My Math Forum The Mediant and new high-order root approximating methods.

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 October 20th, 2017, 07:40 PM #21 Senior Member   Joined: Sep 2016 From: USA Posts: 435 Thanks: 247 Math Focus: Dynamical systems, analytic function theory, numerics I've read through the example and I'm pleasantly surprised. You seem much less cranky than I intially thought. I recommend ditching the video with all the quotes for the first half and instead linking people to your site if you want them to examine your work. That said, you still have not addressed any of the points I brought up. The most important of which are the following. 1. You have not shown any convergence estimates or analysis which backs up your claim that you obtain higher order convergence. Moreover, the fact that you obtain Newton iteration as one of your results, which I suspected earlier you might, leads me to wonder what you are expecting here? In a neighborhood of a true zero of a $C^2$ function, Newton's method can be shown to have quadratic convergence. This means that when your guess is close enough for Newton to become a strong contraction, you expect to gain 2 decimal places per iteration. Your numerical examples don't show anything even remotely close to this. The nature of your method leads me to believe you are describing something similar to the secant method which has linear convergence. This is consistent with your numerics and in the absence of an analysis I'm skeptical of your claims. 2. You haven't addressed the problem of multiple basins of attraction. As predicted, more complicated examples than cube roots make it clear that there are issues with your method that at least need to be considered. For example, in your general cubic solver, you impose $P_1 + P_2 + P_3 = -a_0$ which is completely natural. However, your claim that imposing this condition on each iterate ensures each $P_i$ converges only if they have a common $x$ value is flawed. For example, note that for a quartic equation you would have an expression of the form $P_1 + P_2 + P_3 + P_4 = -a_0$. Suppose $x$ is a true solution to the polynomial equation, then notice that if iteration of $P_1$ is converging to $-x$ and iteration of $P_3$ is converging to $x$, then this constraint will still be satisfied. Since these are separate power functions, its completely reasonable to expect that the intersection of the basin of attraction for $-x$ for $P_1$ and that of $x$ for $P_3$ need not be empty. 3. I don't understand the claim that this math is "simpler" than the results using calculus. As I mentioned earlier, and as your computation has confirmed, both your results and methods such as Newton amount to iteration of a particular rational function. The derivation certainly requires calculus, but the use certainly doesn't. Are you claiming your methods are easier to teach? If so, who do you imagine is in need of high order solvers for smooth functions that doesn't have access to calculus? I don't know any high school curriculum that covers numerical methods. Nor is this common even in university until well after the calculus level. In short, even if all your claims are true, who is your audience for these methods? Thanks from topsquark
October 20th, 2017, 08:02 PM   #22
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Quote:
 Originally Posted by SDK In short, even if all your claims are true, who is your audience for these methods?
this post

indicates at least who he thinks the audience is. The relevant quote

"Hope you find of some interest how to teach children on high-order root-approximating methods..."

Must be some pretty clever children in Venezuela.

 October 20th, 2017, 10:29 PM #23 Math Team     Joined: May 2013 From: The Astral plane Posts: 1,878 Thanks: 760 Math Focus: Wibbly wobbly timey-wimey stuff. @Arithmo: I'll will once again let SDK etal handle the actual math questions regarding your methods...He seems to know the material far better than I. However I do have a question about your purpose in posting here. If your methods are what you say they are then there might well be a "revolution" of sorts. But the question on my mind at the moment, if you will pardon my asking, is why are you posting here? Why haven't you/ aren't you published? I can't think of any major achievement where the author writes a book for sale before sending it in to the Journals. So what is your end-game? Clearly you've been around to the other online Forums and have been poked and prodded and had your methods criticized because you won't let anyone in on the details of why this works. I suspect that this Forum will eventually start doing the same if you keep all the details of your derivations to yourself. What has that gotten you and where is this going? -Dan
October 22nd, 2017, 05:27 PM   #24
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Quote:
 Originally Posted by topsquark @Arithmo: I'll will once again let SDK etal handle the actual math questions regarding your methods...He seems to know the material far better than I. However I do have a question about your purpose in posting here. If your methods are what you say they are then there might well be a "revolution" of sorts. But the question on my mind at the moment, if you will pardon my asking, is why are you posting here? Why haven't you/ aren't you published? I can't think of any major achievement where the author writes a book for sale before sending it in to the Journals. -Dan
Firstly, I cannot imagine any other better place for regaining impulse and get
feed back than a math forum. Even the most "silly questions" and harsch criticism always serves as steps to reach to another level and they bring new ideas, I find that unvaluable and more fruitful that only trying to get published in any journal.

Why haven't you/ aren't you published? you asked:
The methods shown in the videos, my webpage and the book are always accompanied by either the phrase:
1.- "It is striking to realize that these methods does not appear in the math literature since ancient times up to now"

or the phrase:
2.- "It is a shame these methods does not appear in the math literature since ancient times up to now"
And those phrases will always accompany those methods.

