Numerical Methods

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