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

• What is the formula used in Newton's Method?
• Why can Newton's Method fail if $f'(x) = 0$ at some point?
• How does the rate of convergence for Newton's Method behave?

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.