Newton's Method: Convergence, Safeguards, and Extensions

All flashcards

What is a root of a function?

A value α for which $f(\alpha)=0$.

What is the standard Newton iteration?

$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$, provided $f'(x_n)\ne 0$.

What does quadratic convergence mean?

Near the root, the next error is approximately bounded by a constant times the square of the current error.

How can Newton's method be modified for a root of known multiplicity m?

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

What replaces the derivative in Newton's method for a system of equations?

The Jacobian matrix, used to compute a correction by solving a linear system.

Quiz questions

1. For a simple root α, which condition supports local convergence of Newton's method?Answer: $f$ is continuously differentiable near α and $f'(\alpha)\ne 0$.
2. Near a suitable simple root, what error behavior characterizes quadratic convergence?Answer: $|e_{n+1}|$ is bounded approximately by $C|e_n|^2$.
3. For a square system $F(\mathbf{x})=\mathbf{0}$, how is a Newton correction usually computed?Answer: Solve $J_F(\mathbf{x}_n)\,\Delta\mathbf{x}=-F(\mathbf{x}_n)$.
4. To use Newton's method to locate a stationary point of a differentiable function $g$, which equation is solved?Answer: $g'(x)=0$, using $x_{n+1}=x_n-\frac{g'(x_n)}{g''(x_n)}$.
5. When the known root multiplicity is $m$, which update modifies ordinary Newton iteration?Answer: $x_{n+1}=x_n-m\frac{f(x_n)}{f'(x_n)}$.