As all we know, such statements are definitely prohibited in any peer-review journal, and I will never accept to publish the methods without them, then I have an "issue"
As previously said, there are thousands of papers on those peer-review journals stating that Newton's, Halley's and Householder's methods could only be exclusively achieved by the agency of the superb and unique Infinitesimal Calculus, so no way, those phrases will always accompany these new arithmetical methods.

As a consequence, only some particular results (produced by the new methods) have been published in the American mathematical Monthly as well as some reknown books, and webpages.
You can find the links to all those publications in the right side of my webpage:

https://domingogomezmorin.wordpress.com/

Quote:
 Originally Posted by topsquark So what is your end-game? Clearly you've been around to the other online Forums and have been poked and prodded and had your methods criticized because you won't let anyone in on the details of why this works. I suspect that this Forum will eventually start doing the same if you keep all the details of your derivations to yourself. What has that gotten you and where is this going? -Dan
I don't know why yo say: "you won't let anyone in on the details"
I made a book, and it contains all the details, the book is at amazon, and many people who have read it
brought some unvaluable observations to me, even at math-forums, and I am so grateful for that.

My end-game is that these methods will always be accompanied by those two phrases (1, 2), and whoever could be interested in them will have to humbly accept them. may end-game is to be read mainly by young people, not just by those who read journals. Internet is more powerful than journals, they have become the past.

That's it.

Quote:
 Originally Posted by topsquark What has that gotten you and where is this going? -Dan
The methods has been published in the book, so they will stay arround
here for a while and I have no hurry. It is for sure that all this, will find its way through people, mainly young people.

p.d. I know my responses are made with some delay, believe me, we are struggling a battle here in Venezuela, for our lifes and our future.
It is not easy to work over here on these math-issues by these days. So, please accept my apologizes for that.
I'll try to respond another inquire by SDK as soon as I can, I always welcome even harsch criticisms and inquires, as said, I enjoy that.

Last edited by arithmo; October 22nd, 2017 at 05:43 PM.

 October 22nd, 2017, 07:06 PM #25 Math Team     Joined: May 2013 From: The Astral plane Posts: 1,878 Thanks: 760 Math Focus: Wibbly wobbly timey-wimey stuff. My point is that you seem to be more interested in making money off the book rather than sharing your research with the Mathematics community. I have never encountered that kind of attitude in a Mathematician or any other type of researcher before. Are you planning a lecture series? Are you going to present a paper to a Journal? You have not mentioned any of this, except to say "Buy my book if you want more details." (Your web link says nothing about why these methods work, which is what I am talking about.) It may be that you are legitimate and are planning these things. But I have yet to see any evidence that you want to share these methods with the Mathematics community except for profit. This is why I have doubts about your work, not because I think that you are wrong on any specific point. So my question to your response is this: Are you trying to present us with your knowledge or are you just trying to make a buck? -Dan
October 22nd, 2017, 08:03 PM   #26
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Quote:
 Originally Posted by topsquark My point is that you seem to be more interested in making money off the book rather than sharing your research with the Mathematics community. I have never encountered that kind of attitude in a Mathematician or any other type of researcher before. Are you planning a lecture series? Are you going to present a paper to a Journal? You have not mentioned any of this, except to say "Buy my book if you want more details." (Your web link says nothing about why these methods work, which is what I am talking about.) It may be that you are legitimate and are planning these things. But I have yet to see any evidence that you want to share these methods with the Mathematics community except for profit. This is why I have doubts about your work, not because I think that you are wrong on any specific point. So my question to your response is this: Are you trying to present us with your knowledge or are you just trying to make a buck? -Dan

Be honest man. No need to bring out any ridiculous and hypocrite arguments.
Journals earn so much money from subscriptions, huge funds from institutions,
and even worst: they make their bucks from others' work.
Most researchers always begging (licking the boots) for funds.
Professors earn money from their institutions, and oops¡¡¡... also from their math textbooks ¡¡¡.
All of them make their living from maths, they make their daily bucks from nothing else but just from maths.

And I will not criticize them for making their bucks that way.

However, be sure, I certainly do not make my living from maths or any math-work, it is really funny for me to even to hear something like that. I am a structural engineer and structural design and construction is my only way of living.

Thus, just do not even try to make modern mathematicians and researchers appear as "cute angels from heavens", that's truly ridiculous, mainly in this era.

I am happy to share anything for free with good people, with true human beings, and of course I am planning to make something about it by using the cloud, however I will try not to share anything with any of those who hypocritically make their bucks from maths, not having the guts to admit it,
and worst: always use to bring out ridiculous and hypocritical statements to evade the main phrase that will always accompany the new methods:

"It is a real shame these extremely simple high-order arithmetical methods do not appear in the math literature
since ancient times up to now"

The example shown in the web page:
https://domingogomezmorin.wordpress.com/

is a clear evidence of that, they are clearly detailed and summarized there, they all have high-order convergence at any desired rate, any one can check if that really work.
Even worse: You and no one else have seen anything simpler than that in any peer-review paper from any renown journal,
nor any math textbook in the whole math literature, since antiquity up to now.

I know that hurts so much some people ego, sorry for that, I cannot do anything about it but just to resume them in the book.

Last edited by arithmo; October 22nd, 2017 at 08:16 PM.

