User talk:Bo Jacoby

Root-finding algorithm

Hello, and welcome on Wikipedia. I have some questions about your addition to root-finding algorithm. I don't remembering seeing this method before, but that's does not say much as I never really studied the numerical solution of polynomial equations. Do you have some reference for this method (this is required for verifiability)? Is there some analysis; for instance, does the iteration always converge, and is anything known about the speed of convergence? Just a small remark: we sign our contributions on talk pages, but not in the articlesthemselves; see Wikipedia:Ownership of articles#Guidelines. I hope that you continue contributing. Please drop by at Wikipedia:WikiProject Mathematics and feel free to ask me any questions on User talk:Jitse Niesen. Cheers, Jitse Niesen (talk) 22:20, 12 September 2005 (UTC)

Hello Jitse. Thank you very much for your comment on my article on root-finding algoritm. You request a reference for verifiability and some analysis, and you ask whether the method always converge and what is the speed? I agree that such theoretical stuff would be nice, but alas I have not got it. I have got a lot of practical experience with the method. It was taught in an engineering school in Copenhagen for more than 5 years, and the students implemented it on computer and solved thousands of examples. I have not got any nontrivial reports on nonconvergence. So much for verifiability. Does the method always converge ? The answer is no for topological reasons. This is why. Consider initial guess p,q,r,s converging towards roots P,Q,R,S. For reasons of symmetry the initial guess q,p,r,s will converge towards Q,P,R,S. Both solutions are satisfactory, but they are not the same point in four-dimensional complex space. Consider f(t)=(1-t)(p,q,r,s)+t(q,p,r,s), 0<t<1. This line joins the two initial guesses. Note that the iteration function, g, is continuous, no matter how many times we iterate. We don't iterate an infinite number of times. Let A and B be open disjoint sets such that A contains (P,Q,R,S) and B contains (Q,P,R,S) and such that g(f(0)) is in A and g(f(1)) is in B. But no continuous curve can jump from A to B. So for some value of t, 0<t<1, g(f(t)) is outside both A and B, and so the method does not converge everywhere.

I do not think that this immature argument belongs to a wikipedia article.

However, I believe that the method converges 'almost everywhere' in the sense of Lebesque, but I have no proof. Nevertheless, the question of convergence is not the right question to pose. As you only iterate a finite number of times, you will not necessary get close to a solution even if the method converges eventually. So, the method is good for no good theoretical reason! The solutions are attracting fixpoints for the iteration function. That's all.

Bo Jacoby 07:12, 13 September 2005 (UTC)

Please vote

Hello. Please vote at Wikipedia:Featured list candidates/List of lists of mathematical topics. Michael Hardy 23:01, 14 October 2005 (UTC)

Mathematical notation style conventions (non-TeX should match TeX as much as possible)

Hello. Please note the differences between the first and second versions of each of the following:

ln(-1) is a solution to e^x=-1.

ln(−1) is a solution to e^x = −1.

(Proper minus sign instead of nearly invisible hyphen. Spacing on both sides of "=".)

If x=it is

If x = it is

(Spacing.)

e^x=e^it is a point on

e^x = e^it is a point on

(Spacing.)

from point 1 (=1+0i) to e^it.

from point 1 (= 1 + 0i) to e^it

(Spacing. Italicizing i BOTH times, not just the second time.)

at the point -1 (=-1+0i). So e^iπ=-1.

at the point −1 (= −1 + 0i). So e^iπ = −1.

(Proper minus sign. Spacing. Italicizing i BOTH times. Digits, inclduing "1", should not be italicized in non-TeX mathematical notation; neither should punctuation, although that point doesn't arise here.)

And so ln(-1)=iπ

And so ln(−1) = iπ

(Proper minus sign. Spacing. Consistently italicizing i.)

Michael Hardy 19:36, 3 November 2005 (UTC)

Thank you very much ! Bo Jacoby 07:27, 4 November 2005 (UTC)

Hot & Cold Photons?

I've left some comment on the thermodynamic evolution "Talk Page". Let me know if you have suggestions. Thanks:--Wavesmikey 04:38, 26 November 2005 (UTC)

why the difference in notation?

Consider the expression

$\ {n \choose i}p^{i}(1-p)^{n-i}$

Fixing (n,p) it is the binomial distribution of i. Fixing (n,i) it is the (unnormalized) beta distribution of p. The article does not clarify this.

