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 f_{i} at successive integers 
Interpolation[{{x_{1},f_{1}},{x_{2},f_{2}},...}] 
 construct an approximate function with values f_{i} at points x_{i} 
Constructing approximate functions.
Here is a table of the values of the sine function.
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This constructs an approximate function which represents these values.
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The approximate function reproduces each of the values in the original table.
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It also allows you to get approximate values at other points.
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In this case the interpolation is a fairly good approximation to the true sine function.
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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 nonnumerical argument, the approximate function is left in symbolic form.
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Here is a numerical integral of the approximate function.
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Here is the same numerical integral for the true sine function.
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A plot of the approximate function is essentially indistinguishable from the true sine function.
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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 /6.
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The result is close to the exact one.
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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.
If you ask for a value outside of the domain, Mathematica prints a warning, then uses extrapolation to find a result.
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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 x_{i} 
Constructing approximate functions with specified derivatives.
This interpolates through the values of the sine function and its first derivative.
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This finds a better approximation to the derivative than the previous interpolation.
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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>3, 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.
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The approximate function consists of a collection of straightline segments.
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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 twodimensional 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.
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Here is the value at a particular position.
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Here is another array of values. 
To interpolate this array you explicitly have to tell Mathematica the domain it covers.
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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 threedimensional array. 
Mathematica can handle not only purely numerical approximate functions, but also ones which involve symbolic parameters.
This shows how the interpolated value at 2.2 depends on the parameters.
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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.
FunctionInterpolation[expr,{x,x_{min},x_{max}}] 
 construct an approximate function by evaluating expr with x ranging from x_{min} to x_{max} 
FunctionInterpolation[expr,{x,x_{min},x_{max}},{y,y_{min},y_{max}},...] 
 construct a higherdimensional approximate function 
Constructing approximate functions by evaluating expressions.