Skip to main content

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.