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Multivariable interpolation

  1. Jul 22, 2012 #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. jcsd
  3. Jul 22, 2012 #2
    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 [itex]n[/itex] points at which you knew the value of the function, then you can construct an [itex]n-1[/itex] degree polynomial that will pass through each of those points, and this polynomial is unique. Now, you have what looks to be 4 [itex]x[/itex] values and 4 [itex]y[/itex] values. Now, you should be able to construct a polynomial in the two variables [itex]x,y[/itex] that is of degree 3. That is, it has the form: [itex]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}[/itex], where you can find the values of the coeficitnts [itex]a_{i,j}[/itex] 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.
     
  4. Jul 22, 2012 #3
    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!
     
  5. Jul 23, 2012 #4

    chiro

    User Avatar
    Science Advisor

    Hey KV-1 and welcome to the forums.

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