Talk:Root-finding algorithms

message: I did not expect the method to converge for all initial guesses, though that would be great; in fact, the basic result I was wondering about is whether it converges for initial guesses sufficiently close to the exact solution, which apparently it does because the solutions are attractive fixed points of the iteration map. By the way, the first case is called global convergence, and the second case local convergence; I guess I should have been more precise in my question.
I agree, the really interesting question is not about convergence (after an infinite number of iterations), but things like: how often will the method yield a decent result, when applied to polynomials which typically occur in practice, and with starting guesses that are typically used? However, the convergence question is easier to pose and to answer.
I am thinking of putting 'your' method in a separate article and leave a short summary here, just like other methods like Newton's method. What do you think about this? Of course, we then need a name for the method ... any ideas? Perhaps I'll go to the library and see whether I can find anything similar. -- Jitse Niesen (talk) 15:02, 13 September 2005 (UTC)
thread_title: Probably almost global convergence

message: Should we add the method of solving "quadratic" equations using "discriminants" to this article?
i.e.,
ax^2 + bx + c = 0
x1 = (-b + sqrt(b^2 - 4ac)) / 2a
x2 = (-b - sqrt(b^2 - 4ac)) / 2a
: You might want to include a reference to Quadratic equation in the section Root-finding_algorithm#Finding_roots_of_polynomials  Bo Jacoby 09:27, 17 October 2005 (UTC)
thread_title: Quadratic equations

message: There has been some editing on the condition that there exists a and b such that f(a)<0 and f(b)>0. It has been changed to that f(a) and f(b) have opposite signs. Nobody said that aT</sup> 14:25, 18 May 2006 (UTC)
replies: text: Hi KSmrq. You wrote that Typical numerical root-finding methods use iteration. There is no numerical root-finding method that does not use iteration, because a finite rational algorithm does not produce irrational roots from rational coefficients and initial guesses. Do you mind removing the word typical ? Bo Jacoby 09:31, 23 May 2006 (UTC)
replies: text:  Yes, I mind, because your point has no validity in finite precision. All our floating point numbers are rational and bounded. Consider a standard algorithm for finding the real roots of a real quadratic equation; it contains no visible iteration in the usual sense. We may argue that the iteration is hidden within the implementation of the square root function, but that's not a fair argument. In terms of complexity theory, the square root is essentially no different from division, and we don't count division as an iteration unless we are explicitly considering arbitrary precision arithmetic. We can build square root into our hardware, or deploy an algorithm in software with no open-ended iteration yet a good-to-the-last-bit guarantee. The article on methods of computing square roots is a bit of a mess and not especially helpful on this point, but one can consult the professional literature. Also supporting the non-iterative view is the fact that many sophisticated numerical algorithms feel free to use a square root as a constant-time step. For example, the Householder transformation used in a QR decomposition must compute the magnitude of a vector, which means using a square root. Therefore, I claim that "all" is incorrect and "typical" should stand. Which is why I made the change in the first place. --KSmrqT 14:11, 23 May 2006 (UTC)
thread_title: Bisection

message: I converted the "Finding multiple roots" article into a redirect to this article.  I merged the deleted text into this article with minor changes like changing sections into subsections.  You can still reach the talk page of the old article via Talk:Finding multiple roots.  To view the history of the old article, switch to this article itself and select "What links here", then click on "Finding multiple roots (redirect page)", then click the "history" tab.  JRSpriggs 07:01, 20 May 2006 (UTC)
replies: text: support merger. -lethe talk [ +] 13:49, 30 April 2006 (UTC)

text: If by root-finding we mean numerical root-finding using floating-point computations, then I would really like to see a reference to support this approach, because it only seems suitable for exact computations. Unless such a reference is produced, I would not like to see this merged, because it sounds like questionable advice. However, looking for a common root of a polynomial and its derivative is standard practice in exact computations for algebraic geometry, as one example. If this is the intent, perhaps it should be merged with a different article. It doesn't stand well on its own, whatever the merit of its content. --KSmrqT 13:24, 18 May 2006 (UTC)
replies: text: I merged this article into Root-finding algorithm.  Any additional talk about "Finding multiple roots" should be conducted at Talk:Root-finding algorithm. JRSpriggs 08:39, 21 May 2006 (UTC)
thread_title:  Merger with the "Finding multiple roots" article 

