# Constructing a proper Finite Difference Scheme for 3-dim function using data

$$f(x,y)$$ $$f$$ is unknown and I want to estimates its derivatives $$\frac{\partial f}{\partial y}, \frac{\partial f}{\partial x} \text{ and } \frac{\partial f}{\partial x^2}$$ from a data set.

I have 5 levels of $$y: y_1,y_2...y_5$$ and for each of those $$y$$'s I have 5 points in the $$(x,f(x;y=y_i))$$-plane. So bascially I have 5 of 2*5 matrices. For each of those matrices my x's have different values.

How can I construct a Finite Different Scheme to estimates the derivatives listed above?

The reason I struggle is that because my $$x$$'s have different values I don't have more than one point in the $$(y,f(y,x=x_i)$$-plane for any level of $$x$$.

• Hi Kim, i recommend you to take your question to math SE. – Sam Farjamirad Oct 10 at 16:44
• Whether you move the question or leave it here, we need to know more about what you plan to use these estimates for, to give a good answer. For example there is a big difference between numerical methods which use "pointwise" estimates of quantities to set up a solution algorithm, and methods which set up equations using some sort of "average" values integrated over an area or a volume - often using a so-called "weak variational principle" derived from the differential equation you want to solve, rather than using the equation explicitly. – alephzero Oct 10 at 18:05
• There is not much you can do in real world with this few data points. The differences are not big as suggested in the comment above if the problem is not stiff, so every classic method like Euler or Adams-Bashforth gives you a good estimate, again more data points more accuracy – Sam Farjamirad Oct 10 at 18:36