October 23rd, 2017, 12:23 AM   #27
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The goal of any discovery is to talk about it. You seemed quite happy about giving me the answer to my question about how to approximate the solution to the cubic and I thank you for that. On the other hand you gave no information about how you picked the ratios. That is what is missing...the communication of how to derive such a thing on my own. You aren't actually communicating what you are doing: it's more like you are merely showing off.

I don't really care about how the Journals spend their monies. They are the standard method for communicating new information in the field. If you can find another way to do that then fine. But you are doing no such thing. You are grandstanding and giving away very little of substance so that I will want to buy your book.

Quote:
 I am happy to share anything for free with good people, with true human beings, and of course I am planning to make something about it by using the cloud, however I will try not to share anything with any of those who hypocritically make their bucks from maths, not having the guts to admit it, and worst: always use to bring out ridiculous and hypocritical statements to evade the main phrase that will always accompany the new methods
You aren't sharing anything with me. So just how did I end up in the hypocritical category? I am a Physicist and I only dabble in Number theory. No, I am no angel and I don't expect anyone else to be either. It's a rat-race out there. But withholding information does nothing to advance the field. I'm not being hypocritical...I'm not looking for money. I'm not looking to steal anything from you. What I am is being forward, if perhaps a bit insulting. Making a living is what we all need to do. Sure, many in the field have published books...I love reading them. But they also do the lecture circuit (or the relegate that to their grad students) and publish the new material in some form of standard communication. ie. the Journals. You aren't doing any of that here.

So put up or shut up. Prove to me that you are a real researcher. Go back to the October 20th post and show me how to derive the iteration functions.

-Dan

Last edited by topsquark; October 23rd, 2017 at 12:34 AM.

 October 23rd, 2017, 07:42 AM #28 Banned Camp   Joined: Dec 2013 Posts: 1,117 Thanks: 41 All this blablabla is a bunch of lies "You aren't sharing anything with me. So just how did I end up in the hypocritical category? I am a Physicist and I only dabble in Number theory. No, I am no angel and I don't expect anyone else to be either. It's a rat-race out there. But withholding information does nothing to advance the field. I'm not being hypocritical...I'm not looking for money. I'm not looking to steal anything from you. What I am is being forward, if perhaps a bit insulting. Making a living is what we all need to do. Sure, many in the field have published books...I love reading them. But they also do the lecture circuit (or the relegate that to their grad students) and publish the new material in some form of standard communication. ie. the Journals. You aren't doing any of that here." And you finish threatening : "So put up or shut up. Prove to me that you are a real researcher. Go back to the October 20th post and show me how to derive the iteration functions." A real researcher is totally free using his way to communicate. So read what he wrote or shut up. You are not the only one to behave such way. The reality is that you want to teal everything you and claim that you have invented something : you goal is money and glory that`s it. Assume that someone come to the forum with a formula giving the list of the prime numbers. You check the formula : it is working. Why do you want him to give all the process that lead him to the formula? First : you have to applaud loudly and congratulate him Second : you need to do a lot of work to find how he reaches out to the formula. You are going to stay dumb until your death like many in this forum. No proof is needed if the results are here and anyone could check them! Many formulas invented by Ramanujan and many others were never ever proved. But they are working and that is the more important.
October 23rd, 2017, 09:42 AM   #29
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Quote:
 Originally Posted by mobel All this blablabla is a bunch of lies
gee one crank defending another. Color me shocked.

-- I should amend this slightly. Arithmo isn't quite what I'd ordinarily call a crank, but he is disingenuous regarding his purpose here. --

Quote:
 And you finish threatening : "So put up or shut up. Prove to me that you are a real researcher. Go back to the October 20th post and show me how to derive the iteration functions."
If you consider this a threat you must live quite a sheltered life.

Quote:
 Assume that someone come to the forum with a formula giving the list of the prime numbers. You check the formula : it is working. Why do you want him to give all the process that lead him to the formula? First : you have to applaud loudly and congratulate him Second : you need to do a lot of work to find how he reaches out to the formula. You are going to stay dumb until your death like many in this forum. No proof is needed if the results are here and anyone could check them! Many formulas invented by Ramanujan and many others were never ever proved. But they are working and that is the more important.
This is utter nonsense and counter to the entire history of mathematics. We should be satisfied at formulae gifted from the heaven's without any understand behind them or proof that they do as intended?

Totally absurd.

You must not cross very many bridges because I certainly wouldn't feel safe crossing one in a world where formulae are just accepted.

What's going on here is that due to apparently dire financial circumstances arithmo is trying to sell a book. I see no mention of trying to get this research published in more traditional mathematical journals or even any indication that this research is working towards a further goal. Further I've seen indications that his method might not even work. Finally it looks like this method is nearly identical to but inferior to other commonly used methods.

While part of the function of this forum is to facilitate discussion of mathematics it is not intended to provide free advertising for any retail book that hits the market that happens to be about math.

Arithmo has resisted, no argued, about any discussion of his posts and thus it looks like his main interest is in the advertising.

Last edited by romsek; October 23rd, 2017 at 10:22 AM.

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