Question

Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = π/4. The value of the predicted root after the first iteration, up to second decimal, is ______

Answer 1

Concept:

Newton-Raphson method:

It has order of convergence 2 and number of guesses required is 1.

Iterative formula is \({x_{n + 1}} = {x_n} - \frac{{f\left( {{x_n}} \right)}}{{f'\left( {{x_n}} \right)}}\)

Calculation:

x = 10 cos x

⇒ f(x) = x – 10 cos x ⇒ f'(x) = 1 + 10 sin x

\(\therefore At\;n = 0 \Rightarrow {x_1} = {x_0} - \frac{{f\left( {{x_0}} \right)}}{{f'\left( {{x_0}} \right)}}\)

\({x_0} = \frac{\pi }{4}\)

\(f\left( {\frac{\pi }{4}} \right) = \frac{\pi }{4} - 10\cos \frac{\pi }{4} = \; - 6.285\)

\(f'\left( {\frac{\pi }{4}} \right) = 1 + 10\sin \frac{\pi }{4} = 8.071\)

\(\therefore {x_1} = \frac{\pi }{4} + \frac{{6.285}}{{8.071}} = 1.564\)

∴ Value of Iteration after 1^st iteration = 1.56

Mistake Point: While calculation be aware that mod of virtual calculation is in radian, if you are taking angle is π-terms.