Book Contents

New Numerical Methods. The Rational Mean

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Note: Articles, comments and references to these methods can be found at: Book References Page.


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.

As an example follows page 44, with the matricial expression (Sk nxn) that compiles some of the new general arithmetical methods for approximating the nth root of a number P, with any desired convergence rate k (See also page 34 below):

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Follow few selected pages from the book with tidbits on the new methods:

The following three pages deal with the definition of the Rational Mean operation which is the general and unifying principle that rules all the new numerical methods:

The following two pages  contain a brief introduction to the new methods:

The next page describes a very particular method for finding high-order iteration functions for approximating the square root of a positive number P:

The following two pages shows some high-order iteration functions that result from applying the new and extremely easy arithmetical methods for approximating the cube root of a number P.

The new methods are extended to roots of any degree with any desired convergence rate. Some examples on approximating complex roots, as well as the general algebraic equation are given.

By agency of Simplest Arithmetic, these new methods embrace, among many other new iteration algorithms, those celebrated methods that up to now were supposed to be the exclusive achievement of the Infinitesimal Calculus we have inherited.

No derivatives, no Trial-&-Error checking, no geometry, no Cartesian System, but just Simplest Arithmetic.

The matrix form of the new methods is also included and leads the way to many amazing findings: