On 16 jul, 04:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net> wrote:
> I don't think the claim that these methods are in any way > new stands up to scrutiny. > The idea of Farey dissections is clearly not new. > It is mentioned in Hardy and Wright for example.
Frankly, I had so many doubts about answering this response from yours, because I cannot realize if your remarks are the consequence of your ignorance on the methods shown in my webpages, or you just want to disturb people and cause confusion by making false statements. Whatever the case, your comments are nonsense.
You mentioned "Farey Fractions". Do you know what you are talking about?
I have never used any "Farey Fractions" in any of my methods, so I cannot understand why you are mentioning them as the center of the issue. Farey Fractions have been restricted only to reduced fractions, and they operate only between TWO FRACTIONS (even when you can compute many Mediants at the same time it always operate between TWO FRACTIONS). I cannot believe you dare to talk about this issue without even reading any single bit of my methods based on the Rational Mean which is not restricted to reduced fractions and operate on any set of fractions.
> Hurwitz wrote a paper "Ueber die Irrationalzahlen" > in the 1890s which describes a "mediant" method based on Farey > fractions that produces best rational approximations.
The paper you are mentioning deals only with best approximations to any given number, but that is far from being a ROOT-SOLVING ALGORITHM. You are not bringing any reference to a ROOT-SOLVING ALGORITHM by far similar to those shown in my web pages, worst, you are not bringing, at all, any reference to any ROOT-SOLVING ALGORITHM. So, it is clear that you don't know what you are talking about.
The issue on "best approximations" is well explained on my web pages. It is well known that the fundamental law for generating the convergents of continued fractions is the "Mediant" which is the operation which rules Farey Fractions. The fact that the Mediant is a particular case of the Rational Mean does not mean nothing. All those well-known methods for finding best rational approximations that you metined are basically "Continued Fractions of the second degree" (all this is fully defined and explained in my webpages, but it is clear that you have not read any single bit of it)
Indeed, I cannot believe you dare to talk about all this without taking care of what you read and say. Indeed, it seems you even have not read any single bit of my webpages. The only reason I can find for your making false statements and causing confusion is that my comments about the whole root-solving story really hurt you. I'm so sorry for that, but I have to continue doing so, because all that is the crude Truth: It is strinking to realize that such simple and trivial methods do not appear in any book on numbers since Babylonian times up to now. I know that can really hurt some people so much, sorry for that.
> Monkmeyer and Mahler have examined generalizations > of Farey fractions, essentially a higher order > "mediant" method, intended to produce best rational > simultaneous approximations to a set of irrationals.
Again: You are referring to best approximations to any ginven number, those mathematicians you mentioned didn't make any ROOT-SOLVING METHOD. To find best approximations to any given number is far from being similar to a GENERAL ROOT-SOLVING METHOD.
> Can Morin find best rational approximations > to cubrt(2), cubrt(4) with his methods ?
YES, by means of these so TRIVIAL methods everybody CAN, even young students at secondary schools.
It is finally clear that you have never read any single bit of my web pages. These new methods based on the Rational Means embrace --apart from many other new algorithms-- all the well known Bernoulli's, Newton's, Halley's, Householder's methods. Many examples are shown in my web pages, you can compute any root of any number with any convergence speed. The examples are shown in my webpages, do not ask me to explain all them here, just read them.
Your statements are the main reason for the title of my posting: "DEDICATED TO YOUNG MATH STUDENTS..." mainly because young people are usually eager to find new things and try to read before talking about any issue, young people do not pretend to be "experts" but just to learn and to think and that is really wonderful, that is a virtue that many others have lost trough the years that have them passed by.
> He has not been able to do so in the past.- Ocultar texto de la cita - >
That's not true. Again it is finally clear that you have never read any single bit of my web pages.
The only reason I can find for your making false statements and causing confusion is that my comments about the whole root-solving story really hurt you. I'm so sorry for that, but I have to continue doing so, because all that is the crude Truth: It is strinking to realize that such simple and trivial methods do not appear in any book on numbers since Babylonian times up to now. I know that can really hurt some people so much, sorry for that.