Bo Jacoby 10:02, 15 September 2005 (UTC)

This is mentioned only implicitly in the current version, which describes the beta distribution as the conjugate prior for the binomial. You could add a section on occurrence and uses of the beta distribution that would clarify this point further. --MarkSweep✍ 12:50, 15 September 2005 (UTC)

I don't see what makes you think the article is not explicit about this point. You wrote this on Sepember 15th, when the version of September 6th was there, and that version is perfectly explicit about it. It says the density f(x) is defined on the interval [0, 1], and x where it appears in that formula is the same as what you're calling p above. How explicit can you get? Michael Hardy 23:07, 16 December 2005 (UTC)

... or did you mean it fails to clarify that the same expression defines both functions? OK, maybe you did mean that ... Michael Hardy 23:08, 16 December 2005 (UTC)

Yes, precisely! Bo Jacoby 16:54, 31 December 2005 (UTC) See Inferential statistics, where the same simple expression is used for deductive and inductive distributions, and where the limiting cases are: the binomial distribution, the beta distribution, the poisson distribution and the gamma distribution, and, of cause, the normal distribution. I find this unified approch very attractive. Bo Jacoby 09:58, 4 January 2006 (UTC)

Mathematical notation conventions

Hello. Your comments at talk:normal distribution inspire this comment. In editing mathematics articles, you may find it useful to bear in mind the difference in (1) sizes of parentheses and (2) the dots at the end in these two expressions:

$Z(t)=(d/dt)\log(\int {e^{xt}f(x)dx})=\mu +\sigma ^{2}t+...$

$Z(t)=(d/dt)\log \left(\int {e^{xt}f(x)dx}\right)=\mu +\sigma ^{2}t+\cdots$

Michael Hardy 23:56, 8 January 2006 (UTC)

Thank you very much. I totally agree. Please feel free to edit on the spot. Bo Jacoby 07:52, 13 January 2006 (UTC)

Reference

"Bo Jacoby, Nulpunkter for polynomier, CAE-nyt 1988" — could you please write out "CAE-nyt" in full? Is it a journal, a technical report, something else? I have no idea where to find this reference. Thanks. -- Jitse Niesen (talk) 11:26, 11 January 2006 (UTC)

"CAE-nyt" = 'Computer Aided Engineering News', a periodical for "Dansk CAE Gruppe" = 'Danish CAE Group'. I can fax the article to you if you are interested in history. For mathematical reasons you need not read it, because the explanation in the WP-article is better than that of the old article. Bo Jacoby 08:00, 12 January 2006 (UTC)

I found the method from scatch, but I don't know who was the first one to do so. I gave a lecture to 'Dansk Selskab for Bygningsstatik' on December 10th 1991. My lecture was published and a reference to that publication is now added to the WP-article. After the lecture I had some correspondance with Jørgen Sand. He says that the method is the Durand Kerner method, and he gave the following reference, which I have not checked.

Terano, T., el al (1978): An Algebraic Equation Solver with global convergence Property. Research memorandum RMI 78-03, Tokyo.

For topological reasons strict global convergence is impossible, but the method converges almost everywhere, and the convergence is fast. Bo Jacoby 07:49, 13 January 2006 (UTC)

Excellent. I found some references to articles about the Durand-Kerner method. I'll check them when I have some time and see whether this is indeed the same method as described in Jacoby's method. -- Jitse Niesen (talk) 13:53, 13 January 2006 (UTC)

Ohh. I had to look. Cool. How about some pictures of the basin of attraction for this solver? We have those famous pictures of the basin of attraction for the Newton zero finder; I wonder how this compares. In particular, its not clear to me how/why the initial guesses can end up in different basins. linas 05:46, 31 January 2006 (UTC)

The space C⁴ contains 24 open basins of attraction, one for each permutation of the four roots of a degree 4 polynomial. I don't know how to make a picture of that. If an initial guess p,q,r,s is in one of the basins, then q,p,r,s is in another basin. Bo Jacoby 07:33, 31 January 2006 (UTC)

Style remarks

Hi Bo. I have a few style remarks. First is that one should make variables italic, so x instead of x. Second, per the math style manual one should not force PNG images if inline, so one should write $a_{n}$ instead of $a_{n}\,$ which is an image. These are small things, but they are good practice to follow. :) Oleg Alexandrov (talk) 01:17, 21 January 2006 (UTC)

Thanks, Oleg. You've got a point. I need to find out how to make a little not-equal sign in 'math'.

