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Advanced Numerical Integration in
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NIntegrate Introduction
Overview
Design
Features
Strategies, rules, and preprocessors
User extensibility
NIntegrate Integration Strategies
Introduction
Adaptive Strategies
Global Adaptive Strategy
MinRecursion and MaxRecursion
"MaxErrorIncreases"
Example implementation of a global adaptive strategy
Local Adaptive Strategy
MinRecursion and MaxRecursion
"InitialEstimateRelaxation"
"Partitioning"
Reuse of integrand values
Example implementation of a local adaptive strategy
"GlobalAdaptive" versus "LocalAdaptive"
"GlobalAdaptive" is generally better than "LocalAdaptive"
Singularity Handling
User specified singularities
Point singularities
Curve, surface, and hyper-surface singularities
Example implementation of curve, surface, and hyper-surface 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
One-dimensional integration
Multidimensional integration
Cauchy principal value integration
Sampling points visualization
Double-Exponential Strategy
MinRecursion and MaxRecursion
Comparison of double-exponential and Gaussian quadrature
Convergence rate
Example implementation of double-exponential quadrature
"Trapezoidal" Strategy
Example implementation
Oscillatory Strategies
Finite Region Oscillatory Integration
Extrapolating Oscillatory Strategy
Example implementation
Double-Exponential Oscillatory Integration
Generalized integrals
Non-algebraic multiplicand
Crude Monte Carlo and Quasi Monte Carlo Strategies
AccuracyGoal and PrecisionGoal
MaxPoints
"RandomSeed"
Stratified crude Monte Carlo integration
Convergence speed up of the stratified Monte Carlo integration
Global Adaptive Monte Carlo and Quasi Monte Carlo Strategies
MinRecursion and MaxRecursion
"Partitioning"
"BisectionDithering"
Choice of bisection axis
Example: Comparison with crude Monte Carlo
"MultiPeriodic"
Comparison with "MultiDimensionalRule"
Preprocessors
"SymbolicPiecewiseSubdivision"
"ExpandSpecialPiecewise"
"EvenOddSubdivision"
Transformation theorem
"VerifyConvergence"
"OscillatorySelection"
Working with sums of oscillating terms
"UnitCubeRescaling"
Example implementation
"SymbolicPreprocessing"
Examples and Applications
Closed-contour integrals
Fourier series calculation
NIntegrate Integration Rules
Introduction
Integration Rule Specification
"TrapezoidalRule"
Romberg quadrature
"TrapezoidalRule" sampling points and weights
"NewtonCotesRule"
"NewtonCotesRule" sampling points and weights
"GaussBerntsenEspelidRule"
"GaussBerntsenEspelidRule" sampling points and weights
"GaussKronrodRule"
"GaussKronrodRule" sampling points and weights
"LobattoKronrodRule"
"LobattoKronrodRule" sampling points and weights
"ClenshawCurtisRule"
"ClenshawCurtisRule" sampling points and weights
"MultiPanelRule"
"MultiPanelRule" sampling points and weights
"CartesianRule"
"CartesianRule" sampling points and weights
"MultiDimensionalRule"
"MultiDimensionalRule" sampling points and weights
"MonteCarloRule"
"AxisSelector"
Comparisons of the Rules
Number of points in a rule
Minimal number of sampling points
Rule comparison
Examples of Pathological Behavior
Tricking the error estimator
The wrong estimation
Better results
Why the estimator is misled
Phase errors
Index of Technical Terms
NIntegrate References
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