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NIntegrateInterpolatingFunction

As of Version 6.0, NIntegrate natively supports InterpolatingFunction objects.

gives a numerical approximation to an integral with InterpolatingFunction objects in the integrand.

gives a numerical approximation to a multidimensional integral.

MORE INFORMATION

To use , you first need to load the Function Approximations Package using .

uses the function NIntegrate, but it breaks up the domain of integration into sections where the InterpolatingFunction objects are smooth.

If the integrand f does not contain any InterpolatingFunction objects, is equivalent to NIntegrate.

The arguments of the InterpolatingFunction objects may themselves be univariate functions of the integration variables.

If the integrand f is simply an InterpolatingFunction object, it is better to use Integrate because this gives a result that is exact for the polynomial approximation used in the InterpolatingFunction object.

Numerically integrating a multidimensional integral using with InterpolatingFunction objects containing a large number of nodes may take significantly longer than using NIntegrate.

has the same options as NIntegrate.

Basic Examples (1)

A trapezoidal approximation to

:

Since

is not smooth, NIntegrate will generate a warning message:

Using

produces a slightly more accurate answer without any error messages:

In this case the integrand is simply an interpolating function, so you can use Integrate to check:

Needs["FunctionApproximations`"]

A trapezoidal approximation to

:

Out[2]=

Since

is not smooth, NIntegrate will generate a warning message:

Out[3]=

Using

produces a slightly more accurate answer without any error messages:

Out[4]=

In this case the integrand is simply an interpolating function, so you can use Integrate to check:

Out[5]=

threads element-wise over the first argument:

Complex-valued interpolation:

Multidimensional integrals:

Generalizations & Extensions (1)

The arguments of the interpolating function may themselves be univariate functions of the integration variables:

Properties & Relations (1)

A trapezoidal approximation to Sin

:

Accumulate the sampling points used by

:

Plot the sampling points. The function is sampled at the x coordinates in the order of the y coordinates:

Accumulate the sampling points used by NIntegrate:

With NIntegrate, the nonsmooth behavior of

near the points

produces an error message and requires many recursive steps to evaluate accurately:

Increasing the order of the interpolation will produce a smoother function:

With a smoother function, fewer function evaluations are needed by NIntegrate:

If the interpolation is smooth enough, NIntegrate will require fewer function evaluations than

:

Possible Issues (1)

Multidimensional interpolating functions with a large number of nodes may take much longer to integrate using

instead of NIntegrate:

With NIntegrate, only one integral is evaluated, but the nonsmooth behavior generates many recursive steps:

Using

, the integral is broken up into

integrals over a smaller domain, where the integrand is smooth:

Interpolation Integrate NIntegrate

Function Approximations Package

Function Approximations Package

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