Lagmental Vicfred

Newton's Method Is Tangent-Line Root Finding by Vicfred

tags: mathematics, numerical analysis, root finding, interpolation

Solving the linearization at x_n gives a quadratically convergent iteration near a simple root. A small computation will anchor the general statement before the abstraction takes over.

The mathematical object

Numerical analysis replaces an exact object \(x^\ast\) by approximations \(x_n\) with controlled error. Stability asks how rounding or data perturbations affect the answer.

$$ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ e_{n+1}=\frac{f''(\xi_n)}{2f'(x_n)}e_n^2,\qquad e_n=x_n-\alpha $$

One explicit computation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ f(x)=x^2-a\Longrightarrow x_{n+1}=\frac12\left(x_n+\frac a{x_n}\right) $$

Why the identity matters

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)},\\[5pt] \mathsf{C}\;&:\quad e_{n+1}=\frac{f''(\xi_n)}{2f'(x_n)}e_n^2,\qquad e_n=x_n-\alpha. \end{aligned} $$

Where it can fail

Fast local convergence is not a global guarantee. Newton's method can diverge, interpolation can oscillate, and a small residual can coexist with a large forward error.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] e_{n+1}=\frac{f''(\xi_n)}{2f'(x_n)}e_n^2,\qquad e_n=x_n-\alpha \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Mon 23 October 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.