how to create a polynomial interpolation of two variables

how to create a polynomial interpolation of two variables with following data:

x=0:0.5:2;

y=0:0.1:0.4;

z=[13.2689 14.8741 16.4243 17.9337 19.4118]

Answer 1

[John D'Errico]

Sorry, but this is a common mistake made by people.

You really have only ONE variable there, since x and y are linearly related. Therefore you cannot perform a TWO dimensional interpolation.

x=0:0.5:2;
y=0:0.1:0.4;
plot(x,y,'o')

You CAN perform an interpolation of z as a function of x, or z as a function of y. They will be identical mathematically, due to the linear relationship between x and y.

[Torsten]

I think the OP meant a polynomial z = p(x,y) with

p(x(i),y(i)) = z(i) (1<=i<=5)

[John D'Errico]

@Torsten

I fully understand what was meant. HOWEVER. see that the result, since x and y are LINEARLY related, will be the same as if you just performed an interpolation of the form z(x) OR z(y).

Look very carefully at x and y.

The problem is too often then people think they now have a true 2-d interpolation, able to compute an intelligent result for OTHER values of x and y, that do not fall on the line in question. The attempted 2-d interpolation will result in a singular problem.

x=0:0.5:2;
y=0:0.1:0.4;
z=[13.2689 14.8741 16.4243 17.9337 19.4118];

Can we form the linear model p(x,y)?

Pcoeffs = [ones(5,1),x(:),y(:)]\z(:)

Warning: Rank deficient, rank = 2, tol = 3.040471e-15.

Pcoeffs = 3×1

13.3135 3.0691 0

The result of course is a model with a ZERO coefficient on y, and a singularity warning. Essentially the model produced is just:

p(x,y) = 13.3135 + 3.0691*x + 0*y

Could pinv have been used here? Yes, but that is just a smoke screen. The model still is insufficient to intelligently tell you what the interpolant does when x and y do not fall on the line where the data lives.

As I said, you can trivially perform the interpolation z(x) or z(y), with no resulting singularity issues.

[Torsten]

I didn't see that y can directly be substituted by 0.2*x in a two-dimensional polynomial interpolation.

Thanks for the clarification.

[merwan behar]

I want to draw a two-dimensional graph, I can't do it with the surface function .

as the following figure

[John D'Errico]

You don't have two dimensions of information content in your data. Wanting to do something is nice. But wanting the mathemtically impossible is not enough.

As I showed, your data lives only perfectly along a straight line. And that makes it impossible.

You need to get data that provides information in TWO dimensions. Not just one. Again, here is your data in the (x,y) plane:

x=0:0.5:2;
y=0:0.1:0.4;
plot(x,y,'o-')

Do you see that the data lives along a straight line?

Instead, you want data that looks like this:

xgood = [0 1 2 0 1 2 0 1 2];
ygood = [0 0 0 .2 .2 .2 .4 .4 .4];
hold on
plot(xgood,ygood,'rs')

The data in red is data that explores the entire domain. And it allows you to draw the surface you want to see. The data you have is sadly, completely useless in this respect.

You need to understand the difference.