Talk:Householder's method

First equation has problems
The first equation under motivation has a problem. First term should be 1/(f(x)-f(a)), not 1/f(x). There is also a problem with the first sentence of the first paragraph. Has a second what?
 * Yeah, but since f(a)=0, that is really the same expression. I added some words in that respect. Sorry for the second ..., this was a copy-paste from Halley's method.--LutzL (talk) 12:19, 30 December 2007 (UTC)

Why called Householder's?
Why are they called Householder's methods? Crookesmoor (talk) 18:06, 30 October 2008 (UTC)
 * Because obviously some the mathematician named Householder is assiciated with them. The only given source refers to
 * A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, (1970)
 * as the original paper, but I don't have it, so I can't verify the contents.--LutzL (talk) 20:04, 1 November 2008 (UTC)
 * His biography should answer your question! P=)  ~ Kaimbridge ~  (talk) 16:16, 21 January 2009 (UTC)

Questions in Motivation
Some editor was bold and asked

"What is a 'simple zero'? What is a simple pole? Why 1/(x-a) and not 1/(f(x) - f(a))?"

directly in the article. 1) Simple zero was directly explained as a root where the derivative is nonzero. In other words, a root of multiplicity one. 2) A simple pole is in the same manner a pole singularity of multiplicity one. The last question should be elaborated, since f(a)=0 and the approximate behaviour of 1/f(x) was to be discussed.--LutzL (talk) 17:29, 22 February 2010 (UTC)

08:00, 17 May 2010 --- First deletion by Gandalf61
10-gauge00 Some guy who have contributed nothing to this page said that " previous derivation is more concise and clearer, and we don't need another one ", I think it is personal or individual preference to determine which one is more concise and clearer, and if someone who can not understand another derivation, think that is nore complicated and unclearer, and have difficult to understand that derivation, he should spend more time to study mathermatics instead of deleting that derivation of this page. Once again, your difficulty of understanging the another derivation is not my fault and wikipedia's fault. Before deleting the derivation, you should think about your personal ability of maths. Furthermore if somone is eager to say to a derivation sould be deleted in the case that there exists a more compact, easy to understand, more elegant, then it is the same to say that all the article should NOT contain another derviation. If I expended further this thought, if someone said that if a derivation should be deleted by the very reason that there exists more compact, clearer one, I'd like to say, go out to the street, and find someone who is more handsome, earns more money, married more beautiful girl, after finding him, then shoot yourself, by the very reason that there exists more clearer and more concise person. (Gauge00 (talk) 09:52, 17 May 2010 (UTC)).


 * 11-Gandalf61 That "some guy" is me. I deleted your "new derivation" because it simply repeats the special cases from the "Methods of lower order" section, with some tedious algebraic derivations. It offers no insights into why Householders method works in general. And it contains several erros - the correct terms are Taylor expansion, Newton's method and Halley's method, for example. I did not say I could not understand it - I can follow it, but it adds nothing useful to the article. As for the rest of your rant, you need to realise that this type of belligerent and obnoxious abuse is not going to win you either friendship or respect in the Wikipedia community. Gandalf61 (talk) 10:17, 17 May 2010 (UTC)


 * 12-gauge00 It is ridiculous that Taylor expansion, Newton's method and Halley's method are erros. I know I used type Tailor something in the page, I did not corrected it, I know, but I was too busy to check spelling check, grammer check, etc. And If you have noticed them, you should corrected them until I did it instead complaining about them. And please dont say Newton method is not correct but Newton's method is. At least wikipedia did not make Newton method link to red link, the broken link, like Newtooown method . And I am glad to hear you could understand my new derivation. But how on earth didn't you realize than my derviatin contains the method how to calculate the $$ (d^{n}/d^{n}x) (1/(f(x)) $$ Do you know that calculation of $$ d^4/d4(x)(1/f) $$ is so terrible? I will include it in the future beliveing $$ d^4/d4(x)(1/f) $$ would be useful to someone even if you dont need it. And you should realize that my derivation contains a relation between Halley method and Newton method, and also contains between d'th order method and d+1'th method. And you said it adds nothing useful to the article. You should realize that the determination my addition would be usefull or not is NOT the duty or priviledge of you. You are not the sole, unique reader, unique referee. Gauge00 (talk) 11:00, 17 May 2010 (UTC))


