# Help- Newton - Raphson Method

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• Jul 11th 2006, 02:17 PM
fair_lady0072002
Help- Newton - Raphson Method
Consider the function f(x) = x- 4 In x

Show that the equation f(x) = 0 has a solution in the interval (1,2)

Show that for this function f the Newton - raphson method formula can be expressed as xn +1 = 4x n (1-4 In x)/4 - x n (n = 0,1,2....).
• Jul 11th 2006, 04:09 PM
Soroban
Hello, fair_lady0072002!

I'll do the second part . . . it's trickier.
And I believe there's an extra "4" is the answer . . .

Quote:

Consider the function: $f(x) \:= \:x- 4\ln x$

(b) Show that for this function $f$, the Newton- Raphson method formula
can be expressed as: . $x_{n +1} \;= \;\frac{4x_n(1 - 4\ln x_n)}{4 - x_n}$? for $n = 0,\,1,\,2\,\hdots$

Formula: . $x_{n+1}\;=\;x_n - \frac{f(x_n)}{f'(x_n)}$

I'm going to drop the subscripts (temporarily) . . .

We have: . $f'(x) \:=\:1 - \frac{4}{x}$

Then: . $x_{n+1}\;=\;x - \frac{x - 4\ln x}{1 - \frac{4}{x}}$

Multiply top and bottom of the fraction by $x:\;\;x_{n+1}\;=\;x - \frac{x^2 - 4x\ln x}{x - 4}$

Get a common denominator: . $x_{n+1} \;= \;\frac{x}{1}\cdot\frac{x-4}{x-4} - \frac{x^2 - 4x\ln x}{x - 4}$

$x_{x+1}\;= \;\frac{x^2 - 4x - (x^2 - 4x\ln x)}{x - 4} \;= \;\frac{x^2 - 4x - x^2 + 4x\ln x}{x - 4} \;=$ $\frac{-4x + 4x\ln x}{x - 4}$

Factor: . $x_{n+1}\;=\;\frac{-4x(1 - \ln x)}{-(4 - x)}\quad\Rightarrow\quad \boxed{x_{n+1} \;= \;\frac{4x_n(1 - \ln x_n)}{4 - x_n}}$

• Jul 11th 2006, 04:27 PM
fair_lady0072002
My mistake....sorry and thanx
second part should be

x n+1 = 4x n (1- Inx n)/4-x n
• Jul 12th 2006, 06:49 AM
Soroban
Hello again, fair_lady0072002!

I hope this suffices as a solution . . .

Quote:

Consider the function: $f(x) \:= \:x- 4\ln x$

Show that the equation $f(x) = 0$ has a solution in the interval (1,2)

It can shown that $f(x)$ is continuous for all $x > 0.$

We find that: . $f(1) \;= \;1 - 4\ln1 \;= \;1$
. . . . . . and: . $f(2) \;= \;2 - 4\ln 2 \;\approx \;\text{-}0.773$

Since the continuous function changes from positive to negative over the interval,
. . it must achieve $f(x) = 0$ somewhere on the interval.