Question

I am studying numerical analysis and also solving algorithms which is described in book. My problem is about Newton’s method. In general, if some function is given and we have to find root, how can we determine derivative of function in code? or even limit? because as you know Newton’s method involves derivative and makes iteration like this.
Suppose some function f(x) and initial guess,p0, then p(n)=p(n-1)+f(p(n-1))/f'(p(n-1)) here f' denotes derivative of f.

How can I approximate it in code? Thanks a lot.

Answer 1

Depending on how the function is given, you can do a couple of things

1. symbolic differentiation, if you have a symbolic representation of your function
2. Numerical differentiation, if you only have point-value pairs
3. Interpolate with a polynomial and differentiate that (symobolically of course)

All Options are viable. Which of these is most suited to your problem depends on the function and also the time you want to invest in coding and/or reading up on how to do it.

Edit: If you already know the function before execution time, then compute the differential by hand and implement it as a function. You should also already have implemented your f(x) as a function like this

float f (float x) {
    // ...
}

And thus:

float df_dx (float x) {
    // ...
}