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Inverse Interpolating Functions

  • To: mathgroup at
  • Subject: [mg29221] Inverse Interpolating Functions
  • From: "David Park" <djmp at>
  • Date: Tue, 5 Jun 2001 04:21:51 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

Dear MathGroup,

Sometimes we may have an InterpolatingFunction, which is strictly increasing
over its domain. A typical case might be the arc length of a curve. Suppose
we want to calculate an inverse interpolating function. What is the best way
to do this?

When I looked at the structure on an InterpolatingFunction I found that it
contains a list of x points and a list of y points. So I naively thought I
might be able to construct the inverse interpolating function by
interchanging the x and y lists, and taking the limits of the new function
from the end points of the original y list.

Here is a routine to do that. I check that the original y list is strictly

    (And @@ Positive /@ ListConvolve[{1, -1}, #1[[4,2]]] & )] :=
  InterpolatingFunction[{{First[f[[4,2]]], Last[f[[4,2]]]}},
   f[[2]], {f[[4,2]]}, {f[[4,1]], f[[3,1]]}]

Here are several test cases. I plot the error in the inverse interpolation
and the error in the original interpolation.

f = FunctionInterpolation[Sin[x], {x, -(Pi/2), Pi/2}]
g = InverseInterpolatingFunction[f]
Plot[g[x] - ArcSin[x], {x, -1, 1}];
Plot[f[x] - Sin[x], {x, -(Pi/2), Pi/2}];

f = FunctionInterpolation[Sin[x], {x, -Pi/2, Pi/2},
   InterpolationPoints -> 401, PrecisionGoal -> Automatic,
   InterpolationPrecision -> Automatic, InterpolationOrder ->7]
g = InverseInterpolatingFunction[f]
Plot[g[x] - ArcSin[x], {x, -1, 1}];
Plot[f[x] - Sin[x], {x, -Pi/2, Pi/2}];

f = FunctionInterpolation[Sinh[x], {x, -3, 3}]
g = InverseInterpolatingFunction[f]
Plot[g[x] - ArcSinh[x], {x, -10, 10}];
Plot[f[x] - Sinh[x], {x, -3, 3}];

The results seem to be fairly good. But is this just a fluke? Is there a
simple and mathematically sound basis to generate an inverse interpolating
function from a strictly increasing interpolating function?

David Park
djmp at

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