Find the solution obtained at the end of the first iteration by the Newton-Raphson method of the equation f(x) = x³ - x - 1 by taking the initial guess as x₀ = 1.

Concept:

According to Newton - Raphson Method:

• Newton-Raphson method has quadratic convergence,i.e order of convergence = 2
• Newton-Raphson method converges more rapidly than the other methods.
• The method can be used for solving algebraic and transcendental equations and it can also be used when the roots are complex.
• The iteration formula is,

\(X_{(n \ + \ 1)} = X_n - \frac{f(X_n)}{f'(X_n)}\)

Calculation:

We have, 

⇒ f(x) = x³ - x - 1 at x₀ = 1

⇒ f'(x) = 3x² - 1

⇒ \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\)

⇒ X₁ = \(1 - \frac{(1)^3 \ - \ 1 \ -1}{3(1)^2 \ - \ 1}\)

⇒ X₁ = \(1 - \frac{-1}{2}\)

⇒ X₁ = \(1 + \frac{1}{2}\)

⇒ X₁ = \(\frac{3}{2}\)

⇒ X₁ = 1.5

∴ The first iteration by the Newton-Raphson method of the equation, f(x) = x3 - x - 1 by taking the initial guess as x0 = 1  is 1.5