# Multivariable interpolation

1. Jul 22, 2012

### KV-1

Hello!

I am wondering if it is possible to establish a relationship between three sets of points (x,y,z) by interpolating.

Basically i need a function that takes x and y and gives me a z that matches the following points:

130 472 5
130 590 6
130 738 7.5
130 944 10
155 563 5
155 704 6
155 880 7.5
155 1126 10
180 654 5
180 817 6
180 1022 7.5
180 1308 10
205 745 5
205 931 6
205 1163 7.5
205 1489 10
240 872 5
240 1472 8

I want the middle column to be yielded by the equation when the outer columns are fed into it.. ex (first point): f(130,5) = 472

is this possible? if so, are there any calculators that you recommend or methods that don't require very complex math? (I only know calculus).

thanks!!!

2. Jul 22, 2012

### Robert1986

Well, let's say you had a function that is just from R to R. That is, it takes in one variable and spits out a number. If you had $n$ points at which you knew the value of the function, then you can construct an $n-1$ degree polynomial that will pass through each of those points, and this polynomial is unique. Now, you have what looks to be 4 $x$ values and 4 $y$ values. Now, you should be able to construct a polynomial in the two variables $x,y$ that is of degree 3. That is, it has the form: $p(x,y) = a_{3,3}x^3y^3 + \cdots + a_{3,0}x^3 + a_{2,3}x^2y^3 + \cdots a_{2,0}x^2 + \cdots a_{0,3}y^3 + \cdots a_{0,0}$, where you can find the values of the coeficitnts $a_{i,j}$ by solving a system of 16 equations (using the 16 data points you have.) Now, I can show you how to set this up, but I am not sure how well something like this will interpolate. I know that it will interpolate exactly to each of you data points, but I don't know enough theory to be able to predict how it will behave in between those points.

3. Jul 22, 2012

### KV-1

Oh thanks!

Not sure though, I think I found some equations that give me the desired points without having to interpolate though...

It helps to know how this can be done though!

4. Jul 23, 2012

### chiro

Hey KV-1 and welcome to the forums.

Are you aware of integral transforms, especially on multi-dimensional spaces?