message: Q1. Why did you decide to start a new article, instead of adding an extra section to Root-finding algorithm? I prefer to have a few long articles with a couple of people working on them and correcting each other, rather than spreading out our works on many small articles (but this is partially a matter of taste).
A1. No particular reason.  I just thought of this as just another method of finding roots like the other methods which have their own articles.  If you want to merge it into the main article in this category, please feel free to do so.
:support merger. -lethe talk [ +] 13:49, 30 April 2006 (UTC)
replies: text: If by root-finding we mean numerical root-finding using floating-point computations, then I would really like to see a reference to support this approach, because it only seems suitable for exact computations. Unless such a reference is produced, I would not like to see this merged, because it sounds like questionable advice. However, looking for a common root of a polynomial and its derivative is standard practice in exact computations for algebraic geometry, as one example. If this is the intent, perhaps it should be merged with a different article. It doesn't stand well on its own, whatever the merit of its content. --KSmrqT 13:24, 18 May 2006 (UTC)
replies: text: I merged this article into Root-finding algorithm.  Any additional talk about "Finding multiple roots" should be conducted at Talk:Root-finding algorithm. JRSpriggs 08:39, 21 May 2006 (UTC)
thread_title: Merged talk from "Finding multiple roots" -- Response to Jitse Niesen

message: "Finding multiple roots" is not a root-finding algorithm and so it does not belong here. Exact algebra on integers should be finished before the problem is forwarded to approximate root-finding using floating point numbers. Bo Jacoby 10:13, 28 July 2006 (UTC)
thread_title: End merged talk from "Finding multiple roots

message: Hi, while reading the relevant papers of Schönhage and Pan in the last few days I tried to assemble things I understood or knew already into a short description of the method. An accessible formulation of the main result is still missing. If I find a short way to formulate it, I will add it later.--LutzL 17:20, 27 July 2006 (UTC)
thread_title:  Request for comments on [[Splitting circle method]] 

message: Perhaps someone might mention Halley's method - a considerable improvement on Newton's. By comparison, the overhead is calculation of the second derivative of the function at the initial approximation point ... and higher order methods in a similar vein are possible too! If noöne else does it, I may add it. (Note: Halley, of comet fame.) Hair Commodore 20:09, 19 January 2007 (UTC)
replies: text: If you have a reference, go ahead and create an article about it. Put it in this category :Category:Root-finding algorithms when you first start on the article so that it does not get lost in the system. JRSpriggs 06:19, 20 January 2007 (UTC)
replies: text: No - at this precise moment, I don't have a reference for it - but it can be found in the French "section" of Wikipedia!  (Unfortunately, the article in question  - and others associated with it, such as that for Householder's method(s) - does not have a "language translation box".) Hair Commodore 19:33, 21 January 2007 (UTC)
replies: text: Among others, it is described on Xavier Gourdons Newton page, section "Cubic iteration". Use the Postscript-version linked to at the top for a correct display of formulas.--LutzL 06:45, 22 January 2007 (UTC)
replies: text: Could it be another name for Laguerre's method? If you can give me a link to the French article and it is not too wordy, I might be able to translate it to English. JRSpriggs 07:07, 22 January 2007 (UTC)
replies: text: No, Laguerre is "taylored to polynomials". Halley and generally Householder methods are for general real functions. The webpage I gave, although from a french mathematician, is in english. Dates and references are provided. Further reading :fr:Itération de Halley and from there .--LutzL 10:20, 22 January 2007 (UTC)
thread_title:  Halley's method 

message: Conversely, any equation can take the canonical form f(x) = 0, so equation solving is the same thing as computing (or finding) a root of a function.  (from the article, 3rd paragraph)
This is only true of equations which have the form f(x) = g(x).  Implicit equations don't always have a canonical f(x) = 0 form.  —Preceding unsigned comment added by Mgsloan (talk • contribs) 21:51, 9 November 2007 (UTC)
replies: text: Please provide an example. Bo Jacoby 00:02, 11 November 2007 (UTC)
replies: text:  .
Lets say we made this into an equation by setting it equal to 0.3.  This would be equivalent to tracing out a particular slice of this image. There's no way to write this as f(x) = 0 without leaving out huge swaths of the range.  Perhaps the article should specify that they are equations of one variable?  Mgsloan 20:59, 12 November 2007 (UTC)

text: What is the problem in writing  f(x,y)=0. Noone said either that the x in the introduction could not be a vector of variables. The function f could be vector valued as well. So yes, there are root sets that are positive dimensional. If one tried hard enough, one could surely come up with a function with a fractal root set as well. Yours is not fractal, since your function is locally Lipschitz. Although it is a nice picture--LutzL 07:53, 13 November 2007 (UTC)
thread_title:  Root finding is equation solving? 

