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Newton-Raphson Method

This method is generally used to improve the result obtained by one of the previous methods. Let xo be an approximate root of f(x) = 0 and let x1 = xo + h be the correct root so that f(x1) = 0. Expanding f(xo + h) by Taylor’s series, we obtain

f(xo) + hf’(xo) + h2f’’(xo)/2! +-----= 0

Neglecting the second and higher order derivatives, we have

f(xo) + hf’(xo) = 0

Which gives

h = - f(xo)/f’(xo)

A better approximation than xo is therefore given be x1, where

x1 = xo – f(xo)/f’(xo)

Successive approximations are given by x2, x3,----,xn+1, where

xn+1 = xn – f(xn)/f’(xn)

Which is the Newton-Raphson formula.

Example: - Find a real root of the equation x3 -5x + 3 = 0.

Soln: -

Let, f(x) = x3 -5x + 3 = 0

f’(x) = 3x2 - 5

Choosing xo = 1

Step-1:

f(xo) = -1

f(xo) = -2

So, x1 =1 – ½ = 0.5

Step-2:

f(x1) = 0.625

f’(x1) = -4.25

x2 = 0.5 + 0.625/4.25 = 0.647

Step-3:

f(x2) = 0.0356

f’(x2) = -4.139

X3 = 0.647+ 0.0356/4.1393 = 0.6556

Step-4:

f(x3) = 0.0037

f’(x3) = -3.7105

x4 = 0.6556 + 0.0037/3.7105 = 0.6565

The successive approximation 0.6565

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