The Newton-Raphson method helps us locate an approximate real root of the given equation and improves its accuracy by a flow process.
Numerical Methods: Numerical Methods provide various techniques to find an approximate solution to a difficult problem using various operations. Numerical methods are easily adoptable to solving problems using computers as they involve sequential steps.
Here we are calculating
a. The numerical solution of algebraic and non-algebraic equations, and
b. The exact and numerical solution of a system of linear algebraic equations.
Numerical solution of Algebraic and Transcendental Equations: Given an equation f(x) = 0, it is generally not possible to find the roots such that f(x) becomes exactly zero. This topic deals with numerical methods of obtaining approximate roots of the given equation.
Equations involving algebraic quantities like x, x2 , x3,…etc are called algebraic equations.
Example : x3-2x2 + x – 2 = 0, x5 – x6 = 20
Equations that involves non-algebraic quantities like, ex , log |x| , sin x, tan x, …etc are called transcendental equations.
Example : ex – x = 0, x ln |x| - 2 = 0, sin 2x = 5x
The Numerical methods of finding approximate roots of the given equation are a repetitive process known as the iteration process. In every step, the result of the previous step is used, and the process is carried out until we get the result to its desired accuracy. All the numerical methods are only approximate techniques for the solution of any problem and computers play a great role in various numerical methods for obtaining the result to the highest degree of accuracy.
If (x) = 0 is a real valued continuous function of the real variable x, we get the following fundamental property:
When there exist two values a, b such that f(x) has an opposite sign, say f(a)<0, f(b)>0 equivalently f(a), f(b) < 0 then there always exists at least one real root in the interval (a,b).
Geometrically the property means that the graph of y = f(x) intersects the x-axis at least at one point, x0 that lies between a and b which is a real root of f(x) = 0.
This property is useful in locating an initial approximation for a real root of f(x) = 0.
It is important to note that if the values of a, b of x lie near each other, then the numerical iterative methods will give the real root to the desired accuracy more quickly. In some methods, we have to locate an approximate root x = x0 for starting the iterative process. In which case, if the magnitude, say f(a), is nearer to zero compared to f(b), we prefer to take x = x0 = a as the initial approximation.
The two different iterative methods are
1. The Newton-Raphson method
2. The Regular - falsi method