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In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.

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 [

We propose and study in this work two new methods for numerical root approximations, both of which based on iterated maps. In the following sections we present a detailed study of each map, their fixed points and stability, the occurrence of bifurcations and chaotic behavior.

Some common tools of nonlinear dynamics [

In Section 2 we present a new map proposed by one of us (C. C. Dias), named as First Dias Map (FDM), showing the existence of a fixed point for roots in the range

In Section 3, we generalize the FDM by adding a new parameter for studying the stability of its fixed point by defining a new class of maps called Weighted Average Map (WAM). For this class of maps, we investigate the dependence of the fixed point corresponding to the nth root of

Finally, in Section 4, we measure and compare the Mean Convergence Time (MCT) for all the studied maps, for

The map which we will study now was created by Charles C. Dias to extract real roots of numbers numbers, by solving the equation

Comparing this with the Newton-Raphson Method (NRM) equation and Babylonian Method (BABM) [

The base function that appears in the iterated map defined by Equation (1.1) can be derived dividing

and adding

To construct geometrically the FDM time series, the first step is to find the auxiliary equations of the lines

that for

Knowing that their linear coefficients are all

and taking the arithmetic mean between the auxiliary points

shows a numerical development of the FDM series for the parameters

Solving

map function

and solving the last equation we have the range of parameter

The FDM time series have different dynamics depending on the parameters

To measure the rate of divergent orbits, i.e., the sensitive dependence on initial conditions, we can use is characteristic Lyapunov exponent

0 | 3.00000000 | 0.66666667 | 4 | 1.41421378 | 1.41421334 | |

1 | 1.83333333 | 1.09090909 | 5 | 1.41421356 | 1.41421356 | |

2 | 1.46212121 | 1.36787565 | 6 | 1.41421356 | 1.41421356 | |

3 | 1.41499842 | 1.41342913 | 7 | 1.41421356 | 1.41421356 | |

0 | 3.00000000 | 0.22222222 | 20 | 1.25992076 | 1.25992163 | |

1 | 1.61111111 | 0.77051130 | 25 | 1.25992106 | 1.25992103 | |

2 | 1.19081120 | 1.41040608 | 27 | 1.25992105 | 1.25992105 | |

3 | 1.30060864 | 1.18232460 | 28 | 1.25992105 | 1.25992105 |

goes to zero signing the period bifurcations. The yellow to red regions indicate the a positive Lyapunov exponents, the signature of chaos.

The FDM bifurcation diagram, discarded a transient of 10^{3} iterations and plotted the next 500 values of

We also study numerically the FDM return diagrams for different values of the parameter

Instead of adding

Adding

and after collecting

and solving its fixed point equation

Applying the stability criterion [

guarantee fixed point stability, and

A special subclass of WAM is FDM, when

at

For NRM, the derivative of the mapping function at the fixed point is null, satisfying the stability criterion and resulting in the most efficient rate of convergence of the time series near the fixed point. When the starting point

From the definition of the Lyapunov characteristic exponent for a unidimensional map we conclude that the derivative of the mapping function at the fixed point

This section reports the numerical results for the mean convergence time (MCT) for NRM, FDM and WAM, based on the average number of iterates to converge within different precisions^{3} points in the interval

In

Both NRM and FDM belong to the same WAM family, as discussed in Section 3, and the stability of the fixed point

optimal value of the parameter

We can see in

Summarizing the key information about NRM, FDM and WAM, with the numerical results for the MCT within double precision for these maps, for the parameters

The Lyapunov characteristic exponent for a unidimensional map

can be approximate by

For WAM, it is easy to show that

Using the original Lyapunov’s idea, the characteristic exponent

Applying the natural logarithm to both sides of the above equation we have

iterations needed to reduces the error in the second orbit to

valid for any fixed point of a unidimensional map, where the approximated

Applying this model to our more general map (WAM), we have

of the parameters

Map | Estability | MCT (n = 3) |
---|---|---|

NRM | ≈4.6 | |

FDM | ≈52 | |

WAM | ≈26, for |

In the study of iterated maps to extract the real root of real numbers we have applied some common tools from nonlinear dynamics that allowed us to predict the fixed point of the studied maps associated with the nth root of

We conclude, through the geometric argument used to recover the original analytical form of FDM, that both NRM and FDM can be reduced to averages between two terms, one linear and other nonlinear. From this observation, we generalize the original FDM idea to a new family of maps on which we add a new parameter

The mean convergence time (MCT) numerical results indicate that NRM is the most efficient subclass of the more general weighted average map (WAM) proposed in this work, as pointed out in

The main results of this work are obtained for

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.