$x=0\,$, $x\neq 0$, $x=0$, $x?0$  ?

Without 'math' it can be done, but then the font is different:

x = 0, x ≠ 0

I'd like if the same variable takes exactly the same typographical shape thoughout the article. Bo Jacoby 05:49, 21 January 2006 (UTC)

In short, the math display on the web sucks. :) Oleg Alexandrov (talk) 06:17, 21 January 2006 (UTC)

One more style remark

Hi Bo. Just one remark. Writing links as Ordinary_differential_equation#Homogeneous_linear_ODEs_with_constant_coefficients is not a good idea, as they mess up the diffs, as you can see here. Then it is hard to see what changed. I will fix that right now, but a tip for future reference is to remove the underscores. (And by the way, I don't know if it is a good idea to link to sections to start with; those section names can (will) change eventually, and then the link breaks down. But I see, it does not hurt either). Oleg Alexandrov (talk) 00:23, 24 January 2006 (UTC)

Thanks. Its a very good tip. But why didn't you like my other edits to root-finding algorithm ? Please note the Wikipedia:Simplified_Ruleset point 9. Take your time to produce what we agree is a step forwards, rather than to make what I must consider a step backwards. For example I don't think that the words: 'Much attention has been given' belong in an encyclopedia. Bo Jacoby 10:32, 24 January 2006 (UTC)

Splitting circle method

I noticed this comment of yours. I created the article after stumbling across the algorithm's name, but didn't write an explanation as I couldn't figure out much from the sources I found. I created a stub anyway in the hope that someone with more knowledge in this domain will be able to expand it. If you could, the work would certainly be appreciated. Fredrik Johansson - talk - contribs 11:11, 24 January 2006 (UTC)

Thanks to this I read about Jacoby's method. That's a quite interesting algorithm, and remarkably simple to implement. Though it evidently works, it would be nice to have an online reference outside of Wikipedia for verification purposes. You don't have a website where you could put a description? Fredrik Johansson - talk - contribs 11:31, 24 January 2006 (UTC)

Hi Fredrik. Isn't it remarkable that an algorithm 'evidently works' ? Alas my references are all to old to be online. A new website would basicly contain the same information as the WP-article. Look at Talk:Root-finding algorithm for some discussion. There is not much more to be said. Try it and convince yourself that the problem is solved. Bo Jacoby 13:35, 24 January 2006 (UTC)

There's no problem, just an opportunity to make verification more convenient for future readers. Fredrik Johansson - talk - contribs 15:58, 24 January 2006 (UTC)

n-ary operations

I haven't looked at your edit to function (mathematics) on this point, but there are such things as n-ary operations, no matter what the abstract algebra article may say... Randall Holmes 22:40, 29 January 2006 (UTC)

I agree. But the group operation is an example of a binary operation, and not of an n-ary operation for n>2. Bo Jacoby 23:08, 29 January 2006 (UTC)

Query

• JA: Bo, I moved your question to the end (I gave warning in the edit line), as it's best to put new talk at the end, or else people tend to miss it. I'm writing a reply as we speak, well, not just this second, but in a second. Jon Awbrey 15:04, 3 February 2006 (UTC)

Combinations

Hi. Some of your changes to the Combinations page removed info that is relevant, without making it really clear that the material was moved to another article. IMO - it would be better to have a complete, self-contained article on combinations, or to move all of the info to binomial distribution and then have combinations redirect there. Just my $0.02. dryguy 19:20, 8 February 2006 (UTC)

Surely the information is relevant, but it is also stated in binomial coefficient, so it does not need to be repeated everywhere. A link is sufficient. Bo Jacoby 10:44, 9 February 2006 (UTC)

Sorry, I mis-typed. I meant to say move to the binomial coefficient article. In any event, my point was, that some of the info that was moved was highly relevant to the combinations article, and probably best belongs there. If the duplication bothers you, why not pick one of the two articles and place all of the combinations info in one place. I think that the combinations article is now a bit too thin. It could either be restored, or the remaining info moved to binomial coefficient with combinations redirecting to the binomial coefficient article. dryguy 13:32, 9 February 2006 (UTC)

