Newton-Raphson error analysis

5 The equation $$x ^ { 3 } - 5 x + 3 = 0$$ may be solved by the Newton-Raphson method. Successive approximations to a root are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots\).

1. Show that the Newton-Raphson formula can be written in the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where $$\mathrm { F } ( x ) = \frac { 2 x ^ { 3 } - 3 } { 3 x ^ { 2 } - 5 }$$
2. Find \(\mathrm { F } ^ { \prime } ( x )\) and hence verify that \(\mathrm { F } ^ { \prime } ( \alpha ) = 0\), where \(\alpha\) is any one of the roots of equation (A).
3. Use the Newton-Raphson method to find the root of equation (A) which is close to 2 . Write down sufficient approximations to find the root correct to 4 decimal places.

6 It is given that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - x - 1 = 0\) has three roots, one of which is positive.

1. Show that the Newton-Raphson iterative formula for finding this root can be written $$x _ { n + 1 } = \frac { 6 x _ { n } ^ { 3 } + 5 x _ { n } ^ { 2 } + 1 } { 9 x _ { n } ^ { 2 } + 10 x _ { n } - 1 } .$$
2. A sequence of iterates \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) which will find the positive root is such that the magnitude of the error in \(x _ { 2 }\) is greater than the magnitude of the error in \(x _ { 1 }\). On the graph given in the Printed Answer Book, mark a possible position for \(x _ { 1 }\).
3. Apply the iterative formula in part (i) when the initial value is \(x _ { 1 } = - 1\). Describe the behaviour of the iterative sequence, illustrating your answer on the graph given in the Printed Answer Book.
4. A sequence of approximations to the positive root is given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\). Successive differences \(x _ { r } - x _ { r - 1 } = d _ { r }\), where \(r \geqslant 2\), are such that \(d _ { r } \approx k \left( d _ { r - 1 } \right) ^ { 2 }\) where \(k\) is a constant. Show that \(d _ { 4 } \approx \frac { d _ { 3 } ^ { 3 } } { d _ { 2 } ^ { 2 } }\) and demonstrate this numerically when \(x _ { 1 } = 1\).
5. Find the value of the positive root correct to 5 decimal places.

It is given that \(f(x) = x^3 - k\), where \(k > 0\), and that \(\alpha\) is the real root of the equation \(f(x) = 0\). Successive approximations to \(\alpha\), using the Newton-Raphson method, are denoted by \(x_1, x_2, \ldots, x_n, \ldots\).

1. Show that \(x_{n+1} = \frac{2x_n^3 + k}{3x_n^2}\). [2]
2. Sketch the graph of \(y = f(x)\), giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for \(|x_2 - x_1|\) to be greater than \(|x_1|\). [3]

It is now given that \(k = 100\) and \(x_1 = 5\).

1. Write down the exact value of \(\alpha\) and find \(x_2\) and \(x_3\) correct to 5 decimal places. [3]
2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). By finding \(e_1\), \(e_2\) and \(e_3\), verify that \(e_3 \approx \frac{e_2^2}{e_1}\). [3]