 * 13-LutzL Your addition is badly readable. The principal ideas of the calculations are barely exposed. And where something is exposed, it is largely exaggerated. For instance, I cannot see a derivation of the method. Because that derivation would have to explain the convergence orders. What is contained is a badly explained procedure for computing the derivatives of 1/f from those of f. But the notation is unusual and very hard to read. There may be some insight in your additions, but it is hard to detect in your formula desert.--LutzL (talk) 11:45, 17 May 2010 (UTC)


 * 14-gauge00 If I understanded correctly your saying as that my article has no derivation part of the method, could I said that my derivation was one of the mathematical induction? Case 1 is true; using case 1, case 2 is true; using case 1, case 2, case 3 true. is the type of the mathematical induction (Gauge00 (talk) 20:54, 17 May 2010 (UTC))


 * 15-Gandalf61 I have placed a note at Wikipedia talk:WikiProject Mathematics asking other interested editors to add their views to this discussion. Gandalf61 (talk) 12:01, 17 May 2010 (UTC)


 * 16-gauge00 Questions; please specify the place where I exaggerated; where is the badly explaned part in the procedure for computing the d of 1/f. What notation is the notation you said that it is the unusual very hard to read; and i'd like to know whether of not you still do not know the reason why I did the derivatives of 1/f at first to get :$$-fD^{(n)} = C(n, 1)D^{(n-1)}f^{\prime} + C(n, 2) D^{(n-2)} f^{\prime\prime} + C(n, 3) D^{(n-3)} f^{\prime\prime\prime} + ... + C(n, n) Df^{(n)} $$? (Gauge00 (talk) 12:12, 17 May 2010 (UTC))
 * 17-Paul Carpenter There's no doubt the presentation of the alternative derivation is very poorly presented. It could be made neater, more concise and significantly clearer. None of that dictates that it shouldn't be there. Pending the dramatic improvement of that section (and probable demotion to subsection of ==Derivation== ) I don't see why it should be completely removed. However, repeating the contents of the methods of lower order is entirely unnecessary and does indeed add nothing to article. Paul Carpenter (talk) 13:33, 17 May 2010 (UTC)

14:34, 17 May 2010 --- 2nd deletion by RGBury

 * (20-gauge00) Funny RDBury deleted my another version of derivation; So I reverted to previous version promptly thinking it is impossible to talk something without it. Probably RDBury thinks that it is possible to talk without contents, How Funny!! And it is interesting he deleted original derivation first, and then deleted my version, and then inserted original derivation. I think that he DID NOT read original derivation and my derivation. He seem he just deleted something that is controversal. He seem the controversial part was the original derivation. HaHa Funny (talk) 14:54, 17 May 2010 (UTC))


 * (21-RDBury) There seems to be a consensus to remove the new section do I did. In general, multiple derivations should be avoided on WP per WP:NOTTEXTBOOK. In this case neither derivation was referenced but the original was simpler and resulted in the formula given in the Method section. Lower order methods are already discussed in their respective articles so there is no need for detailed derivations of them here. To me, the highest priority for this article is finding references, not adding new material.


 * (22-gauge00) By this reason did you deleted original version? Did you ever read the my derivation?? (Gauge00 (talk) 15:02, 17 May 2010 (UTC))


 * (23-gauge00) To you, the highest priority for this article is finding references, not adding new material, and I respect your personal prefernece, Good! you prefer references to adding. But why you forced your policy to wikipedia? And why on earth you started to talk about references? Did someone in this talk page say about the references of my derivation? And you said There seems to be a consensus to remove the new section , you are not apt to conclude about the ongoing talk. Who told you to conclude about this talks? Your father did? And I want to know did you ever adding something to wiki?? You seem just deleting machine. (Gauge00 (talk) 15:14, 17 May 2010 (UTC))


