Question

The Newton-Raphson method is used to solve the equation f(x) = x³ – 5x² + 6x – 8 = 0. Taking the initial guess as x = 5, the solution obtained at the end of the first iteration is ________.

Answer 1

Concept:

Newton Raphson Method:

• Converges at a very fast rate and its order, k = 2.
• Its iteration follows this:

\({{x}_{n+1}}={{x}_{n}}-\frac{f\left( {{x}_{n}} \right)}{{f}'\left( {{x}_{n}} \right)~}~\) (Where f(x) is a given polynomial and f^’(x) is its derivative)

Its Solution majorly depends on x(n) value and can be obtained in very less iteration if f^’(x_n­) is large.

Calculation:

Given f(x) = x³ – 5x² + 6x – 8 = 0

Then f’(x) = 3x² – 10x + 6

Initial guess given is x₀ = 5

Then its 1^st iteration is:

\({{x}_{1}}={{x}_{0}}-\frac{f\left( 5 \right)}{f'\left( 5 \right)}=5-\frac{22}{31}=4.2903\)