Iterative process

Definition

An iterative process is a method that involves repeatedly applying a set of rules or operations to approximate a desired result. It is often used to find solutions to equations, especially when exact solutions are difficult or impossible to obtain analytically.

5 Must Know Facts For Your Next Test

1. Newton's Method is a prime example of an iterative process used in calculus to find roots of functions.
2. The key formula for Newton's Method is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.
3. Convergence of Newton's Method depends on the initial guess being sufficiently close to the actual root.
4. If the derivative $f'(x)$ is zero at any point during the iterations, Newton's Method will fail.
5. Newton's Method can converge quadratically, meaning the number of correct digits approximately doubles with each iteration.

Review Questions

Related terms

Convergence: The property that an iterative process approaches a finite limit as the number of iterations increases.

Derivative: A measure of how a function changes as its input changes; symbolically represented as $f'(x)$.

Root:

A solution to the equation $f(x) = 0$, where $f$ is a given function.