message: The Bernoulli referred to is Daniel for clarity.  His method is described in the book Mathematics and Computers by Stibitz and Larrivee (1957),pp 77-79.--Billymac00 (talk) 03:34, 13 April 2010 (UTC)
thread_title:  Bernoulli method 

message: Why isn't fixed point iteration mentioned in this article ?  — Preceding unsigned comment added by 168.96.204.43 (talk) 17:45, 29 June 2011 (UTC)
replies: text: What do you have in mind? Bo Jacoby (talk) 06:40, 30 June 2011 (UTC)

text: Probably the user meant iteration x_{k+1} := g(x_k)  to solve the equation  x = g(x) , as described e.g. in . I second the request for it to be included in the article. I think this method is important to know to see a broader picture: for example, the gradient descent can be interpreted as the simple fixed point iteration method applied to find a zero of the first derivative, in which the step size  s  should be chosen small enough to guarantee the convergence condition |s g'(x)| < 1. AVM2019 (talk) 21:56, 14 September 2020 (UTC)
thread_title:  Fixed Point Iteration 

message: Please.  Notice this edit.  In particular:
* Don't capitalize the initial letter of a word just because it's in a section heading.
* Things like "Cohn–Schur" require an en-dash, not a hyphen.
* Similarly, ranges of pages, as in "pp. 31–43" require an en-dash, not a hyphen.
* Do not indiscriminately italicize everything in non-TeX mathematical notation.  Italicize variables, but NOT digits, parentheses and other punctuation, or things like sin, cos, det, log, max, sup, etc.  Also, capital Greek letters should not be italicized.  The point is of course to be consistent with TeX style.  I found lots of stuff like f(x)3 instead of f(x)3.
All of this is codified in WP:MOS and WP:MOSMATH. Michael Hardy (talk) 20:27, 3 January 2014 (UTC)
thread_title:  Spelling, etc. 

message: Isn't there an algorithm that starts with a similar equation with a known solution and approximates the equation, finding a new solution each step, using the previous one as a guess? I read about that but couldn't find it again.— Preceding unsigned comment added by 187.35.227.178 (talk) 00:32, 21 November 2014‎
:Please sign your posts in talk pages with four tildes (~~~~).
:Apparently, you refer to the homotopy continuation method. This method is limited to polynomial equations, because of the need of a proper notion of "similar equation". As far as I know, although it could work, it is not useful and not used for single univariate equations. D.Lazard (talk) 08:55, 21 November 2014 (UTC)
replies: text:  Homotopy continuation methods, such as Newton homotopy methods, are a bit more general than that. While they require some differentiability conditions, methods such as the hybrid method in  can handle nonpolynomial equations. --Mark viking (talk) 12:36, 21 November 2014 (UTC)
thread_title:  Couldn't find an algorithm 

message: (...) a limit (the so-called "fixed point") which is a root
This phrase is misleading as the roots of e.g. f(x)=x^2+1 are not fixed points of the same function (i.e. they are not zero). Helder 23:06, 9 July 2015 (UTC)
replies: text: Good point. I have fixed this. D.Lazard (talk) 07:19, 10 July 2015 (UTC)
thread_title:  Roots vs fixed points 

message: :re: this revert with explanation "(Reverted 2 edits by 199.34.4.20 (talk): Primary source, too recent for the existence of secondary sources. No reason for giving a spcial emphasis to one paper among tausends. (TW))"
I do not understand why the McDougall Wotherspoon modification of Newton's method is not worth mentioning under this section, and moreover why any of the other "thousands" of variants on Newton's method should also not be mentioned. Especially if, as is in the case of the McDougall variant, they are published in a scholarly pier reviewed journal. When looking for a root-finding algorithm to use in practice I for one would appreciate knowing about a method which: has a better convergence rate,  the same calculation cost, is more likely to converge, and remains as relatively simple to implement as Newton's method. 
Text cut out of the section:
 Another method by McDougall and Wotherspoon is a simple modification on Newton's method to achieve  convergence while still only using one function and derivative evaluation per step. McDougall, Trevor J. and Wotherspoon, Simon J. "A simple modification of Newton’s method to achieve convergence of order ", Applied Mathematics Letters, Volume 29, March 2014. p. 20-25. Retrieved on 18 November 2015.
199.34.4.20 (talk) 20:49, 19 November 2015 (UTC)
thread_title:  Newton's method (and similar derivative-based methods) 

message: Would it be worth adding the golden section search below the bisection search?  — Preceding unsigned comment added by Jeremyminton (talk • contribs) 12:22, 18 December 2015 (UTC)
replies: text: Golden section search is a search for an optimum rather than a root.  Glrx (talk) 19:59, 23 March 2016 (UTC)
thread_title:  Golden Section Search 

message: There are a lot of algorithms.  Would it make sense to make a table of them clarifying their differences, such as how many roots they find at once, whether they are guaranteed to converge, convergence speed, etc.?  — Preceding unsigned comment added by 71.167.63.118 (talk) 02:02, 21 March 2016 (UTC)
thread_title:  Table of algorithms?