FunctionInterpolation
 Usage
Notes
 Further Examples
FunctionInterpolation can be used to simplify a combination of InterpolatingFunction objects. It can also be used to make an approximate function to a desired precision and accuracy goal from a function that is costly to evaluate. Here is a function with argument t that gives the x value of the intersection of the graphs y = x and y = Exp[t x].
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If it is necessary to compute it repeatedly, it is useful to construct a faster approximate function.
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Now the approximation can be used without having to call FindRoot again, as in a plot, for example.
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There happens to be a solution for the equation in Mathematica.
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The approximation was computed so that the default accuracy and precision goals ( digits) would be met if possible. The plot indicates that these goals were fairly well met.
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FunctionInterpolation provides an easy way to combine InterpolatingFunction objects with other functions to produce a new approximate function. The function int is an InterpolatingFunction object that depends on the parameter  .
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The function fint is an InterpolatingFunction object that interpolates over a two-dimensional domain, with t varying over the domain of int and x positive.
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Its values approximate the function.
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The function int3 is a three-dimensional InterpolatingFunction object.
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A "slice" can be taken along a complicated curve or surface. Here is an InterpolatingFunction object that represents the values of int3 on the cone  .
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This InterpolatingFunction object that represents the values along a section of a helix.
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Let's clean up by getting rid of the symbols defined in these examples.
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