On 16 jul, 18:24, gwh <ghug...@cei.net> wrote: > On Jul 14, 10:30 pm, arithmeticae <djes...@gmail.com> wrote: > > > If you really like to analyze the most simple high-order root-solving algorithms then you should take a look at: > > >http://mipagina.cantv.net/arithmetic/rmdef.htm > > > It is striking to realize that these new extremely simple artihmetical algorithms do not appear in any text on numbers since Babylonian times up to now. > > Maybe not in "any text on numbers", but back in 1945 I purchased a > copy of "Handbook of Engineering Fundamentals", by Eshbach, and the > cube root extraction scheme described there was precisely the same as > the scheme described on one of the links given on the above website. I > used that method lots of times in my engineering career when I needed > more precision than my trusty log-log duplex decitrig slide rule was > able to give me. > > Regards, > > Grover Hughes
Both you and I certainly know that you will not answer to my simple challenge, i mean, to show any simple numerical example on Eschbach's method, so sci.math audience could be able to realize that you are telling the truth, mainly because you certainly know that the new simple arithmetical methods shown in my webpages and book have no precedents, at all.
However, I will do a favor to you and your friend firstname.lastname@example.org. Yes I will show to you a very simple example on computing square roots by agency of the RATIONAL MEAN (notice that i am not talking about NEITHER FAREY FRACTIONS NOR PELL'S EQUATION)
THE VERY SIMPLE EXAMPLE FOLLOWS. I am sure you will also be able to say that you have read what follows in so many books on numbers (PUN INTENDED, OF COURSE), worst when considering that we are talking at this moment about SIMPLE SQUARE ROOTS:
All this comments comes from the contents of the webpage: http://mipagina.cantv.net/arithmetic/rmdef.htm and the book: LA QUINTA OPERACI?N ARITM?TICA. Arithmonic Mean. ? Domingo Gomez Morin. Copyright. All rights reserved. 2006 -----------------------------------PRELIMINARY NOTE----------------------------------- The Rational Mean of the fractions: f1=a1/b1 and f2=a2/b2 is:
Rm[f1, f2] = (a1+a2)/(b1+b2)
By agency of such a simple arithmetical operation you can produce all the Householder expressions for the Nth root of any number P. Notice that if you change the form of the fraction a1/b1= (x/x)*(a1/ b1)= (x*a1)/(x*b1) then you will get another result (provided that x is not equal to 1):
--------------------------------END OF NOTE----------------------------------------
Higher-order rational process based on the Rational Mean:
FUNDAMENTAL PRINCIPLE: Any two fractions whose product is P represent two rational approximations -by defect and excess-- to the square root of P.
If departing from those two fractions you can compute two mean values whose product is also P then you have another two closer approximations to the root. By continuing this process you will get a root-solving algorithm for the square root of P, moreover, by using the Rational Mean you will get a higher-order root-solving algorithm.
Starting with a set of two fractions f1, f2 whose product is f1*f2 = P. For example: f1 =x/1 f2=P/x
Compute the following two rational means:
Rm[(x/x)*f1, f2] = (P+x^2) / (2x) (Newton)
Rm [(P/P)*f1, (x/x)*f2]= (2Px) / (P+x^2)
It yields, two expressions whose product is trivial and equal to P and are closer to the square root of P. You can use each of those new functions as independent iterating functions both of them converging quadratically.
If you don't like quadratic convergence then compute another two similar rational means by previously assigning those new functions to f1 and f2, as follows:
f1 = (P+x^2) / (2x) f2 = (2Px) / (P+x^2)
The two new rational means yields: Rm [(x/x)*f1, f2] = (x^3 + 3Px) / (P +3x^2) (Halley) Rm [(P/P)*f1, (x/x)*f2]= (P^2 + 3Px^2) / (x^3 + 3Px)
two expressions whose product is trivial and equal to P, both of them multiply by THREE the number of exact digits in each iteration.
Two expressions whose product is trivial and equal to P, both of them multiply by FOUR the number of exact digits in each iteration.
By continuing this process, in the next step you will get two functions which multiply by five the number of exact digits in each iteration.
And so on.
Believe it or Not!. Based on the evidence at hand, this so na?ve, trivial, natural and simple rational process has no precedents since Sumerians times up to now. We have not used neither any Cartesian-decimal system, nor any derivatives, nor infinitesimal calculus, at all I think, many experts on the history of mathematics should cogitate on the very long story on root-solving. Indeed, it is disturbing to realize these so simple rational processes based on the rational mean do not appear in any book on numbers since ancient times up to now.
All this is fully explained in my book: LA QUINTA OPERACI?N ARITM?TICA. Arithmonic Mean ? Domingo Gomez Morin. Copyright. All rights reserved. 2006