Function Approximations Package Symbol

NIntegrateInterpolatingFunction

NIntegrateInterpolatingFunction[f, {x, x_min, x_max}]
gives a numerical approximation to an integral with InterpolatingFunction objects in the integrand.

NIntegrateInterpolatingFunction[f, {x, x_min, x_max}, {y, y_min, y_max}, ...]
gives a numerical approximation to a multidimensional integral.

MORE INFORMATION

To use NIntegrateInterpolatingFunction, you first need to load the Function Approximations Package using Needs["FunctionApproximations`"].

NIntegrateInterpolatingFunction 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, NIntegrateInterpolatingFunction 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 NIntegrateInterpolatingFunction with InterpolatingFunction objects containing a large number of nodes may take significantly longer than using NIntegrate.

NIntegrateInterpolatingFunction has the same options as NIntegrate.

EXAMPLES

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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:

Needs["FunctionApproximations`"]

A trapezoidal approximation to

:

In[2]:=

Out[2]=

Since

is not smooth, NIntegrate will generate a warning message:

In[3]:=

Out[3]=

Using NIntegrateInterpolatingFunction produces a slightly more accurate answer without any error messages:

In[4]:=

Out[4]=

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

In[5]:=

Out[5]=

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[

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

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: