# Two dimensional interpolation polynomial

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Jaime De La Mota Sanchis on 25 Sep 2019
Commented: David K. on 30 Sep 2019
Hello everyone.
I have a file which size is 681*441, it contains values measured at 300321 points. I want to find the interpolation polynomial. I have looked for functions which return what I am seeking; but I have found none. griddedinterpolant does not return the polynomial in question, but only values in a more precise grid. On the other hand, the function polyfit does provide with the nodes of a polynomial of any desired order, but it seems to work only in one dimension.
Can someone please tell me if there is a way to do as in the function polyfit but returning a two dimensional polynomial?
Thanks.
Jaime.

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Jaime De La Mota Sanchis on 25 Sep 2019
Thanks for the offer; this is the file in question. In it, I have generated a function of X and Y and I have it's value in a rectangle.
I hope it is clear; wind is the matrix with the values that I want to interpolate.
close all
clear
clc
x = 0:440;
y = 0:680;
[X,Y] = meshgrid(x,y);
wind=X+Y.^2;
surf(X,Y,wind,'EdgeColor','none','LineStyle','none','FaceLighting','phong')
David K. on 25 Sep 2019
Hm, actually I do not know if I understand your question. I thought you had data and was looking for the polynomial equation that fit it, but you already have said equation. Is this just a test example and you plan to apply to actual data to find the equations for?
You are also mentioning interpolation and I am not sure why gridded interpolant ot interp2 would not work with what you need it for.
Jaime De La Mota Sanchis on 25 Sep 2019
Yes, it is a test example; I know that the data is generated from the polynomial X+Y.^2, therefore the result also has to look similar. I am doing this to make sure that the code works before going to my data, where I have no means to check the results.
About the functions that you mention, they do not return the coefficients of the polynomial, they only provide interpolated data as far as I understand. I constructed this code before posting, but as said, the result is a more precised grid, not the polynomial coefficients
close all
clear
clc
[X,Y] = ndgrid(1:10,1:10);
V = X.^2 + 3*(Y).^2;
F = griddedInterpolant(X,Y,V,'cubic');
[Xq,Yq] = ndgrid(1:0.5:10,1:0.5:10);
Vq = F(Xq,Yq);
mesh(Xq,Yq,Vq)
figure
mesh(X, Y, V)

David K. on 25 Sep 2019
First off, I looked a little bit on wikis to find https://en.wikipedia.org/wiki/Multivariate_interpolation which might have what you want but I am not sure. However, the way I wanted to do it is with fmincon. I attached the objective function I created. Basically it creates a function of the form
a1x^2+a2y^2+a3xy+a4x+a5x+a6
then it loads in the X,Y, and wind data that you have and determines how closely it fits by a basic difference and summing.
This is utilized by fmincon as such:
a = fmincon(@objfun, rand(6,1), eye(6),ones(6,1).*20)
I decided to make the initial conditions random just because. Then, because this is a constrained minimum search, I just made it so that no value could go over 20, it would also likely work with large constraints as well but I doubt they will come into affect much. I tried using other optimization algorithms that did not require constraints but they did not work as well. For the example you gave the output should be
a = [0 1 0 1 0 0]
It generally gets pretty close, sometimes the constant value messes up though.

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David K. on 26 Sep 2019
save('testData.mat', 'wind','X','Y')
it should work fine.
Or, we can remove saving in general to make it a little nicer. Change the function as seen
function output = objfun(coeffs,wind,X,Y)
% create the function ax^2+by^2+cxy+dx+ey+f
currPoly = @(x,y) coeffs(1).*x.^2 + coeffs(2).*y.^2 + coeffs(3).*x.*y + coeffs(4).*x + coeffs(5).*y + coeffs(6);
currGuess = currPoly(X,Y); % eval the coefficients
error = abs(wind-currGuess);
output = sum(error,[1,2]); % sum the error
Then change the way you are calling it as such:
a = fmincon(@(x) objfun(x,wind,X,Y), rand(6,1), eye(6),ones(6,1).*20)
This will actually also be faster since calling a load function multiple times is pretty slow.
Jaime De La Mota Sanchis on 27 Sep 2019
I am very sorry, but the code still does not work. The error is
Error in objfun (line 10)
output = sum(error,[1,2]); % sum the error
Error in @(x)objfun(x,wind,X,Y)
Error in fmincon (line 535)
initVals.f = feval(funfcn{3},X,varargin{:});
Caused by:
Failure in initial objective function evaluation. FMINCON cannot continue.
I have tested in two diferent computers and I still get the same error.
I really don't understand what is wrong. I generate x, y, X, Y and wind as I have done before (see the attached file) and thencalling fmincon from the comand window.
The sequence is as follows:
first, I run grid_generator_2.m and then I call fmincon from the comand window as
>> a = fmincon(@(x) objfun(x,wind,X,Y), rand(6,1), eye(6),ones(6,1).*20)
I can only think of one reason for this not to work. In the function objfun, there is the following:
output = objfun(coeffs,wind,X,Y), x takes the place of the coeff and x is defined as x = 0:440; Perhaps should I use a vector of zeros of length 6 instead?
David K. on 30 Sep 2019
That is so weird. I added the fmincon call to your grid gen function and ran it on mine. It looks like we have the exact same thing. Perhaps the second argument for sum doesnt work in your version? try just replacing it with sum(sum(error)) I guess.

John D'Errico on 26 Sep 2019
I don't think you realize that an interpolation polynomial that passes exctly through 300321 points will be impossible to evaluate in double precision arithmetic. In fact, it may require a precision that is on the order of many thousand of decimal digits to get any thing out if it. Possibly hundreds of thousands of digits will be required.
And then, expect complete crapola anyway. Why? Because the polynomial might actually predict the points, but give you meaningless garbage between the points!