The Rational Mean.

 

High-Order Arithmetical Root-Approximating Methods

It took thousands of years for mathematicians to finally achieve modern high-order root-approximating methods for algebraic equations, i.e.: Newton’s, Householder’s methods. In the mean time, many attempts were made always by the agency of trial-&-error methods, geometry, and finally the cartesian system and infinitesimal calculus. All through the math history, there are no traces of any general natural arithmetical root-approximating method exempted from either any trial-&-error checks or geometry. Notice that even Newton’s method is based entirely on geometry. This work shows that by means of the Simplest Arithmetic one can develop all those renown methods (among many others) in a very simple way, without any need of trial-&-error checks, nor geometry, nor derivatives, nor infinitesimal calculus:

The Arithmonic Mean

The Arithmonic Mean is an arithmetical operation, a particular case of the Rational Mean (Generalized Mediant), and it can be used for generating high-order root-approximating  functions (Iteration functions), as an example:

Given an initial set A of three fractions whose product is trivial and equal to P (three approximations by defect and excess to the cube root of the number P), all of them arranged in three sets as follows:

The product of the fractions in the set is trivial because the numerators and denominators cancel to each other so the resulting product is obvious.

By computing the Arithmonic Mean for each set, it yields another set of three fractions whose product is also trivial and equal to the number P.

The arithmonic mean requires to previously modify the form of the fractions, by equating pairs of denominators and numerators according to a very simple rule, keeping their values the same, and subsequently applying the Rational Mean (Generalized Mediant: The sum of the numerators and denominators) :

Equating the first two denominators and the last two numerators for the set A₁,
the first two numerators and the last two denominators for the set A₂, and
the first two denominators and the last two numerators for the set A₃, always keeping the values of the fractions the same:

The Rational Mean (Generalized Mediant: Sum of all the numerators and denominators) of the three fractions for each set  A₁,  A₂,  A₃,  are:

Summarizing, the corresponding arithmonic means of A₁,  A₂,  A₃ are:

A new set of three fractions whose product is also trivial and equal to P. Notice that numerators and denominators are trivially canceling each other, producing the obvious product: P. Three new approximations by defect and excess to the number P.

By repeatedly applying this procedure, it yields high-order approximations to the cube root. Moreover, it can be extended to yield high-order iteration functions similar to those of Newton’s, Halley’s, Householder’s  methods.


High-order Root-Approximating Methods

In order simplify the above procedure let’s arrange the initial set of three approximations to the cube root of P, by using the following set of three expressions:

x,    x,    P/x²

Then, by  repeatedly computing the three arithmonic means as stated above, then in each iteration step it will  yield a new set of three expressions whose product is always trivial and equal to the number P, that is, sets of rational functions for approximating the cube root of P at any convergence rate, as can be seen in the following table:


The product of all the rational functions in any row is P, and the convergence rate of each function in the table corresponds to its row number: quadratic (2nd row-Iteration 1), cubic (3rd. row-Iteration 2), quartic(4th. row-Iteration3, quintic, …
Each of them can be used as an independent iteration function for approximating the cube root of P, as for instance the rational function located in row: 5, column: 3:

quintinc funcion.jpg

which has quintic convergence rate.  The reader can check that all the functions in the second column of the table corresponds to those that result from applying Newton’s, Halley’s and Householder’s methods to the equation:

  f(x)=x³-P

Just by agency of the Simplest Arithmetic, without requiring neither any  Trial-&-Error checks, nor Geometry, nor Derivatives, nor Infinitesimal Calculus. Moreover, in the book these methods are extended to the n-th root of any number and the general algebraic equation.

Considering the very long story of root-solving methods, this is really striking ¡¡¡
A very rich but unexplored field. Actually, there are no precedents on this matter since ancient mathematics up to now, so from the evidences at hand, what you will read in the book does not appear in any other math-book abroad.
Really striking, indeed, because this means that all those modern high-order root-approximating methods that have been consecrated in academic journals as an exclusive and superb result of Infinitesimal Calculus can be trivially developed by means of the Simplest Arithmetic, even worse, the new methods could have been easily developed by any inexperienced young student and even children at any stage of the whole history of mathematics and at any place: Greece, Europe, Asia, China, Middle East.
As a consequence, this not just a matter of new algorithms but about the whole history of mathematics and philosophy.

The new methods are accompanied with their corresponding convergence proofs. There is an uncountable number of variants for these methods, the book shows some of them  also embracing the well-known  Bernoulli’s and Lucas’ root-approximating methods, among some new others.

The methods are also simplified and arranged in matricial form and extended to complex numbers and the general algebraic equation.

As for an example, all the iteration functions for approximating the Cube Root of any positive number P, with convergence rate: k,  can be generated by using the following matrical expression:cuberootmatrix

For k=3  the resulting rate of convergence is 3:

cuberootmatrix1

It brings all the iterated functions that correspond to the row entitled: ‘Iteration 2‘  in the table previously shown above for the Cube root of P.  The numerator and denominator of each iterated function are correspondingly located in the first and third row of the resulting matrix. Tne same applies for k = 2, 4, 5, 6, 7,…

A similar expression is easily generated for roots of any degree at any desired convergence rate.


Generalized Continued Fractions

Apart from other stuff that might be of interest to the reader, the book also includes a new type of Generalized Continued Fraction for the number Pi, e,  Golden Mean, among others, i.e.:

This special continued fraction serves as the book’s cover.

The book also includes a chapter on some new geometrical constructions (partitions), as well as a chapter on mathematical foundations of music, specifically on consonance.


Sound Waves and Consonance

The chapter on Music includes some observations on the mathematical fundaments of music and consonance.


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Additional stuff in this webpage:

Synthesizer Open Project (Not included in the book)

At the end of the sidebar, you can find a link for a Synthesizer programming code elaborated using  wolfram language (Mathematica) that can be freely downloaded and improved. This code is not included in the book.  It allows to play chords of up to six sounds at any frequency, including piecewise functions for the attack, decay, sustain and release parameters for each sound. It also incorporates interpolate functions with up to 12 harmonics which can be edited and saved at any time for modeling musical instruments timbre at any frequency. The reason for including this Synthesizer is due to the fact that almost all current synthesizers are based exclusively on the customary frequencies of musical notes, so it is so hard for any researcher to make any analysis on consonance with any frequencies chosen at will.

synthesizer


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