# Approximate Functions and Interpolation

In many kinds of numerical computations, it is convenient to introduce

*approximate functions*. Approximate functions can be thought of as generalizations of ordinary approximate real numbers. While an approximate real number gives the value to a certain precision of a single numerical quantity, an approximate function gives the value to a certain precision of a quantity which depends on one or more parameters.

*Mathematica* uses approximate functions, for example, to represent numerical solutions to differential equations obtained with

NDSolve, as discussed in

"Numerical Differential Equations".

Approximate functions in

*Mathematica* are represented by

InterpolatingFunction objects. These objects work like the pure functions discussed in

"Pure Functions". The basic idea is that when given a particular argument, an

InterpolatingFunction object finds the approximate function value that corresponds to that argument.

The

InterpolatingFunction object contains a representation of the approximate function based on interpolation. Typically it contains values and possibly derivatives at a sequence of points. It effectively assumes that the function varies smoothly between these points. As a result, when you ask for the value of the function with a particular argument, the

InterpolatingFunction object can interpolate to find an approximation to the value you want.

Interpolation[{f_{1},f_{2},...}] | construct an approximate function with values at successive integers |

Interpolation[{{x_{1},f_{1}},{x_{2},f_{2}},...}] |

| construct an approximate function with values at points |

Constructing approximate functions.

Here is a table of the values of the sine function.

Out[1]= | |

This constructs an approximate function which represents these values.

Out[2]= | |

The approximate function reproduces each of the values in the original table.

Out[3]= | |

It also allows you to get approximate values at other points.

Out[4]= | |

In this case the interpolation is a fairly good approximation to the true sine function.

Out[5]= | |

You can work with approximate functions much as you would with any other

*Mathematica* functions. You can plot approximate functions, or perform numerical operations such as integration or root finding.

If you give a non-numerical argument, the approximate function is left in symbolic form.

Out[6]= | |

Here is a numerical integral of the approximate function.

Out[7]= | |

Here is the same numerical integral for the true sine function.

Out[8]= | |

A plot of the approximate function is essentially indistinguishable from the true sine function.

Out[9]= | |

If you differentiate an approximate function,

*Mathematica* will return another approximate function that represents the derivative.

This finds the derivative of the approximate sine function, and evaluates it at

.

Out[10]= | |

The result is close to the exact one.

Out[11]= | |

InterpolatingFunction objects contain all the information

*Mathematica* needs about approximate functions. In standard

*Mathematica* output format, however, only the part that gives the domain of the

InterpolatingFunction object is printed explicitly. The lists of actual parameters used in the

InterpolatingFunction object are shown only in iconic form.

In standard output format, the only part of an

InterpolatingFunction object printed explicitly is its domain.

Out[12]= | |

If you ask for a value outside of the domain,

*Mathematica* prints a warning, then uses extrapolation to find a result.

Out[13]= | |

The more information you give about the function you are trying to approximate, the better the approximation

*Mathematica* constructs can be. You can, for example, specify not only values of the function at a sequence of points, but also derivatives.

Interpolation[{{{x_{1}},f_{1},df_{1},ddf_{1},...},...}] |

| construct an approximate function with specified derivatives at points |

Constructing approximate functions with specified derivatives.

This interpolates through the values of the sine function and its first derivative.

Out[14]= | |

This finds a better approximation to the derivative than the previous interpolation.

Out[15]= | |

Interpolation works by fitting polynomial curves between the points you specify. You can use the option

InterpolationOrder to specify the degree of these polynomial curves. The default setting is

InterpolationOrder, yielding cubic curves.

This makes a table of values of the cosine function.

This creates an approximate function using linear interpolation between the values in the table.

Out[17]= | |

The approximate function consists of a collection of straight-line segments.

Out[18]= | |

With the default setting

InterpolationOrder, cubic curves are used, and the function looks smooth.

Out[19]= | |

Increasing the setting for

InterpolationOrder typically leads to smoother approximate functions. However, if you increase the setting too much, spurious wiggles may develop.

ListInterpolation[{{f_{11},f_{12},...},{f_{21},...},...}] |

| construct an approximate function from a two-dimensional grid of values at integer points |

ListInterpolation[list,{{x_{min},x_{max}},{y_{min},y_{max}}}] |

| assume the values are from an evenly spaced grid with the specified domain |

ListInterpolation[list,{{x_{1},x_{2},...},{y_{1},y_{2},...}}] |

| assume the values are from a grid with the specified grid lines |

Interpolating multidimensional arrays of data.

This interpolates an array of values from integer grid points.

Out[20]= | |

Here is the value at a particular position.

Out[21]= | |

Here is another array of values.

To interpolate this array you explicitly have to tell

*Mathematica* the domain it covers.

Out[23]= | |

ListInterpolation works for arrays of any dimension, and in each case it produces an

InterpolatingFunction object which takes the appropriate number of arguments.

This interpolates a three-dimensional array.

Out[25]= | |

*Mathematica* can handle not only purely numerical approximate functions, but also ones which involve symbolic parameters.

Out[26]= | |

This shows how the interpolated value at 2.2 depends on the parameters.

Out[27]= | |

With the default setting for

InterpolationOrder used, the value at this point no longer depends on

.

Out[28]= | |

In working with approximate functions, you can quite often end up with complicated combinations of

InterpolatingFunction objects. You can always tell

*Mathematica* to produce a single

InterpolatingFunction object valid over a particular domain by using

FunctionInterpolation.

Out[29]= | |

Out[30]= | |

Out[31]= | |

FunctionInterpolation[expr,{x,x_{min},x_{max}}] |

| construct an approximate function by evaluating expr with x ranging from to |

FunctionInterpolation[expr,{x,x_{min},x_{max}},{y,y_{min},y_{max}},...] |

| construct a higher-dimensional approximate function |

Constructing approximate functions by evaluating expressions.