Newton Raphson Works Great, Until It Doesn't - Part 1

Most people would likely have chose the option of 0, because that is the closest root to $0.45$ .

However, if you applied the Newton Rahpson method, you would noticed that the approximated roots swing around some. In particular, if $f'(x)$ is close to 0, then the next approximate root will be quite far away (see $n=2$ below).

Here is the table of values, if you followed the method accordingly:

Hence, with a starting approximation of 0.45, we actually end up with the root of -1!

$$ Using Ms. Excel spreadsheet, we can obtain the root will converge to $-1$ for initial value $x_0=0.45$ . Take a look the picture below. $$ Newton Raphson $$ Newton’s (or the Newton-Raphson) method is one of the most powerful and well-known numerical methods for solving a root-finding problem. Newton’s method is an extremely powerful technique because it is rapidly convergent than any other methods. In this case, it takes only $14$ iterations to yield the root of $f(x)=x(x-1)(x+1)$ . Iteration should be stopped if $x_n=x_{n+1}$ or $|x_n-x_{n+1}|=\epsilon$ , where $\epsilon$ is the tolerance value and it is chosen as close as possible to $0$ for yielding the best accuracy of the root of function. $$

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$\text{\# }\mathbb{Q.E.D.}\text{ \#}$

The problem doesn't require any table of iterations. We simply apply Newton's method once. We first find $f'(0.45)$ :

$f(x)=x^3-x \implies f'(x) = 3x^2 - 1$

Plugging in $x=0.45$ gives us $f'(0.45) \approx -0.3975$ . Now we need to find the equation of this tangent line and determine it's x-intercept. We know it goes through the point $(0.45,f(0.45))=(0.45,-0.3588)$ . Using this information we determine the equation of the tangent line and then its x-intercept:

$y=-0.3975x-0.179925 \implies x_{int} \approx -0.452$ .

Since we know that $f(x)=0$ at $x=0,1,-1$ , and the first iteration of Newton's method yields a negative x-intercept, then after a sufficient number of iterations, the root will converge to $\boxed{x=-1}$ . And we're done.

just simply like this x+1 = 0 so that x = -1 .. -1 is the answer