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THE GENERALIZED MEDIANT (RATIONAL MEAN) AND NEWTON'S METHOD
It is absolutely disturbing and worrying to realize that the extremely simple arithmetical methods shown at: New arithmetical root-solving methods, which embraces Newton's, Bernoulli's, Halley's and Householder's methods (among many other new ones) do not appear in any text on numbers since Babylonian times up to now. —Preceding unsigned comment added by Arithmonic (talk • contribs)
What the heck?!
OK, so I came here looking to see if there was any kind of semi-layman explanation for this thing I only otherwise saw in unexplained computer code using programming functions I'm currently unfamiliar with. And I was disappointed.
Laying all the degree-level mathspeak aside for a moment... given that said code amounted to about six lines of no more than 20 characters' width, and carried out only one or two otherwise simple-looking operations per line (and thus DIDN'T seem to use any actual calculus - just subtraction and division, for the most part, but with the reasoning behind it obfuscated), can someone please put an easily-understood explanation of how you carry out this approximation either in the article or here on the talk page? In plain English? As in, if your input value is X, your initial guess is Y, and the output is Z (or indeed, starting values are X0, Y0 and Z0, increasing up to Xn etc as you iterate), how do you arrive at the first estimated value of Z, and then iterate towards a more precise one from there?
First person to use the words "function", "derivation", "tangent", any letter followed by a degree-minute mark, or an unexposited reference to another "theorem" or "method" gets a boot to the head, with the size and weight of the boot and the spikiness of the sole in direct proportion to how many of those offenses are committed all at once.
It is, after all, supposed to be a simple and easily conducted approximation of the root. If I can arrive at a better approximation, faster, just using guesswork and long division, without having to break out a dictionary of mathematics to understand what the Dickens is being discussed, then something's surely gone wrong.
An example incorporating a functional ISO BASIC program that only uses fundamental mathematical operators gets additional credit.
Multitudinous thanks in advance. 193.63.174.211 (talk) 14:09, 13 January 2014 (UTC)
- (For example, say I am Teh Thick, and want to try and work out what the root of 6.25 might be, knowing only that it must be more than 1 (as 1x1 = 1) but less than 4 (as 4x4 = 16) and having never done my 25x tables ... how does this method end up converging towards 2.5, any better than it would with my own amateur way of just cutting the range into halves and recalculating to see which side of the line it falls, if it doesn't in fact land on it exactly?) 193.63.174.211 (talk) 14:15, 13 January 2014 (UTC)
...JESUS CHRIST. OK, forgive my profanity, but I just found this on an external site:
Newton's square root equation
The equation to use in this method is:
√ N ≈ ½(N/A + A) , where:
N is a positive number of which you want to find the square root
√ is the square root sign
≈ means "approximately equal to..."
A is your educated guess
Therefore, if, say N = 121 and you guess at A = 10, you can enter the values into the equation:
√ 121 ≈ ½(121/10 + 10) = ½(12.1 +10) = ½(22.1) = 11.05
That is pretty close to the correct answer of 11.
This means, as far as I can tell, that you make an initial guess (Xn), and if it is not correct, then you make another guess (Xn+1) which is the mean of (your guess added to (the object number divided by your last guess)). And then you keep going until a satisfactory level of precision is achieved. THAT'S ALL IT SEEMS TO BE.
Why must things therefore be so complicated?! That's four lines of text. Four. And a single line of equation using very plain algebraic language.
Or have I got it all wrong? 193.63.174.211 (talk) 14:25, 13 January 2014 (UTC)
- The difference is that what you quote is merely a recipe for applying the method in one particular case, whereas the Wikipedia article seeks to explain the general method, which is applicable in a far wider range of cases. It is completely impossible to explain the method without using concepts such as "function" and "tangent", because they are fundamental to the method. And if you don't want an explanation of the method, but are just after a set of easy instructions on how to use it to find a square root, then you are in the wrong place, because Wikipedia is not a "how to..." guide. JamesBWatson (talk) 14:37, 13 January 2014 (UTC)
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- 193.63.174.211, perhaps you would be better served if you got yourself the collected works of Viete of 1591, as published around 1650. There he has a chapter on numerical approximations of roots of equations up to degree 4 (or even higher?) avoiding any minus sign. Lots of special cases for that reason, and the idea of a function was also still in its infancy, derivatives and tangents were still unknown.--LutzL (talk) 18:18, 13 January 2014 (UTC)
- Source: F. Cajori (1929): A History of Mathematics, page 137--LutzL (talk) 16:49, 14 January 2014 (UTC)
- Derivatives were still unknown? Yes. Tangents were still unknown? No. There was already a significant body of mathematics relating to tangents to curves, some of it going back at least as far as Euclid, to my knowledge. JamesBWatson (talk) 16:52, 14 January 2014 (UTC)
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- For more information about the algorithm you mentioned (applying Newton's method to calculating square roots), please see Methods of computing square roots#Babylonian method and the lead of that article. JRSpriggs (talk) 09:00, 14 January 2014 (UTC)
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...
OK, well, I suppose that would sort of help, but still, is it necessary to have to go read centuries-old fundamental texts on this sort of thing when the thing that sent me looking up how it worked was a/ otherwise very simple, just with some functions whose actual operation was obscured, b/ implemented because the semi-scripting language was so simplistic that it didn't itself include support for square root calculations, so I doubt it used calculus.
Also, is there no hope of a simplified, first-principles sort of explanation, but with the attached caveat that "this only applies in these (XYZ) (general?) cases, and the detailed method described below should be followed to ensure universal success"? How much of what's written here was even known to Newton?
