# Numerical Methods

## Definition

Numerical methods are a class of procedures/algorithms that **repeatedly perform certain operations to refine the solution** of a mathematical problem.

Numerical methods may or may not take an initial guess to solve a mathematical problem. We can classify numerical methods into two categories based on whether they require an initial guess or not.

### Direct Methods

Direct methods **don't require any initial guess** and can solve a problem in a finite number of steps(the number of steps required cannot exceed a certain threshold). These methods solve a system of equations and determine all the roots/solutions at the same time. Examples include Gauss-Elimination method, Gauss-Jordan method, LU factorization method etc.

### Indirect/Iterative Methods

Indirect methods or Iterative methods **require one or more initial guesses** and iteratively refine the initial guess to solve the problem. The number of steps required to find the solution can't be predetermined but generally, we can get a good enough approximation within a few steps after which the procedure is stopped. Indirect methods converge nearer to the solution as the number of steps increases.

#### Bracketing Methods

Some indirect methods require **two initial guesses between which the solution exists**, these are called bracketing methods, examples are Bisection method and Regula-Falsi method. Bracketting methods converge faster if the length of the bracketing interval is smaller.

#### Open Methods

Indirect methods whose **initial guesses need not enclose any solution** are called open methods. Open methods may require one or two initial guesses. In the case of two initial guesses, there is no requirement for the interval to contain a solution. Examples include Netwon-Raphson method, Secant method and Fixed point iteration method. Open methods converge faster if the initial guess(es) is nearer to the solution.