The Newton-Raphson method, explained in a Platonian dialogue

As you all know, I like the Newton-Raphson method. And so, explaining it covers quite a few articles on this website. Still, there is more we can do to explain this method. There is always more. To do that, I once again enlisted my academic colleague ChatGPT, and asked for this:

The Prompt

Write a step-by-step instruction for the Newton-Raphson method, but in the style of a Platonian dialogue

The Platonian dialogue explaining the Newton-Raphson method

Socrates: My dear friend, have you ever pondered upon how one might find the root of a complex equation?

Glaucon: Indeed, Socrates, it is a task that often vexes me.

Socrates: Fear not, for I shall guide you through the steps of a method known as Newton-Raphson, a path that will lead us to the root.

Glaucon: I would be most grateful, Socrates.

Socrates: First, we must imagine that we have an equation, let us call it \(f(x)\). The root we seek is a value of \(x\) where \(f(x)\) equals zero. Do you follow?

Glaucon: Yes, Socrates, that seems clear.

Socrates: Very well. To start our journey, we must first make a guess as to where the root might be. This we shall call \(x_0\). Like a blind man seeking light, we must take a step, even if we do not yet see the path.

Glaucon: I see, we are stepping into the unknown. What next?

Socrates: We then seek guidance from the derivative of the equation, which we shall call \(f'(x)\). The derivative, like a wise mentor, shows us the way by revealing the slope of the curve at a certain point.

Glaucon: I understand, Socrates. The derivative guides our path.

Socrates: Precisely, Glaucon. Now, with the aid of our mentor, we apply the following formula:
\[x_1 = x_0 – f(x_0)/f'(x_0)\]

Just as a student learns from his teacher, our new guess, \(x_1\), benefits from the wisdom of the old, \(x_0\).

Glaucon: I see, so the new guess is closer to the root?

Socrates: Indeed, Glaucon. But one lesson is seldom enough to grasp the whole truth. Just as we must repeatedly question our beliefs, we must iterate this formula:
\[x_{i+1} = x_i – f(x_i)/f'(x_i)\]

With each iteration, we draw ever closer to the root, just as through questioning, we draw ever closer to the truth.

Glaucon: And when do we stop this process, Socrates?

Socrates: We stop when we can come no closer to the root, when further questioning adds no more to our understanding. Then we have found our answer.

Glaucon: Thank you, Socrates, for this enlightening discourse. The path to the root, much like the path to the truth, is indeed a fascinating journey.