 * (24-gauge00) To RDBury You said that In this case neither derivation was referenced but the original was simpler and resulted in the formula given in the Method section, then you mean that my derivation has not resulted in the formula given in the Method section? Do you ever remember that I defined $$ D = 1/f(x) $$? Painting your dog with spotted black paint makes the animal a cow? (Gauge00 (talk) 15:29, 17 May 2010 (UTC))


 * (25-Gandalf61) Four different editors have responded to you in this thread. All four of us agree that your new section does not improve the article. That is called consensus, and is one of the fundamental policies of Wikipedia. Another fundamental policy is civility. It would be a good idea if you read and thought about our civility policy, and changed the aggressive tone of your posts. Even allowing for the fact that English is probably not your first language, your tone comes across as extremely rude. Gandalf61 (talk) 15:55, 17 May 2010 (UTC)


 * (26-gauge00)  If you are doing math stuff, or sometimes did make a paper or something, please make it more clear. If you said that Four different people, you sould said that the names, one by one, like you did in your papers, as if you wanted those papers to be accepted. You did just said, 4 people. I dare to ask you to name one by one. And I rereaded talk page, LutzL (talk), the creator of this page, did not said that my article does not contain usefulthings, he just said my article is too bad to understand, bad notation etc. Again, for the clearness, you should specify the names who said my article contains nothing new. If you do not, you are making perjury. And you must not think that 4, 5 people can make one consensus, mainly because the number of wikipedia users are more at least 10. Anyway it is interesting that you have difficulty in understaing my article, how did you know my article contains nothing? And I'd like to know when did you said that my article does not improve the article? And you said that my E is rude, then isn't it rude that you deleted by article at the first? Deleting whole article at a glance is a good ettiqutte? gauge00 (123.141.179.226 (talk) 19:08, 17 May 2010 (UTC))

(27-gauge00) Interesting is that there are so many people that have the characteristics that can be seen at the professors, I mean, for me they seem they did the same thing when for example, professors grade  their student's homeworks, I mean, wikipedia, there are so many grading machines. Somebody added something, then the grading machines, try to check its valueness and grade them. Funny and interesting. They should know that the articles added are not their student homework!! It is a good fortune fot students to meet a good professor, the one, who does not throw away student's homework, treet their homework politely, and corrected theiir errors. And I hope that they have to know that the grading is not their whole job and that the grading should be done only with the uni. (Gauge00 (talk) 19:26, 17 May 2010 (UTC))

16:18, 17 May 2010 --- 3rd deletion by R.e.b
(30-gauge00) --- R.e.b deleted my article. You did not write any thing on talk page. You should know that there are lots of people who can delete my article, but the right should be gone to the people of talk page, including me. I think the people who complained about my page have privilege of deleting the article. (Gauge00 (talk) 19:26, 17 May 2010 (UTC))

(31-gauge00) --- I did not major math, and I am not related to the math world. Anyway I have great difficulty of understanding the derivation section; I admit I thought that what is Pade approximation, should I accustomed to it to derive the Householder's; I dare to think at present, lots of people who visited this page by the advertisement of WikiProject Mathematics mightly have the same difficulties of mine. I think that is the reason why talk page has so small visiters. And I am curious ..that the other talk writers have understanded the derivation section. RDBury said that In this case neither derivation was referenced but the original was simpler and resulted in the formula given in the Method section, so I sure RDBury surely understanded it, and he is fond of simpleness, as Einstein was. I agree that the orignal derivation is more simpler, becuase it has fewerer number of lines. Three times simpler, because original version has 1/3 lines of mine.(Gauge00 (talk) 20:18, 17 May 2010 (UTC))

(32-RDBury) --- I've reported this to WP:ANEW. Gauge00 seems to be the only person in favor of keeping the new material but has repeatedly reverted attempts to remove it.--RDBury (talk) 20:27, 17 May 2010 (UTC)


 * (33-gauge00) --- User:Paul_Carpenter said that None of that dictates that it shouldn't be there, and I don't see why it should be completely removed. You are saying that the Paul is not faver of keeping the new material. Answer me.(Gauge00 (talk) 20:54, 17 May 2010 (UTC))


