The Rational Mean. The Arithmonic 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 Artihmetical root-approximating method exempted from neither any trial-&-error checks, nor geometry. Notice that even Newton's method is based entirely on geometry.
This book shows that by means of the Simplest Arithmetic one can develope all those re-known methods (among many others) in a very simple way, without any need of trial-&-error checks, nor geometry, nor derivatives, nor infinitesimal calculus, as you will see in the book.
The Arithmonic Mean is 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 trivial and equal to the number P.
The arihmonic 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 Rational Mean (Sum of all the numerators and denominators) of the three fractions for each set A₁, A₂, A₃, are:
That is, the 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 will yield 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.
As a matter of fact, if the initial set of approximations to the cube root of P is:
x, x, P/x²
Then, the successive arithmonic means yields three new sets in each iteration whose product is always trivial and equal to the number P:
The reader can check that for any row in the table the product of the functions is P, and that the convergence rates of each function in the table corresponds to its row number, that is: quadratic (2nd row), cubic (3rd. row), quartic, quintic, ...
Each of them can be used as an independent iteration function for approximating the cube root of P:
All of them are independent High-Order iteration functions, and the reader can check that all the functions in the second column of the table corresponds to those that produced when applying Newton's, Halley's and Householder's methods to the equation:
Just by agency of the Simplest Arithmetic, and as previously said, without any need of trial-&-error checks, nor geometry, nor derivatives, nor infinitesimal calculus. Moreover, these methods are extended in the book to complex numbers and the general algebraic equation.
Considering the very long story of root-solving methods, this is really striking, indeed.
This is a very rich and unexplored Number-Theory & Numerical-Analysis 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.
The book can be found at: amazon.com/author/b01h4s0gfo
The new methods are fully explained and detailed with convergence proofs, also embracing among many new others, the well known Bernoulli's and Lucas' root-approximating methods, and are also arranged in matricial form and extended to complex numbers and the general algebraic equation.
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.
more preliminar information on the methods can be found at my youtube channel:
Kindly notice that this a private project only supported by your kind contribution when purchasing the book. Indeed, considering the vast scope and philosophical implications of these new methods, I would be so happy to freely distribute these methods abroad, however, I am located at Venezuela and things here are hard enough to allow me to assign a price for this work at amazon.com.
Final remarks on the new methods:
A new general and unifying arithmetical concept: The Rational Mean, which allows to generate, among many other new algorithms, those celebrated Lucas’s, Bernoulli’s, Newton’s, Halley’s, Householder’s root-approximating methods which up to now were considered the exclusive achievement of Infinitesimal Calculus. No derivatives, no trial-and-error checkings, no geometry, no cartesian system, but just Simplest Arithmetic.
These new high-order methods also embrace complex roots and the general algebraic equation, and from the solid evidence at hand, these arithmetical methods have no precedents in the mathematics literature.
Indeed, it is really striking to realize that even ancient mathematicians had at hand the elementary tools for constructing these high-order algorithms. Actually, these new findings compel us to cogitate on the reasons mathematicians were forced to create the "elevated" infinitesimal system that we have inherited.
The book also includes the new Generalized Continued Fractions for approximating the maximum and the minimum modulus root of the general algebraic equation, number Pi, number e, Golden Mean, and others with improved convergence rate.