## 2

## Non-Linear Equations

The aim of this chapter is to discuss the most useful methods for finding the roots of any equation having numerical coefficients. Polynomial equations of degree ≤ 4 can be solved by standard algebraic methods. But no general method exists for finding the roots of the equations of the type *alogx* + *bx* = *c* or *ae*^{-x} + *b* tan *x* = 4, etc. in terms of their coefficients. These equations are called transcendental equations. Therefore, we take help of numerical methods to solve such type of equations.

Let f be a continuous function. Any number *f* for which *ξ* is called a root of the equation *f*(*x*) = 0. Also, *ξ* is called a zero of function *f*(*x*).

A zero *ξ* is called of multiplicity *p*, if we can write

*f* (*x*) = (*x* − *ξ*)^{p} *g*(*x*),

where