There's a massive problem with a lot of the math pages on WP where even when they're discussing things that otherwise seem simple and are casually tossed into conversation by people who are familiar with them (without any explanation, as if everyone should know it) are written up in such impenetrable, jargon-heavy, eye-watering formula infested copytext that someone who doesn't already have at least foundation degree maths (I have (UK) AS-level, and some half remembered shreds of calculus from a mandatory module within a vocational diploma) may as well just not even bother. Which seems rather more like an explanation that would belong in a textbook, not a general encyclopaedia. Isn't there an expectation (socially / de facto if not actually mandatory / de jure) of, say, an easily understood - even if minced into near oblivion - layman's version in the lede paragraph? WP may not be a how-to guide, but it's not a niche textbook either; I don't need to be a history graduate to understand the history pages, can even understand at least some of the physics (yes, I know it's just "applied maths"), chemistry, etc. I've been familiar with print encyclopaedias from well before I'd even heard of the internet, and it never seemed that hard, even poring over the 6pt text as a pre-teen. And I think my "mistake" (I'm still not entirely sure what mistake I made other than I made one, btw) shows that some clarification may be in order, because a person of otherwise normal intelligence came to look up the method, got frustrated trying to figure it out here, went and googled for something else, and settled on one that is apparently only applicable to a limited set of circumstances without having any idea what those were, or that it was even the case.
I mean, having looked up the explanation given on the other webpage, and comparing it with the first paras of this article itself, I can see the resemblance and figure out what is what ... but some bits are obscure enough that although there is the odd bit of bluetext, including for "function" it's hard to know what to click first or if that page will be any less confusing.
x0 and x1 are easy enough to get (thanks to the returning shreds of the aforementioned school/college room maths, between a quarter and half a lifetime ago), as the initial "wild" guess and the improved estimate of the root; "f" and "function" are placeholders for "y" and "known variable"; and "f'" is... er...
No, wait, what?
OK, perhaps f =/= y. And these aren't so similar. Unless f(x) is "y2 - x" and... no, still can't make it work.
\*abandons attempt*
Look, instead of the snarky bickering and such, is it possible to just get a simple-as-possible explanation, if that's even a thing? Even if just here on the talk page? Even if it comes with a qualification that it only holds "in a particular circumstance" (what this ephemeral set of conditions may be, I have no idea at the moment; I can already see that the numbers in question have to be "reals", which without clicking through to another possibly jargonariffic page I'm going to take a stab as meaning "positive and with no imaginary component", so what's the other requirement?) ??
If not, I might have to make a proposal that pages like this come with a "warning: abandon all hope of understanding this unless you're already qualified in the field, or have four hours to kill looking up all the references" banner ;-) ... which, y'know, would be fair. Some things take time to learn. I'm just having trouble figuring out why it would be the case here.
PS all things like tangents etc can be worked out from fundamental principles, right? And whilst I wouldn't go so far as to demand a detailed explanation of those inline with the text, maybe there's a way to write out the method that incorporates the longhand working-out of said tangent without having to name it as such? (IDK ... wibbling a bit at this point if I'm honest) 193.63.174.211 (talk) 09:29, 15 January 2014 (UTC)
- But the first principles explanation is just that of the article: you got a function and an iteration point, you compute a good or even the best linear approximation to that function in that point. Then compute the root of that linear approximation, replace the iteration point with it, and repeat the whole process. For the same task you can take different functions, for the square root of a you have f(x)=x2-a or f(x)=x-a/x or f(x)=1-a/x2 that all lead to a different iteration formula, and different global behavior of the iteration, but all have the same final quadratic convergence behavior (if convergence occurs).--LutzL (talk) 12:08, 15 January 2014 (UTC)
Section structure of the article
There are, apart from the justifiable request for more references and the questionable request for more inline citations, other problems with the current structure:
- The lead now has a long, unnecessary paragraph with formulas that are duplicated in the directly following description section. It would be sufficient to mention the "solution of the linear approximation", or "root of the tangent" in the lead.
- The description section appears a bit lengthy. If one goes to that length, a short derivation of the error terms based on the quadratic approximation in the root could be added, but leaving the proof-level estimates for the "analysis" section.
- The history section is now very well written (missing time periods for the persian/arab mathematicians). But current consent seems to be to relegate historical remarks to the end of the article?
- "Practical considerations" is a horror story on why not to use the method. In this prominent place it seems highly demotivational. Its main intent is also duplicated in the "failure analysis" section.
- "Examples" and "pseudo code" are currently the last sections. In the spirit that the article should progress from popular science to university level, they should be placed somewhere before the section on the analysis of the method and applications to higher dimensions and complex numbers.
- Analysis of multiplicities and over-relaxation is also an advanced topic that has its logical place after the analysis section. A counter-example for a double root (f(x)=x²) can be placed and explained in the example section. There should only be one place discussing multiplicites, and it should have better references (if at all) than an (incompletely cited, in multiple ways) exercise in some text book.
--LutzL (talk) 09:39, 5 May 2014 (UTC)
- You have good reasons to make changes, so go ahead and make some. Glrx (talk) 21:52, 8 May 2014 (UTC)
The .gif in the description section is not in English.
Just thought I'd make everyone aware, it says "funktion" and "tangente". — Preceding unsigned comment added by 5.66.18.14 (talk) 23:36, 2 May 2015 (UTC)
Assessment comment
The comment(s) below were originally left at Talk:Newton's method/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
The structure of the sections practical considerations, counterexamples and analysis needs an overhaul. Add a bit on line search to make method robust. Newton-Kantorovich theorem. Section on systems needs to be expanded greatly; can grow in a separate article. -- Jitse Niesen (talk) 12:46, 24 May 2007 (UTC) |
Last edited at 12:46, 24 May 2007 (UTC). Substituted at 02:23, 5 May 2016 (UTC)