Finite Difference Method on a function with multiple elements of the same array

First time posting here, so I apologize for any missing info upfront. I am working on a program in VBA that calculates a function (which itself calls another function), then calculates the derivative of that function, then uses that to form a Jacobian matrix to calculate the solution via the Newton-Raphson method. My question relates to taking the derivative of a function, $$f$$.

First, the function has the general form: $$f(i, h, q_0, A, x, v(x, X_{min}))$$. $$i$$ is a counter variable, and $$h, q_0, A$$ and $$X_{min}$$ are constant single value inputs. The remaining arguments are all members of the same array $$x$$ or another function $$v(x, X_{min})$$. The form of the function changes based on the value of $$i$$. The inputs from the array may be, depending on the value of $$i$$:

1. $$x_i$$ and $$x_{i+1}$$
2. $$x_i$$, $$x_{i-1}$$, $$x_{i+1}$$
3. $$x_{i}$$ and $$x_{i-1}$$

The inputs from $$v(x)$$ may be, again depending on i:

1. $$v(x_i)$$
2. $$v(x_{i-1})$$ and $$v(x_i)$$
3. $$v(x_{i-1})$$

Here is an example of the function in question: $$f = \frac{1}{h}\bigg[\frac{q_0}{A} * (x_{i + 1} - x_i) + v(x_{i - 1}, X_{min}) * x_{i - 1} - v(x_i, X_{min}) * x_i)\bigg]$$

I am using this form of the finite difference method: $$f′(x) = \frac{−f(x+ 2∆x) + 8f(x+ ∆x) −8f(x−∆x) + f(x−2∆x)}{12\Delta x}$$

My question is, for each time I use this function, do I need to apply the $$∆x$$ to each instance of the array element passed to the function? Ie, for the case where $$x_i$$ and $$x_{i+1}$$ are called, do I need to add or subtract $$∆x$$ to both $$x_i$$ and $$x_{i+1}$$? This would give $$x_i$$ $$+/-$$ $$1$$ or $$2∆x$$, as well as $$x_{i+1}$$ $$+/-$$ $$1$$ or $$2∆x$$. Or, according to this method, is it better to calculate the finite difference around $$x_i$$ only?

And further, for the case where both $$v(x_{i-1})$$ and $$v(x_i)$$ are passed to the function $$f$$, would adding and subtracting $$∆x$$ to each instance of the $$x$$ array passed into the $$v$$ function be required? While I'm at it (and I hope this is overkill), would I also need to calculate the finite difference centered around the values returned by $$v(x)$$ as well?

Alternatively, is there simply a better way to determine the derivative of this function than the way I'm going about it? I hope I have provided enough info. I didn't include the code, as I think that is a separate question, to be explored once I have an answer to this question.

Thank you!!

• I edited part of your question to use mathjax formatting for the equations. Let me know if I messed anything up and see if you can extend this to the rest of your question Jul 20 '21 at 18:42
• Thank you, I will return to this later today and try to format the rest in mathjax as you suggested. Jul 20 '21 at 19:01

With $$f(x, y(x))$$ while one could technically split up $$\frac{\partial f}{\partial y}$$ and $$\frac{\partial y}{\partial x}$$, it isn't necessary. You can just directly perturb both $$x$$'s at the same time to get the total derivative on $$f$$ w.r.t. $$x$$. But, in your equations, $$f$$ does not actually take $$v$$ as an input. It might call $$v$$ with different inputs (the same way it calls $$-$$, $$v$$ isn't special). So, it really is just $$f(x)$$ here.
When it comes to perturbing vector inputs like $$x$$, you will need to do it one component at the time, meaning you will have to call $$f$$ many times here. Each perturb $$x_i$$ gives you one column of your Jacobian. If you happen to know that a certain $$x_j$$ isn't going to affect $$f_i(x)$$ at all, you can of course skip all those computations and just fill in the 0 in your jacobian.