How do I obtain the derivatives after interp2 with the 'spline' option?

Hello,

I need to obtain the first derivatives at the points evaluated by interp2.

thanks in advance

facundo

Answer 1

[Star Strider]

To get a numerical derivative, use the gradient function.

With a matrix input, request two outputs. See Contour Plot of Vector Field for an example.

Answer 2

[John D'Errico]

interp2 does not return derivative information at the interpolation points.

If you want to generate that at some general interpolation point, it would seem simplest to just use a finite difference approximation for the partial derivatives at that point. I would strongly suggest a central finite difference for accuracy, as long as spline interpolation is involved. If you were using linear interpolation, then just use a forward difference.

That is, for reasonably small dx, just evaluate the interpolant at two points.

df/dx ~ (f(x + dx, y) - f(x - dx, y))/(2*dx)

You can do the same for the derivative in y, holding x fixed at the current value..

df/dy ~ (f(x,y + dy) - f(x,y - dy))/(2*dy)

So to compute the gradient vector [df/dx,df/dy] just requires 4 function evaluations.

This is NOT an exact computation of the derivative at any point. It may even be moderately far off, if the function is highly nonlinear, but then any such estimate can be wild on a massively nonlinear surface.

If instead, you wanted to infer the derivatives at every node of the lattice used by interp2, then gradient will NOT provide the exactly correct derivatives as implicitly used by interp2. At best, it would return only an approximation. You could use the same solution I suggest above at each node of the lattice, as long as you used forward or central differences along the boundary of the lattice.

If you truly wanted the EXACT derivatives, you are probably out of luck, unless you were willing to code the spline equations up yourself. I suppose you could throw my derivest code at the surface to estimate the partial derivatives as well as possible. At least it tries to provide an error estimate too, to tell you if it thinks there is a problem.

[X,Y] = meshgrid(-1:.1:1);
Z = exp(X+2*Y);

So a very simple surface. Now lets estimate the gradient of the surface at (x,y) = (1.23,-0.35), fist using a central difference estimate.

delta = 1.e-8;
xy = [1.23,-0.35];
diff(interp2(X,Y,Z,xy(1)+ delta*[1 -1],xy(2),'spline'))/(2*delta)
ans =
         1.68949371159854
         
diff(interp2(X,Y,Z,xy(1),xy(2)+ delta*[1 -1],'spline'))/(2*delta)
ans =
         3.39632481027863

How would I do so using my derivest tools?

fun = @(xy) interp2(X,Y,Z,xy(1),xy(2),'spline');
[grad,err] = gradest(fun,xy)
grad =
         1.68949373130569          3.39632447633558
err =
      1.68153581563919e-14      6.47785565541463e-12

So my gradest utility was a little more accurate, at least in the computation of the derivative of the splined surface. You cannot know the true derivative of a general unknown function without the functional form.

[facundo sepulveda]

Thanks John, I was not aware of derivest.

[J. Alex Lee]

in general, should it not be possible to evaluate the derivative of the underlying interpolation model, which in principle could be to operate on whatever coefficients are calculated by interp2? You would just need access to them.

The analog would be the function that caculates the polynomial coefficients of the derivative of another polynomial specified only by its own coefficients.

[Stephen23]

https://www.mathworks.com/help/curvefit/postprocessing.html

https://www.mathworks.com/matlabcentral/answers/276471-derivative-of-piecewise-polynomial-using-ppval

Answer 3

[Bruno Luong]

Checkout this File exchange

I haven't visited this file for long time. At the time I wrote the interface is not very intuitive, I would probably do differently now.