Though numerical methods were known long back, they became more popular after the introduction of high speed digital computers. Many of the real world physical problems can be modeled mathematically by means of differential equations or by a set of algebraic equations. Most of the existing analytical methods can solve linear problems and  some non-linear problems in simple form. However, there is no general analytical method which can solve any given differential or algebraic equation. Similarly there are formulae to find the roots of a polynomial of degree less than or equal to four but there exists no formula to find the roots of a polynomial of degree greater than four. And the algebraic manipulations  will be much more complicated if transcendental equation has to be solved instead of a polynomial equation. Solving large linear algebraic system is another typical example which cannot be solved with a pen and paper if the order of the system is moderately large or very large. In all these  situations numerical methods help us to find an approximate solution using computers. However, the solutions obtained from numerical methods are approximate as there are two kinds of errors in these solutions:

Truncation Errors

Round off Errors

The errors in these approximate solutions are expressed as