There is a discussion going on regarding merging of the two articles, as you also suggest, but not everybody is in favour of a merge. The present cleanup is a compromise. I don't mind at all that an article is thin, if it contains a definition and a link to more detail in another article. The concept of 'combination' is equivalent to 'subset', so there is not much to be said, I think. Bo Jacoby 14:30, 9 February 2006 (UTC)

Multiset

Hello. Please note my recent edits to that article. Michael Hardy 00:40, 9 February 2006 (UTC) Thank you, Michael. Bo Jacoby 10:31, 9 February 2006 (UTC)

Style

Bo, just one remark, and I may have said it before. Per the math style manual, variables should be italic. Thanks. Oleg Alexandrov (talk) 16:02, 8 March 2006 (UTC)

Ordinal fraction listed for deletion

An article that you have been involved in editing, Ordinal fraction , has been listed at Wikipedia:Articles for deletion/Ordinal fraction . Please look there to see why this is, if you are interested in it not being deleted. Thank you.

Note

You may want to take a look and comment at Wikipedia talk:WikiProject Mathematics#Problem editor.

The moral of the story is that please modify articles only on topics you are very sure about, and only when you have good published references for whatever you are writing.

Also, if a couple or more of editors tell you to drop something, then drop it, especially if you are not completely sure you perfectly understand the topic at hand. Oleg Alexandrov (talk) 05:09, 17 August 2006 (UTC)

Exponentiation

I moved your comment about the article to Talk:Exponentiation because discussions about article content belong on article talk pages. I assure you I don't have any personal grudge with you. The article Kepler's laws of planetary motion seems much improved due to your editing. CMummert · talk 14:17, 12 January 2007 (UTC)

Thank you! Bo Jacoby 15:31, 12 January 2007 (UTC).

Mathematics CotW

I am writing you to let you know that the Mathematics Collaboration of the week(soon to "of the month") is getting an overhaul of sorts and I would encourage you to participate in whatever way you can, i.e. nominate an article, contribute to an article, or sign up to be part of the project. Any help would be greatly appreciated, thanks--Cronholm144 17:46, 13 May 2007 (UTC)

Thanks! Bo Jacoby 18:04, 19 May 2007 (UTC).

definition of a subgroup

A subgroup is a pair (S, *) closed under the operation * and selection of unity, which is what we call a 'nullary operation". This means that the only unity allowed in the construction of a subgroup is the unity of the group. --VKokielov 18:37, 4 June 2007 (UTC)

By the way, 0 is never part of the multiplicative group of a field. --VKokielov 18:42, 4 June 2007 (UTC)

Thanks. The pair ({0},·) is closed under the operation of multiplication · . The element 0 satisfies 0·x=x for x in {0}, because 0·x = 0·0 = 0 = x. The group ({0},·) is not a proper subgroup of a larger group of complex numbers, but still it is a group. It is isomorphic to ({1},·) . Bo Jacoby 22:18, 4 June 2007 (UTC).

thanks

Thanks for such a precise and detailed answer to my question about statistical significance on the Talk:Standard deviation page. Luzhin 17:13, 23 July 2007 (UTC)

you are welcome, my friend. Bo Jacoby 20:24, 23 July 2007 (UTC).

Please don't use Wikipedia for self-promotion

Howdy, I noticed the little disagreement over at Wikipedia:Reference_desk/Mathematics#What_is_the_addition_equivalent_of_a_factorial.3F and it struck me as odd than a mathematical disagreement would get people saying "please take it elsewhere." In looking into this further, I came across things like Wikipedia_talk:WikiProject_Mathematics/Archive_16#Problem_editor and Wikipedia:Articles_for_deletion/Ordinal_fraction. It appears there has been an ongoing problem for quite some time with you trying to promote your own nonstandard notation. I'm not much of a subject matter expert in this field, so I can only go by what other people have written, but do you agree with this assessment? I must remind you once again that editors are expected to cooperate with each other, and this includes observing Wikipedia's policies and guidelines. Friday (talk) 18:56, 6 August 2007 (UTC)

\cdots on Negative binomial distribution

Does the Negative binomial distribution article really need product dots between each variable? I've never seen it written like this is statistics books, or many other articles.--Vince | Talk 07:55, 2 November 2007 (UTC)

no, it is not strictly necessary, but it makes the formulas safer to read, and it makes no harm. It avoids confusion where f(k) does not imply multiplication while p(1-p) does imply multiplication. Bo Jacoby 23:53, 2 November 2007 (UTC).

