Thread: Approximation function vs Interpolation function

1. Approximation function vs Interpolation function

Hi all,

Currently at collage we are using approximation and interpolation functions, but I cannot really understand what is the difference between them.
Can someone please explain to me the difference and when to use one instead of the other?

Any help would be appreciated.

2. Re: Approximation function vs Interpolation function

They're so similar that it's a little dangerous to say anything without seeing how your specific course has them defined.

That said, I think in general the distinction is that, although both interpolation and approximation functions are functions, interpolation functions are about finding a function that fits a set of points, whereas approximation functions are about finding a function that estimates another function. You interpolate on points, but approximate on functions.

If you're hunting for a function that produces 3 when t = 1, and 5 when t = 2, you interpolate on the points (1, 3) and (2, 5), probably choosing f(t) = 2t + 1 as your interpolation function.

If you're hunting for a function that estimates the function g(t) = t + t sin(t) for values of t near 0, you might choose $\displaystyle f(t) = t + t^2$ as your approximation function (another reasonable choice of an approximation function to g near t = 0 would be h(t) = t).

In general connotation, "interpolate" is about "a reasonable choice for something that's missing", but "approximate" is about "a good estimate of something that exists".

3. Re: Approximation function vs Interpolation function

What johnsomeone has described as an interpolating function is better called a collocating function.

A collocating ( = co - locating) function matches the values of a given function exactly at specified points.

An interpolating function provides a means of obtaining values of some function between known values.
The exact function and its values will be known at specified points.

For example you may have a table of sines tabulated at 1 degree intervals and use an interpolating function (which is not a sine) to obtain the values at half degree points.

A collocating function (polynomical) may often be used for this purpose.

An extrapolating function is similar to interpolating functions, except that it finds values beyond known ones.

For example your table of sines may only extend to 30 degrees and you need to find the sine of 44 degrees.

Finally an approximating function is a much more general term and used where you perhaps do not have a table of the correct function, or the ability to calculate it.
However you can derive some other function that you can prove is never further from the real one than some criterion,

That is Vfunction - Vapprox < e over the entire range of interest

For example we use Theta = sin(theta) = tan(theta) for small angles as approximating functions.

Vfunction may never actually equal Vapprox at any point in the range of interest.