 * (34-gauge00) --- For me, RDBury is a liar; I thought Paul Carpenter perfers keeping, but RDBury neglected him. (Gauge00 (talk) 21:07, 17 May 2010 (UTC))

(35-gauge00) --- RDBury said in WP:ANEW that '''This editor is apparently not fluent in English. However edit comments and comments left on the article's talk page have been rude and sarcastic. It seems that despite a consensus from several more experienced editors that the material added is not appropriate, this editor is insisting on restoring the material despite the prevailing opinion''' If I corrected understanded Paul's statement, I am not the only one who wanted to keep the material, and as I said you neglected Paul, so this example show why you should not act like a referee in wikipedia. Excluding Paul means that you, RDBury, are manipulatjng talk opinion data; and If these my words are correct, you are also not fluent in English. If I am wrong, please correct me. And I'd like to say that if your field is not related with numerical analysis, you are not apt to the captain of this talk page. As you know numerial A is so general and fundamental that one cannot dictate others, as I can not boast I can drive a car.(Gauge00 (talk) 21:26, 17 May 2010 (UTC))

(36-gauge00)  ---  gauge00 makes a monologue I think my derivation of Householder Method was a kind of mathematical induction. That why I started Newton method, and Halley method, and derived 3rd order relation, and, nth order. I dont know why so many people wanted to delete my derivation, they are zealous about my derivation? No, No,... Then is my derivation was the so original, revolutionary that they copy derivation and publish it on journals?? What a insane!!! Anyway why so many people wanted to delete my page? To save internet traffic in downloading Householder's method? NoNo.. In order not for their students not to download the derivation of the Householder's method, and to make them do their homework on their hand? NoNo... What makes me angry more is that why on earth RDBury who does not seem numerical analysis experts acted like a referee of talk page. Please dont act like that if your field is different from Numerial Analysis. In physics, anybody does not act like an quantum mechanics experts. One thing I'd like to say to RDBury, is that, you can make baby at your will at any time, but you can not kill that baby no matter what condition is. You can give money anybody at your will, but you can not get others' money at your will. As you know this is robbery. At the same base, you can creat any material at your will, but you should be careful in deleting something of others material. Are you listening? In jury system, 100% consensus is needed to make someone guilty, on the contraray, only one persion is needed to make him not-guilty. So you can not state that if many people wanted to delete your page, the I can delete, mechanism is not good practive. Making something is on your will, but destroying is not on your will, especially in the case of the something is not yours. you should not said stuff like 'more experienced editors' something, mainly because you did not meet them, any of them. If you dont know them, why making suck a false, unrefernecened words, like, 'more experienced editors'? And experienced on what??? What??? And once again, my derivation was one of 'mathematical induction'. (Gauge00 (talk))

02:49, 18 May 2010 --- 4th deletion by Savonneux

 * (40-Savonneux) --- Im not even sure where to reply in this mess but is it necessary to have an impenetrable (i.e. tedious) derivation tacked on? The one already there illustrates the concept. This doesn't add anything except a lot of scrolling. --Savonneux (talk) 02:46, 18 May 2010 (UTC)


 * (41-gauge00) --- Specify the one that you said The one already there. Please before deletion, think about that if it looks like just a scrolling, it not the fault of the derivation, but could be the fault of yours. And this talk is on going, the talk users could know more about the this talk history, please give them the priviledge of the deletion. I'd dont like a guest who came my house, lying in my bedroom. It seem you came right before, and arrogantly deleted my drivation, didn't you. (Gauge00 (talk) 02:58, 18 May 2010 (UTC)) It is funny right when some looks bad, deleting it? Anyway do you have any relation with math stuff? Your contributions look they all do not have math materials. Can you do the derivative of (1/f)?


