2

$\sqrt{2}$ can be approximated by bisection method to $f(x)=x^2-2$ on $[0, 1]$.

I expected this to be linear convergence, but thought it might not be.

The definition of the order of convergence of $\lim\limits_{n\to\infty} p_n=p$ is the positive number $\alpha$ that satisfies

$$\lim\limits_{n\to\infty}\frac{|p_{n+1}-p|}{|p_n-p|^\alpha}>0$$

In this case, $p_n$ is the midpoint of the $n$-th interval and $p$ is $\sqrt{2}$.

I doubt this limit exists because it seems to be possible that $p_n$ is accidentally very close to $\sqrt{2}$ and then $p_{n+1}$ is relatively far from it.

Dinosaur
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  • You're halving the interval, so you get one more bit of precision at each step. – PM 2Ring Apr 08 '24 at 01:36
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    Actually what you consider is a good question. Say $x^{\ast} = (0.101001000100001 ...)_2$ in binary representation, and use bisection on interval $[0, 1]$ to find it, then each time, we recover one bit. But the limit

    $$\lim_{k\to\infty} \frac{|x^{\ast} - x_k|}{|x^{\ast} - x_{k-1}|}$$ does not exist, although it "almost" equals $\frac{1}{2}$, but it will be drop to small values infinite many times. You can replace the limit with $\limsup_{k\to\infty}\frac{|x^{\ast} - x_k|}{|x^{\ast} - x_{k-1}|}$, which is telling you the sequence should at least converge with some speed.

     – Yimin Apr 08 '24 at 02:05
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    unfortunately, almost all textbooks are avoiding to make a clear statement in the "limit" definition for convergence order. You can use limsup for bisection case. For many other methods, you need to get an asymptotic rate by $\lim_{k\to\infty} \sqrt[p]{\frac{|x^{\ast} - x_k|}{|x^{\ast} - x_{k-p}|^{p\alpha}}}$ for some fixed $p$ instead of using $p=1$. – Yimin Apr 08 '24 at 02:15
  • @Yimin I would like to know more details. Can you recommend a textbook? – Dinosaur Apr 08 '24 at 02:27
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    I do not know any textbook making this statement clear, I have a note for that, but I guess you can always use limsup instead to understand this. This is not an important part in the course of numerical analysis tho. The note is at https://math-5630-6630.readthedocs.io/en/latest/root_finding.html, be aware, I cannot guarantee correctness for this, I made that for class supplemental material use. – Yimin Apr 08 '24 at 04:40
  • I cover several related issues in this answer: https://math.stackexchange.com/a/4370177/307944 Bisection is mentioned as an example of an algorithm whose order of convergence is at least linear. It is usually hard to apply the definition of order $p$ convergence, but it is frequently possible to show that the order of convergence is at least $p$. – Carl Christian Apr 10 '24 at 13:34

1 Answers1

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Bisection is a simple and wasteful method. All the better methods use the available data about function values and build a model from them. The root of the model is then taken as the new midpoint. The cost for the better convergence is that the function has to be differentiable to some order and the root simple.

In the bisection method, the next root approximation is computed without referring to the previous knowledge of the function, the next midpoint is just the arithmetic midpoint of the interval. Only in the selection of the next search interval do the signs of the function values enter. This works for any continuous function, but with the slow speed of such an universal procedure. It is very hard to generate a function that is continuous but not (piecewise) differentiable as a computer function, giving bisection a similar place as bubble sort has among the search algorithms, educational but not practical.

The order of convergence, although formulated for the point sequence and its distance to the root, is in most proofs of it about the progression of the function values. At simple roots, these two measures are equivalent (via mean value theorem). The success of such an analysis depends on the point sequence being constructed from the function values. This is not the case for the bisection method, which is the cause of it being somewhat exceptional in that regard. Thus as a similar measure the progression of the lengths of the bounding/bracketing interval is taken instead. This will always leave the feeling of not being true to the principles.

Lutz Lehmann
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