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IntegrateNDSolve

NIntegrate

FilledSmallSquareNIntegrate[f, x, xmin, xmax] gives a numerical approximation to the integral .

FilledSmallSquare Multidimensional integrals can be specified, as in Integrate.

FilledSmallSquareNIntegrate tests for singularities at the end points of the integration range.

FilledSmallSquareNIntegrate[f, x, , , ... , ] tests for singularities at each of the intermediate points . If there are no singularities, the result is equivalent to an integral from to . You can use complex numbers to specify an integration contour in the complex plane.

FilledSmallSquare The following options can be given:

FilledSmallSquareNIntegrate usually uses an adaptive algorithm, which recursively subdivides the integration region as needed. In one dimension, GaussPoints specifies the number of initial points to choose. The default setting for GaussPoints is Floor[WorkingPrecision/3]. In any number of dimensions, MinRecursion specifies the minimum number of recursive subdivisions to try. MaxRecursion gives the maximum number.

FilledSmallSquareNIntegrate usually continues doing subdivisions until the error estimate it gets implies that the final result achieves either the AccuracyGoal or the PrecisionGoal specified.

FilledSmallSquare The default setting for PrecisionGoal is usually equal to the setting for WorkingPrecision minus 10 digits.

FilledSmallSquare If an explicit setting for MaxPoints is given, NIntegrate uses quasi Monte Carlo methods to get an estimate of the result, sampling at most the number of points specified.

FilledSmallSquare The default setting for PrecisionGoal is taken to be 2 in this case.

FilledSmallSquare You should realize that with sufficiently pathological functions, the algorithms used by NIntegrate can give wrong answers. In most cases, you can test the answer by looking at its sensitivity to changes in the setting of options for NIntegrate.

FilledSmallSquareN[Integrate[ ... ]] calls NIntegrate for integrals that cannot be done symbolically.

FilledSmallSquareNIntegrate has attribute HoldAll.

FilledSmallSquare Possible settings for Method are GaussKronrod, DoubleExponential, Trapezoidal, Oscillatory, MultiDimensional, MonteCarlo, and QuasiMonteCarlo. GaussKronrod and MultiDimensional are adaptive methods. MonteCarlo and QuasiMonteCarlo are randomized methods, appropriate for high-dimensional integrals.

FilledSmallSquare See The Mathematica Book: Section 1.6.2, Section 3.9.1, Section 3.9.2 and Section 3.9.3.

FilledSmallSquare Implementation Notes: see section A.9.4.

FilledSmallSquare See also: NDSolve, NSum.

FilledSmallSquare Related packages: NumericalMath`ListIntegrate`, NumericalMath`CauchyPrincipalValue`, NumericalMath`GaussianQuadrature`.

Further Examples

IntegrateNDSolve