Can Multivariable Interpolation Relate Three Sets of Data Points?

[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!

 

[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.

 

[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!

 

[chiro]

Hey KV-1 and welcome to the forums.

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

 

[CellCoree]

Hello! Yes, it is possible to establish a relationship between three sets of points by using multivariable interpolation. This involves finding a function or equation that can accurately predict values for the middle column (z) based on the values of the outer columns (x and y). This type of interpolation is known as trivariate interpolation.

There are several methods that can be used for multivariable interpolation, including polynomial interpolation, spline interpolation, and radial basis function interpolation. These methods can be implemented using mathematical software or programming languages such as MATLAB or Python.

As for calculators, there are online tools available that can perform multivariable interpolation, but they may be limited in terms of the number of data points that can be used. It may be more efficient to use a mathematical software or programming language for larger datasets.

In terms of complexity, multivariable interpolation does involve advanced mathematical concepts, but with a basic understanding of calculus, you should be able to grasp the fundamentals and use available tools to perform the interpolation. I would recommend further research and practice in this area to improve your understanding and skills.