integration

Multidimensional integration

Cauchy Principal Value IntegrationCauchy Principal Value Integration

Sampling Points Visualization

Double Exponential StrategyDouble Exponential Strategy

MinRecursion and MaxRecursion

Comparison of Double Exponential and Gaussian Quadrature

Convergence Rate

Example Implementation of Double Exponential Quadrature

"Trapezoidal" Strategy"Trapezoidal" Strategy

Example Implementation

Oscillatory Strategies

Finite Region Oscillatory Integration

Extrapolating Oscillatory Strategy Extrapolating Oscillatory Strategy

Example Implementation

Double Exponential Oscillatory IntegrationDouble Exponential Oscillatory Integration

Generalized Integrals

Nonalgebraic Multiplicand

Crude Monte Carlo and Quasi Monte Carlo StrategiesCrude Monte Carlo and Quasi Monte Carlo Strategies

AccuracyGoal and PrecisionGoal

MaxPoints

"RandomSeed"

Stratified Crude Monte Carlo Integration

Convergence Speedup of the Stratified Monte Carlo Integration

Global Adaptive Monte Carlo and Quasi Monte Carlo StrategiesGlobal Adaptive Monte Carlo and Quasi Monte Carlo Strategies

MinRecursion and MaxRecursion

"Partitioning"

"BisectionDithering"

Choice of Bisection Axis

Example: Comparison with Crude Monte Carlo

"MultiPeriodic""MultiPeriodic"

Comparison with "MultidimensionalRule"

PreprocessorsPreprocessors

"SymbolicPiecewiseSubdivision""SymbolicPiecewiseSubdivision"

"ExpandSpecialPiecewise"

"EvenOddSubdivision""EvenOddSubdivision"

Transformation Theorem

"VerifyConvergence"

"OscillatorySelection"

"UnitCubeRescaling""UnitCubeRescaling"

Example Implementation

"SymbolicPreprocessing"

Examples and ApplicationsExamples and Applications

Closed-Contour Integrals

Fourier Series Calculation

NIntegrate Integration RulesNIntegrate Integration Rules

Introduction

Integration Rule Specification

"TrapezoidalRule""TrapezoidalRule"

Romberg Quadrature

"TrapezoidalRule" Sampling Points and Weights

"NewtonCotesRule""NewtonCotesRule"

"NewtonCotesRule" Sampling Points and Weights

"GaussBerntsenEspelidRule""GaussBerntsenEspelidRule"

"GaussBerntsenEspelidRule" Sampling Points and Weights

"GaussKronrodRule""GaussKronrodRule"

"GaussKronrodRule" Sampling Points and Weights

"LobattoKronrodRule""LobattoKronrodRule"

"LobattoKronrodRule" Sampling Points and Weights

"ClenshawCurtisRule""ClenshawCurtisRule"

"ClenshawCurtisRule" Sampling Points and Weights

"MultipanelRule""MultipanelRule"

"MultipanelRule" Sampling Points and Weights

"CartesianRule""CartesianRule"

"CartesianRule" Sampling Points and Weights

"MultidimensionalRule""MultidimensionalRule"

"MultidimensionalRule" Sampling Points and Weights

"LevinRule""LevinRule"

Levin-Type Collocation Method

Specifying Oscillatory Kernel

"MonteCarloRule""MonteCarloRule"

"AxisSelector"

Comparisons of the RulesComparisons of the Rules

Number of Points in a RuleNumber of Points in a Rule

Minimal Number of Sampling Points

Rule Comparison

Examples of Pathological BehaviorExamples of Pathological Behavior

Tricking the Error EstimatorTricking the Error Estimator

The Wrong Estimation

Better Results

Why the Estimator Is Misled

Phase Errors

Index of Technical Terms

NIntegrate References

Tutorial Collection Tutorial Collection

Advanced Numerical Integration in Mathematica
Mathematics and Algorithms

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