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I have to use the bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on $[0,1]$.

However I simplify I get,

$$x^{2x(1-x^{2}+x)}-e^{x^{2}-1}=0$$

Which is undefined at $0$. How would I go about approaching this problem?

Michael Hardy
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Gamecocks99
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  • I think the problem meant on the interval $(0,1)$. To use the bisection method you have to choose two points on either side of the root as initial guesses. In your case, $x_l = .2$ and $x_r=.4$ will work. You then calculate their midpoint which is $.3$. Now you test whether $f(.3)$ is more or less than $0$. And depending on that you replace either the left-hand or right-hand guess with .3 and repeat until desired accuracy. Bisection method algorithms usually assume that roots are discretely spaced so that you can choose a small initial neighborhood around your root to avoid complications. –  May 01 '14 at 00:43
  • The bisection method is rather slow, so how many iterations you want? – IAmNoOne May 01 '14 at 00:52
  • the interval you are using should be continuous but the function is undefined at $0$, means the function is not continuous in the given interval. –  Aug 16 '19 at 15:53

2 Answers2

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I think the interval may be wrong as $\ln (0)$ is undefined from the start, but regardless the solution is $x=1$

Ellya
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  • I think the point of the question was to use the bisection method to approximate the root, not simply blurt out a root. This kind of question is all about the how and less about the what. – amWhy May 01 '14 at 15:41
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We are being asked to use the Bisection Method to find the root of:

$$2x(1-x^{2}+x)\ln(x)=x^{2}-1, x\in [0,1]$$

If we plot these functions, we see two points of intersection:

enter image description here

Performing the Bisection Method using $20$ digits of accuracy using $a = 0.05, b = 0.35$ converges in $62$ steps to the root:

$$x = 0.32796778533181880526$$

Repeating this for the next root, performing the Bisection Method using $20$ digits of accuracy using $a = 0.8, b = 1.0$ converges in $31$ steps to the root:

$$x = 0.99999999990686774254$$

Amzoti
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  • This is an interesting bisection method you use but do not explain, especially for the second root, which is simply 1, and hence there are no 20 digits of accuracy in your answer. – Artem May 02 '14 at 14:08
  • No changes in the answer. Sorry, but this is clear -1. – Artem May 03 '14 at 02:47