The Chord Method is:

$x^{(k+1)} = x^{(k)} - {g(x^{(k)}) \over g'(x^{(0)})}$

The question is to compute the cube root of 2, using the Chord Method. Carry out the first few iterations, using x(0) = 0.6.

$g(x) = x^3 -2$

$g'(x) = 3x^2$

$x^{(1)} = 0.6 - {(0.6)^3 -2 \over 3(0.6)^2}$

$x^{(1)} = 0.6 - {(0.6)^3 -2 \over 1.08 }$

$x^{(1)} = 2.2518$

$x^{(2)} = 2.2518 - {(2.2518)^3 -2 \over 1.08 }$

$x^{(2)} = -6.4685$

$x^{(3)} = -6.4685 - {(-6.4685)^3 -2 \over 1.08 }$

$x^{(3)} = 245.9868$

And then things get really really bad.

The problem is when usign Newton's Method it converges to a value of around ~1.2599; and

"The two methods converge under essentially the same conditions."

According to my notes. So shouldn't it also converge using the Chord Method as it does using Newton's? Am I missing something?

  • $\begingroup$ For the Chord method to converge, the first guess, $x^{(0)}$, needs to be sufficiently close to the root. $\endgroup$ – Stephen Montgomery-Smith Feb 22 '15 at 5:13
  • $\begingroup$ Isn't that the same for Newton's method though, it does say under 'under essentially the same conditions.'? $\endgroup$ – RaenirSalazar Feb 22 '15 at 5:16
  • $\begingroup$ Yes, Newton's Method generally also requires that the first guess be sufficiently close to the root. However when solving polynomial equations, most of the time most anything will work as a first guess. $\endgroup$ – Stephen Montgomery-Smith Feb 22 '15 at 5:18
  • $\begingroup$ Are you sure that the denominator is always $g'(x_0)$? That would make it very sensitive to the initial approximation. $\endgroup$ – marty cohen Feb 22 '15 at 5:39
  • $\begingroup$ I would expect Newton's to be a bit more robust since for each iteration it uses an improved x value while this method does not. if you choose x = 1.2 as your initial condition the chord method converges fine. $\endgroup$ – bobbym Feb 22 '15 at 6:19

For solving $g(x)=0$, the iterates generated by the chord method are $$x_{k+1} = x_k - {g(x_k) \over g'(x_0)}$$ while those given by Newton method are $$x_{k+1} = x_k - {g(x_k) \over g'(x_k)}$$ so the two methods will behave the same (or almost the same) if $g'(x_k) \approx g'(x_0)$ (that is to say when $g(x)$ is almost linear from $x_0$ to the solution) or when the solution is close to $x_0$.

If you want to use the chord method (what I should almost never suggest), what you can do is to use it for $2$ or $3$ iterations (so we have $x_0,x_1,x_2$) and repeat the process defining the new $x_0$ as being equal to the last $x_2$.

I any manner, when it works, the chord method as a linear convergence while Newton method has a quadratic convergence (which means that it will require much less iterations).

If you really do not want to update the derivative, I suppose that you could approximate it by finite difference using $$g'(x_k)\approx\frac{g(x_k)-g(x_{k-1})}{x_k-x_{k-1}}$$

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  • $\begingroup$ The question was "Show how to use the Chord method to compute the cube root of 2. Carry out the first few iterations, using x(0) = 0.6" so I do not believe it is expected that I update the denominator, I guess I got hung up on taking "will behave the same on essentially the same" as being that if using the same initial guess and g(x) will always converge. $\endgroup$ – RaenirSalazar Feb 22 '15 at 14:25
  • $\begingroup$ The approximation using the finite difference scheme is known as the Secant Method. $\endgroup$ – Stephen Montgomery-Smith Feb 23 '15 at 0:06

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