0
$\begingroup$

Apply Newton-Raphson method to find the solutions to the equation $x^3-5x=0$ starting with an initial guess of $x_0 = 1$.

While using Newton Raphson method, the value doesn't converge to a specific number. Rather, every iteration either gives $1$ or $-1$. Why does this happen? Any graphical interpretation for this?

$\endgroup$
  • $\begingroup$ All you have to do is draw a sketch. There is a similar question here, which has some pictures. $\endgroup$ – TonyK May 4 at 12:16
3
$\begingroup$

For the graph of $f(x)=x^3-5x$, we have $f'(x)=3x^2-5$ $$f(1)=-4$$ $$f'(-1)=4$$ $$f'(1)=-2=f'(-1)$$

The tangent line at $x=1$ is $y+4=-2(x-1)$ which is $y=-2x-2$.

The tangent line at $x=-1$ is $y-4=-2(x+1)$ which is $y=-2x+2$

Geometrically, what has happened is you are trapped in the following cycle.

enter image description here

Starting from $(1,-4)$, by traveling along the tangent, we intercept the $x$-axis at $x=-1$. From $(-1,4)$, by trveling along the tangent, we intercept the $x$-axis at $x=1$.

$\endgroup$
  • $\begingroup$ $f'(1)=f'(-1)=-2$. Isn't it? $\endgroup$ – pi-π May 4 at 12:39
  • $\begingroup$ yes, that's a typo, thanks. $\endgroup$ – Siong Thye Goh May 4 at 12:40
2
$\begingroup$

The basic reason for this is that the graph of $f(x) = x^3 -5x$ has some symmetry. Specifically, $f(x)$ is an odd function.

The intuitive visualization of the Newton-Raphson method is that you are finding the tangent line to the curve at the initial guess point, and then finding the x-intercept of this line. You are then just repeating this, using the x-intercept you found as the next ‘guess’ point.

The first tangent line we find is given by $-2x -2$, which intersects the x-axis at $x=-1$, and because of the symmetry of the graph (the slope at $-x$ is the same as the slope at $x$), the tangent line here will intersect back at $x=1$. So we are stuck in a never-ending loop.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.