FunctionApproximations`
FunctionApproximations`

# NIntegrateInterpolatingFunction

As of Version 6.0, NIntegrate natively supports InterpolatingFunction objects.

NIntegrateInterpolatingFunction[f,{x,xmin,xmax}]

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

NIntegrateInterpolatingFunction[f,{x,xmin,xmax},{y,ymin,ymax},]

gives a numerical approximation to a multidimensional integral.

# Examples

open allclose all

## Basic Examples(1)

A trapezoidal approximation to :

Since is not smooth, NIntegrate will generate a warning message: Using NIntegrateInterpolatingFunction 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:

## Scope(3)

NIntegrateInterpolatingFunction 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[π x]:

Accumulate the sampling points used by NIntegrateInterpolatingFunction:

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 f[x] near the points x= 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 NIntegrateInterpolatingFunction:

## Possible Issues(1)

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

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

Using NIntegrateInterpolatingFunction, the integral is broken up into integrals over a smaller domain, where the integrand is smooth: