Talk:Rate of convergence

message: That book, unfortunately not only forgets that one must have μ<1, they actually have exercises where they ask to prove x_n=1/n converges linearly . So I think that book is overall confused on that topic. Oleg Alexandrov 20:36, 16 Jan 2005 (UTC)
replies: text: Cheers, I didn't notice that, so it is a deliberate choice of Burden and Faires. All other books require μ < 1 though, so I'm keeping that definition. It's a pity, since the Burden and Faires book seems pretty nice overall, and apparently also used as a textbook at my university. -- Jitse Niesen 11:55, 17 Jan 2005 (UTC)
replies: text:  I think the point of Burden and Faires is that they are really after quadratic convergence. Therefore, anything slower than that they think is all and the same damn slow linear convergence. :) Oleg Alexandrov 16:11, 17 Jan 2005 (UTC)
thread_title: Burden and Faires definition again

message: In reference to my simultaneous edit to the article: The new formulation with "converges with at least order q" was taken from the Suli and Mayers book. The previous formulation ("the order of convergence is the largest q") has the disadvantage that it is not obvious that such a q exists. Feel free to revert as it is a bit nit-picking. (PS: Removing Burden and Faires from the references was the right solution) -- Jitse Niesen 12:21, 20 Jan 2005 (UTC)
replies: text: I like your change. Cheers, Oleg Alexandrov  | talk 05:41, 21 Jan 2005 (UTC)
thread_title: Further tweaks to extended definition

message: In the basic definition, it seems that what one is really after is not the lim, but the lim sup.  I think that this would actually be equivalent to the extended definition.  Does that make any sense?
--128.100.151.121 22:58, 11 February 2006 (UTC)
replies: text: Yes, using "lim sup" instead of "lim" would make the basic definition equivalent to the extended definition. But I find it simpler that way, without lim sum. No? Oleg Alexandrov (talk) 00:25, 12 February 2006 (UTC)
thread_title: Question regarding extended definition

message: Hello everyone, first of all, I don't understand why the order of converging page was directing everyone to the rate of convergence page.  They are clearly two different things.  They both measure how fast a sequence converges but they are two different way of measuring convergence.  Order of convergence is how fast the terms of a sequence get smaller compared to its previous terms.  Rate of convergence measures how fast a sequence/function converges compared to a known sequence/function.  In addition, this page previously held information about order of convergence as opposed to rate of convergence.  So I created the Order of convergence page and swapped both pages with each other because the info on the page titled rate of convergence was actually about the order of convergence. It took me a while to figure out the confusion as well. It is not easy. I have also added a few more examples. In case you are wondering, I used the Burden & Faires 8th edition. I understand from previous discussions on the rate of convergence page that some of you are already familiar with this book. The two definitions are entirely different with the rate of convergence being defined on page 35 and order of convergence being defined on page 75.
A Real Kaiser 06:38, 28 October 2007 (UTC)
replies: text: I am familiar with the Burden & Faires book. I think it is rather non-standard in how it defines things. I undid your edits and I made order of convergence redirect here. Oleg Alexandrov (talk) 15:29, 4 April 2008 (UTC)
thread_title: Swapping Pages

message: There is a page on the "Big Oh" notation in Wikipedia but the page on rates of convergence does not have the definition using the "Big Oh" notation. I propose we add a section to the rates of convergence page that gives a definition using "Big Oh" notation and then link to the "Big Oh" notation page for more information. The definition would be as follows:
Suppose {Bn}n=1infinity is a sequence known to converge to zero, and {An}n=1infinity converges to a number A. If a positive constant k exists with 
abs(An-A) <= k*abs(Bn), for large n
then we say that {An}n=1infinity converges to A with rate of convergence O(Bn). It is indicated by An = A + O(Bn).
Reference: Numerical Analysis by Richard L. Burden and J. Douglas Faires.  —Preceding unsigned comment added by 132.235.37.25 (talk) 18:35, 26 September 2008 (UTC)
replies: text: There are two different contexts in which you can talk about the speed of convergence. The first one is iterative methods where n denotes the iteration number, the second one is numerical integration where n denotes the number of grid points. In the first context, "linear convergence" means roughly O(C−n), and in the second context, "linear convergence" means O(1/n). The first context is what's treated in the article now, the second context is what you're talking about. If you can address all this without confusing the reader, please go ahead. -- Jitse Niesen (talk) 21:38, 29 September 2008 (UTC)

text: I wasn't very happy in how the two contexts were separated, so I rewrote bits with the hope that it's clearer now. -- Jitse Niesen (talk) 00:58, 30 November 2008 (UTC)
thread_title:  Rate of Convergence "Big Oh" notation.  

message: On top of talking about Big O notation, I would also like to propose a section that emphasizes O( 1/n^p ) convergence, which is concerned in finding largest number p such that for some sequence {an} | ( 1 <= n <= infinity ) which converges to a, an=a + O(  1/n^p ).
This is similar to above definition, but is more useful in some cases.
For example if an = (n+1)/(n^2), bn = (n+1)/(n^3) it can be demonstrated that although both {an} and {bn} converge to zero as n -> infinity, {bn} converges with rate bn = 0 + O( 1 / n^2 ), while an = 0 + O( 1/n ).  This means bn goes to zero much faster than an.
Reference: Numerical Analysis by Richard L. Burden and J. Douglas Faires.
:There are two different contexts in which you can talk about the speed of convergence. The first one is iterative methods where n denotes the iteration number, the second one is numerical integration where n denotes the number of grid points. In the first context, "linear convergence" means roughly O(C−n), and in the second context, "linear convergence" means O(1/n). The first context is what's treated in the article now, the second context is what you're talking about. If you can address all this without confusing the reader, please go ahead. -- Jitse Niesen (talk) 21:38, 29 September 2008 (UTC)
replies: text: I wasn't very happy in how the two contexts were separated, so I rewrote bits with the hope that it's clearer now. -- Jitse Niesen (talk) 00:58, 30 November 2008 (UTC)
thread_title:  Emphasize O( <math>1/n^p</math> ) method 

message: The citation which references the convergence order of the secant method is incorrect. In particular, instead of providing a 1.681... decimal value (should be 1.618... at the least), it should note that the order is equal to the golden ratio. Thus, the last part of the citation would be more specific and correct if it were written as:
: \alpha =  the golden ratio = \frac{1+\sqrt{5}}{2} \approx 1.618 . JasonYerardiMath (talk) 16:34, 12 September 2012 (UTC)
replies: text: Corrected the numerical value. Please note that the letter phi is already a link to the golden ratio.--LutzL (talk) 15:14, 13 September 2012 (UTC)

text: A root where the function is regular. Since the function has to be at least 1,1-smooth to get this result, it means that the function value is zero and the derivative value different from zero.--LutzL (talk) 06:30, 19 April 2014 (UTC)
thread_title:  Correction of Citation for Convergence of Secant Method;  

message: What's a regular root? —Tamfang (talk) 03:58, 19 April 2014 (UTC)
replies: text: A root where the function is regular. Since the function has to be at least 1,1-smooth to get this result, it means that the function value is zero and the derivative value different from zero.--LutzL (talk) 06:30, 19 April 2014 (UTC)
thread_title: regular root

message: This sentence should be strengthened to "converges logarithmically" because (1/n -1/[n+1])/(1/[n+1]-1/[n+2])=(n+2)/n, which converges to 1. I will make the change now.144.35.45.98 (talk) 16:43, 6 November 2015 (UTC)
thread_title:  "Finally, the sequence {dk} converges sublinearly." 

message: In the definition the meaning of the constant M is not explained. Joel Brennan (talk) 22:20, 23 April 2018 (UTC)
thread_title:  Definition of convergence with order <math>q</math>. 

message: The definition for linear convergence should have a "limsup" not a "lim", because the lim does not necessarily exist. For example, you could have a sequence where the distance to the optimium decreases by a factor of 1/2 at odd iterations, but by 1/3 at even iterations. That would still be linearly convergent with rate 1/2, but the lim does not exist.
2001:16B8:60E3:6200:AC65:A419:7775:7CFF (talk) 11:42, 23 April 2020 (UTC)
replies: text: 2001 16B8 194.250.110.49 (talk) 08:59, 19 November 2023 (UTC)
thread_title:  Linear convergence is wrong 

message: There appears to be some disagreement among sources about whether or not q=1 is allowed for the order of convergence. In https://link.springer.com/content/pdf/10.1007/BF00939805.pdf they specify that q>1 (in their paper q is called \tau). But in http://fourier.eng.hmc.edu/e176/lectures/NM/node3.html it is given as p \geq 1. Should we include a note that says both definitions are in use? The-erinaceous-one (talk) 05:20, 1 August 2020 (UTC)
replies: text: I went ahead and included note in the article that clarifies the difference with a reference to https://link.springer.com/content/pdf/10.1007/BF00939805.pdf. The-erinaceous-one (talk) 07:04, 1 August 2020 (UTC)
thread_title:  Differening definitions of Convergence Order 

message: I know the definition of order of convergence with lim sup rather than lim, because in many cases the limit does not exist, and one is mainly interested in getting the inequality  |x(n+1) - L| ≤ C |x(n) - L|q, which BTW can also (and actually should) be used to define convergence of order q. In case the sequence might jump to the exact limit, which happens, e.g., when the bisection method is applied to find a root of the form r = m/2^n, one cannot divide by (x(n) - L). — MFH:Talk 14:02, 28 March 2022 (UTC)
thread_title:  limit or limit superior? 

message: what is mew?if it can include 0 and infinity then there can be any order of convergence? is this true? can someone clarify within this page. thanks Jarfuls of Tweed (talk) 11:48, 13 December 2022 (UTC)
replies: text: nvm, it's clear, it is just explained farther down Jarfuls of Tweed (talk) 11:54, 13 December 2022 (UTC)
thread_title:  do we assume that mew I. the first paragraph is any non-zero number? 

message: :
The bolded bit was inserted anonymously in October 2008, leaving the first "used in" with no citation to call its own.  (The citation itself was added by  half a year earlier.)  I'm going to remove the addition; if you check the book and find that it also supports , do add it back. —Tamfang (talk) 23:59, 10 November 2023 (UTC)
thread_title:  hijacked citation?