 * (42-gauge00) --- You said that The Methods of computing square roots has a bunch of algorithms that are historically significant without their own pages. This page is just Newton's method and Shifting nth root algorithm for finding $$\sqrt[3]{x}$$ both of which apply to nth roots not just cubics. gauge00 started to say.... in the articleMethods of computing cubic roots, my algorithm contains $$ (k + a_1 10^{-1} + a_2 10^{-2} + a_3 10^{-3} + ...)^3 $$ expansion series, in the article, the deleted article; Then do you think the expansion could be used to get the 4th root, and 5th root, 6th root also??? Why you can't see the superscipt $$3$$ on the $$ (k + something + ...)^3 $$ Dont make statement when do not know something. (Gauge00 (talk) 03:16, 18 May 2010 (UTC)) It is extremely entertaining that RDBury, the nominator of Articles for deletion/Methods of computing cubic roots and the its supportor, Savonneux, came here also to Householder's method.


 * (43-Savonneux) --- The contents of that AFD changed considerably from nomination till the end, and yes it had examples of methods that already had articles. It's also a completely different issue that is closed. Meh. --Savonneux (talk) 04:05, 18 May 2010 (UTC)


 * (44-gauge00) --- The expansion series was the esseential part to compute cubic root, and it was there from the beginning. the expansion could be used to solve cubic root, only cubic root.  So You did false statement, it means, you did not understand my method, as most othes did not. From this basis, I do uncertain your math. (Gauge00 (talk) 04:24, 18 May 2010 (UTC)) The shifting nth root algorithm is so naive and elementary to me, I learned it in middle school.  Why on earth I did make an article such elementary shifting nth root?????  My cubic root algorithm is differenet from that. You should know... In adition, you seem you are not related with math, why did you come here?? For me, RDBury and you are scavenging forests to find some target to kill poor animals. Is it your job? or duty? RDBury acts like a policeman in the middle of street. Do you know how bad it is when non-experts came along and deleted my articles?


 * (45-gauge00) --- Hey, Savnneux, your name sounds like a French, I know a few french words, gateux, chateau, eau, plateaux, surely I sould not drop the manteaux, any way what I want to say is that I added some material in Methods of computing square roots, in a section of 'Babylonian method and arithmetic mean'. I'd like to show how arithmetic mean is related to the Babylonian method. The reason I said this here is that you and RDBury, the reference checking machine, would be very happy to hear that I added something material without any references. Surely you two would visited that page, and contribute that page by deleting. (Gauge00 (talk) 05:11, 18 May 2010 (UTC))


 * (46-gauge00) --- User:Gandalf61 was the supportor of the deletion of the Methods of computing cubic roots, maybe came along with me, like a stoker, haha. RDBury, savoneux, Gandalf61, three came along here following me. These three guys came here, and deleted my derivation; at first, by Gandalf, at 2nd by RGBury, 4th by savoneux. The cooperations of you three guys works very well.  And Gandalf said that individual methods already have their own articles, please if you dont understand, dont say anything. Where did you find my digit by digit method in the wiki?? Where? If you can, give the link to me. By giving that link, you can stop me.  (Gauge00 (talk) 05:46, 18 May 2010 (UTC))

07:45, 18 May 2010 --- 5th deletion by Gandalf61
Gandalf61, who deleted at the first, and had placed a note at Wikipedia talk:WikiProject Mathematics, deleted once again, leaving message rv - consensus is that new section should be removed - please accept this and stop edit warring. (Gauge00 (talk) 09:20, 18 May 2010 (UTC))


 * (61-gauge00) --- Gandalf61 said in (25-Gandalf61) that Four different editors have responded to you in this thread. All four of us agree that your new section does not improve the article. That is called consensus gauge00 once agin asks you to specify the four different editors. You Gandalf61, 1st deletor, and RDBury, 2nd deletor, LitzL if you want (13-LutzL), and who?? (Gauge00 (talk) 10:54, 18 May 2010 (UTC))