Durand-Kerner-Weierstrass method

Thank you for considering my proposed change regarding subscripts. I believe the two formulations are equivalent. I found the alternate slightly easier to implement for arbitrary n, as each iteration relies only on results from previous iterations, rather than the iteration in progress. This makes it easier to halt iteration if changes fall below a specified precision.

May I impose on you to comment on the suitability of this algorithm to polynomials with complex coefficients?

John Matthews JMatthews 09:01, 2 November 2007 (UTC)

Thanks for your comment. The two formulations are not quite equivalent. In the original case the newly computed values of the approximate are used as soon as possible. In your formulation they are not used until after the loop. Both formulations provide useful algorithms. The original formulation is the one consistent with the numerical example.

Regarding the halting condition: You do not want the computer to loop infinitely at any rate. So in very exceptional cases you must stop the iteration even if the roots have not been found with the specified accuracy. How many iterations are you prepared to perform in that case? Why not use the same number of iterations in the normal case? So don't bother testing against a precision, but iterate in the normal case the same number of times as you iterate in the worst case. Have a nice day. Bo Jacoby 09:18, 2 November 2007 (UTC).

Thank you for this thoughtful analysis. In this particular case, I am implementing a generic procedure. The user may have instantiated the code with a more or less precise numerical type, depending on space and time constraints. I'm not sure I see the value in iterating beyond the useful precision of the specified type. John Matthews JMatthews 11:54, 2 November 2007 (UTC)

Hi John. You will find that the 'old' version of the algorithm is slightly space-saving as only one version of the variables p,q,r,s is needed.

Yes, I see this. Of course, for a given polynomial order, n, each iteration takes O(n²) effort, while the array copy takes only O(n). It is still an appealing optimization. John Matthews JMatthews 19:06, 4 November 2007 (UTC)

You will also find that the criterion for halting the iteration loop is a little complicated, involving first the computation of a size of the last step taken, and secondly an emergency break preventing the program from looping infinitely in exceptional cases. If you try to solve x²+1=0 using real initial guesses the algorithm will loop forever. That's why real initial guesses are avoided. But no matter which initial guesses you choose some equation exists that remains unsolvable using these initial guesses. So, theoretically, you cannot safeguard against the infinite loop. That's a hard fact of life. So you must stop the program after a finite number of iterations. If your experiments show that the roots usually have stopped moving after 5 iterations, you may choose to limit the number of iterations to, say, 20. Having done this you may contemplate the cost and the benefit of performing these 20 iterations every time. The cost is the time of doing 15 needless iterations in the normal situation, and the benefit is the simplification of coding the halting criterion. It may seem insensitive and brutal and un-gentlemanlike to keep beating on roots after they have stopped moving, but actually it makes no harm or pain. Have fun! Bo Jacoby 00:22, 3 November 2007 (UTC).

My exit condition looks at both max_count and change. The former is O(1); latter is only O(n). I'm looking at scaling max_count as a function of n, now. Yes, it is indeed fun! John Matthews JMatthews 19:06, 4 November 2007 (UTC)

Yes, and the program becomes clearer if you simply omit the change part of the exit condition. You need the max_count part anyway. If one exit condition is sufficient then the other one is unnecessary from a logical point of view, if not from an optimization point of view. For sufficiently small values of n the algorithm is fast no matter what, and for sufficiently big values of n the algorithm is too slow no matter what. So why complicate the program for the sake of the intermediate values of n only? Scaling max_count is a good idea. I believe that max_count=7*n is sufficient, but I am not sure. It depends on the 'typical' equations to solve. Your report will be interesting. Bo Jacoby 12:26, 6 November 2007 (UTC).

I'm getting excellent results up to order 38 with the floating point precision I have available. Optimizing the max_count parameter proved unreliable, and the exit condition is really quite simple. Rather than reference my site, I'll post a link here. Thanks! John Matthews JMatthews 18:29, 7 November 2007 (UTC)