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We propose and demonstrate an original geometric argument for the ancient Babylonian square root method, which is analyzed and compared to the Newton-Raphson method. Based on simple geometry and algebraic analysis the former original iterated map is derived and reinterpreted. Time series, fixed points, stability analysis and convergence schemes are studied and compared for both methods, in the approach of discrete dynamical systems.

The oldest known algorithm for successive numerical approximations to the square root of a real number was created by the Babylonians (1950 BC-648 BC) as reported by the Greek mathematician Heron of Alexandria [

Recent applications of iterated maps in numerical analysis have been found in literature, using and extending the techniques of dynamical systems to the study of numerical algorithms and number theory [

In this work, we study and compare two methods that are based on iterated maps, and some common tools from nonlinear dynamics [

From the point of view of discrete dynamic systems BABM is a one-dimensional iterated map defined over the set of real numbers

The Babylonian method (BABM) is based on the iterated map defined by

where

The BABM can be seen as particular case of the more general NRM used to evaluate a zero of the function

The geometric construction used by NRM can be reduced to the following geometric path: 1) take an initial value

The numerical approximations generated by the NRM method are in general represented by the iterated map

where

which is exactly the same BABM iterated map function.

There is no historical report showing any geometric argument perhaps used to construct the BABM, but in this section we propose a simple and original one, in the same spirit of that used to construct the NRM.

0 | 3.000000000000000 | 4 | 1.414213780047197 |

1 | 1.833333333333333 | 5 | 1.414213562373111 |

2 | 1.462121212121212 | 6 | 1.414213562373095 |

3 | 1.414998429894802 | 7 | 1.414213562373095 |

The root of the function (1.2) is found by approximation, from the initial condition, doing arithmetic mean between two numbers. We assume that the root is between two points, do the arithmetic mean between these two points we are closer and closer to the root. These two points are

Given an initial condition we start the geometric path construction. The first step is to find out the equation of the first auxiliary line

that generates the

Drawing the straight line

The convergence analysis of time series generated by BABM will be done analytically and graphically by determining the its fixed point and testing its stability. By inspecting Equation (1.1) we see its general form

where

In general, the first values of the series

When a series converges asymptotically to a single fixed value

For the analytical determination of a fixed point

that for BABM is

whose solution for

and the existence of this fixed points is the starting point to use the map for square root extraction. For the stability of the fixed point of the map, we have to ensure that

which is the general condition for stability of fixed points of any onedimensional map [

and according to the stability criterion (1.10), that at the point fixed (1.9) is

and its fixed points

Another important tool to analyze the orbit of a map and its evolution in time is called return diagram or cobweb. A cobweb is built with all the values of

the tangent line at the fixed point of the map. The linear term has a greater weight ensuring that

what guarantees that the fixed point is stable. Other important information we can obtain from this figure is the stability of the fixed point, since the map function has a minimum exactly at fixed point, and thus the derivative of the map is zero at this point. According to the stability criterion this is necessary and sufficient condition for the fixed point to be stable.

The use of iterated maps to solve the fundamental mathematical problem of square root estimation by numerical approximations was revisited and some tools from nonlinear dynamics were used to predict their stable fixed points and test the behaviour of the corresponding time series over a large region of parameter

The main result of this paper is fulfilled once we have proposed and demonstrated an original geometric argument to the underlying geometry in the Babylonian square root method, the oldest known and one of the most efficient methods to solve this classical and current problem. The proposed argument is very simple and intuitive, and can be easily extended to other similar maps, and perhaps its basic idea could be useful for constructing new iterated maps, from the geometrical point of view.

This work was partially supported by the Brazilian agency Conselho Nacional de Desenvolvimento Cientfico e Tecnológico―CNPq and Universidade do Estado de Santa Catarina―UDESC.