 * Paul Carpenter had also said above: "repeating the contents of the methods of lower order is entirely unnecessary and does indeed add nothing to article". You are beginning to sound petty and obsessive. I suggest you take a Wikibreak and calm down a bit. Gandalf61 (talk) 12:06, 18 May 2010 (UTC)

preparation for blocking
104th I never know about blocking system of wiki, even though somehow where exists some mechanism to settle stallmate. Anyway as I wrote in this talk page, I did 4th reverted, including not-login state reverting, and I was warned of blocking from someone, even though I do not know the blocking period. Anyway I prepared blocking state by copying my derivation to my name space, to those who want to know about my derivation. My derivation of Householder's method can be found at User:gauge00/Householder. (Gauge00 (talk) 04:56, 18 May 2010 (UTC))

This article needs pictures
Notation can be pretty opaque sometimes if you don't already understand the idea. Pictures can often be groked in a single glance. Just a suggestion. —Preceding unsigned comment added by 98.210.32.163 (talk) 07:13, 26 May 2010 (UTC)

"Derivation new"
The article now has two sections that are substantially the same. Someone should probably work on getting this down to one section. Also, I notice that both version of the derivation need some copyediting, especially the new one. Sławomir Biały (talk) 11:53, 31 May 2010 (UTC)
 * I'd wait some time so gauge00 has time to cool down. The problem with any extension of the article is the missing serious sources. I've seen at least the iteration formulas in some articles, but didn't bookmark them.--LutzL (talk) 15:38, 31 May 2010 (UTC)
 * User:Gauge00 added the "Derivation new" section 10 days ago, but has not come back to clean it up. Since he seems to have abandoned his work on this article, I have removed the "Derivation new" section. Gandalf61 (talk) 12:54, 1 June 2010 (UTC)
 * As we all have noticed, User:Gandalf61 has so much impudance to conclude something on his narrow minded, no basis ground, at his tempting will. Such a beast once upon a time, negelected Carpenter's opinion on his will. Eh?? Did you concluded that gauge00 did nothing means  he must have abandoned his addition on Householder's method? This behaviour is your normal behavior? Haha. DO NOT CONCLUDE SOMETING AS YOU LIKE. I waited also 10 days or so util someone who has more experience or more priviledges, except for you, DOES something on my Derivatation new; someting like, merging two versions, or deleting my one, or deleting original one, EXCEPT you. It is extremely annoying that you User:Gandalf61 deleted my one by one reason that gauge00 seems to forget it??? Anyway I will revert.  And it is funny that someone who earns coins of undergradudates by lecturing maths to them, came here the Householder's method, so basic and elemantary method, and DOES delete and comment on this numerical page. Is it your hobby to play baseball with elementary students of 10 years age? You surely make home runs all the time when you swing your bat!!!  Get out of here, I think a mumerical method is not on the field of mathametics, as RDBury has found out it a few days ago. Do not act as if you are a math specialist in front of  such a common and so elementary numerical method. (Gauge00 (talk) 04:46, 2 June 2010 (UTC))
 * addition The name of User:Gandalf61 contains 61. What does it mean? You was born in 1961, or if not, you started your undergraduge in 1961? What does it mean? I asks it because you, surely, have conclude someting on my 00 tag of my name of gauge00. I mean I'd like to emphasize that the 00 of my name has no meaning. (Gauge00 (talk) 04:46, 2 June 2010 (UTC))


 * Gauge00, I have no problems with your reversion of my edit. However, this sort of abusive posting at talk pages is not tolerated at Wikipedia. I suggest you read our policies Civility and No personal attacks and moderate your attitude accordingly. If you continue to be abusive towards me, I will report your behaviour to Wikiquette alerts. Gandalf61 (talk) 08:11, 2 June 2010 (UTC)


 * Still no work by Gauge00 on merging the two derivations, and no-one else seems to be interested either, so I have removed the new derivation again. Gandalf61 (talk) 13:12, 10 June 2010 (UTC)

gagege00 added citation needed on One could determine the Padé approximant starting from the Taylor polynomial of f using Euclid's algorithm. because I could not understand the method how to determine Pade coeffs using Euclid algorithm. Let me give an example. Since (1+x)/(1-x) = 1 + 2x + 2x^2 + ..., we know the Pade approximation of (1,1) order of 1+2x+2x^2 was (1+ax)/(1+bx),where a=1, b=-1. Then what should I do use the Euclid method to get a=1, b=-1??? Is there any body who can give an example? Why a GCD determination alogrithm, the Euclid algorithm, is related to the Pade coeffs? (Gauge00 (talk) 07:39, 12 June 2010 (UTC))

gauge00 commented '''I waited many days w/o touching original deri, because I'd like to repect the original one. However Gan61 is so diligent to touch others work, I deleted original, JUST TO SEE Gandalf61's response''' while adding article page. At this User:Gandalf61 accused me to the Wikiquette_alerts (Gauge00 (talk) 15:54, 13 June 2010 (UTC))

