Question

2. (a) We want to find the root $x^$ of the function $f(x)$; that is, we need $f(x^) = 0$. This can be done using Newton's method, making use of the iterative formula $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ Show that the sequence of iterates \{$x_n\$} converges quadratically if $f'(x) \neq 0$ in some appropriate interval of $x$-values near the root $x^*$. [9 points] (b) We can get Newton's method to find the $k$-th root of some number $a$ by making it solve the non-linear equation $f(x) = x^k - a = 0$. Show that the iteration scheme then becomes $$x_{n+1} = \left(1 - \frac{1}{k}\right)x_n + \left(\frac{a}{k}\right)x_n^{1-k}$$ [5 points] (c) In the case of the square root, for which $k = 2$, show that the error satisfies the relation $$|\sqrt{a} - x_{n+1}| = \frac{|\sqrt{a} - x_n|^2}{|2x_n|}$$ Is this the behaviour you would expect? [6 points]

          2. (a) We want to find the root $x^*$ of the function $f(x)$; that is, we need
$f(x^*) = 0$. This can be done using Newton's method, making use of the
iterative formula
$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$
Show that the sequence of iterates \{$x_n\$} converges quadratically if
$f'(x) \neq 0$ in some appropriate interval of $x$-values near the root $x^*$.
[9 points]
(b) We can get Newton's method to find the $k$-th root of some number $a$ by
making it solve the non-linear equation $f(x) = x^k - a = 0$.
Show that the iteration scheme then becomes
$$x_{n+1} = \left(1 - \frac{1}{k}\right)x_n + \left(\frac{a}{k}\right)x_n^{1-k}$$
[5 points]
(c) In the case of the square root, for which $k = 2$, show that the error satisfies
the relation
$$|\sqrt{a} - x_{n+1}| = \frac{|\sqrt{a} - x_n|^2}{|2x_n|}$$
Is this the behaviour you would expect?
[6 points]

$2. (a) We want to find the root x^* of the function f(x); that is, we need f(x^*) = 0. This can be done using Newton's method, making use of the iterative formula xn+1 = xn - (f(xn))/(f'(xn)) Show that the sequence of iterates {xn$ converges quadratically iff'(x) ≠0in some appropriate interval ofx-values near the rootx^*. [9 points] (b) We can get Newton's method to find thek-th root of some numberaby making it solve the non-linear equationf(x) = x^k - a = 0. Show that the iteration scheme then becomesxn+1 = (1 - (1)/(k))xn + ((a)/(k))xn^1-k[5 points] (c) In the case of the square root, for whichk = 2, show that the error satisfies the relation|√(a) - xn+1| = (|√(a) - xn|^2)/(|2xn|)Is this the behaviour you would expect? [6 points]$

Added by Joe A.

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