Hi, what is it that you are after? The electronic equivalent of blood? But you are right, strangely enough the Pade-pages have no mention of the euclidean algorithm, even if it is simple enough.


 * $$\frac{p(x)}{q(x)}=s_n(x)+O(x^{n+1})$$

translates by definition into
 * $$p(x)=s_n(x)\,q(x)+x^{n+1}\,r(x)$$

which can be understood as a representation of the remainder sequence in the extended gcd-computation of $$s_n(x)\text{ and }x^{n+1}$$. In your example, you get
 * $$\begin{align}

&&r_0&=x^3,&u_0&=1,&v_0&=0\\ &&r_1&=1+2x+2x^2,&u_1&=0,&v_1&=1\\ q_1&=x/2-1/2,& r_2&=x/2+1/2,&u_2&=1,&v_2&=-x/2+1/2\\ q_2&=4x,&r_3&=1,&u_3&=-4x,&v_2&=2x^2-2x+1 \end{align}$$ where $$r_k=u_k\,x^3+v_k\,s_2$$, so that
 * $$\begin{align}

s_2(x)+O(x^3)&=\frac{r_1}{v_1}=s_2\\ &=\frac{r_2}{v_2}=\frac{x+1}{-x+1}\\ &=\frac{r_3}{v_3}=\frac{1}{2x^2-2x+1} \end{align}$$ giving a diagonal of the Pade table. --LutzL (talk) 09:48, 14 June 2010 (UTC)

Title should be Schröder's method
I believe this method is due to Schröder. The name "Householder's method" appears to be due to the article "Newton’s method and high order iterations" by Sebah and Gourdon, but Householder did not claim to invent the method. Householder credits Schröder for the method. The Wolfram MathWorld article on Schröder's method cites appropriate references, including the book by Householder that describes this particular instance of Schröder's method (Theorem 4.4.2). — Preceding unsigned comment added by 129.55.200.20 (talk) 19:55, 20 March 2012 (UTC)

Assessment comment
Substituted at 02:13, 5 May 2016 (UTC)

Higher order convergence?
What this article badly needs, but does not have, is a proof of the statement
 * $$| x_{n+1} - a | \le K \cdot {| x_n - a |}^{d+1} $$, for some $$K > 0.\!$$

And in particular a way to determine the value of K so that I can tell just how useful these methods actually are. JRSpriggs (talk) 05:14, 1 May 2018 (UTC)

How to write the equation
Wouldn't it be better to write the equation under "Method" as


 * $$x_{n+1} = x_n + d\; \frac {\left(f\right)^{d}(x_n)} {\left(f\right)^{d-1}(x_n)} $$

rather than taking recropicals of the function in the numerator and denominator? Bubba73 You talkin' to me? 01:12, 11 May 2024 (UTC)


 * Other sources give a different equation!! Bubba73 You talkin' to me? 01:22, 11 May 2024 (UTC)


 * The derivative of a reciprocal is not the same as the reciprocal of the derivative. — Q uantling (talk &#124; contribs) 23:43, 11 May 2024 (UTC)


 * The notation was not clear to me. So you take the derivatives of the reciprocal of the function, right? Bubba73 You talkin' to me? 02:21, 12 May 2024 (UTC)
 * Yes. The superscript $(d)$ and superscript $(d−1)$ indicate the number of times to differentiate.  — Q uantling (talk &#124; contribs) — Q uantling (talk &#124; contribs) 15:32, 13 May 2024 (UTC)