Francisco I. Chicharro

Drawing Dynamical and Parameters Planes of Iterative Families and Methods

The complex dynamical analysis of the parametric fourth-order Kims iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).

Complex dynamics of derivative-free methods for nonlinear equations

The dynamical behavior of two iterative derivative-free schemes, Steffensen and M4 methods, is studied in case of quadratic and cubic polynomials. The parameter plane is analyzed for both procedures on quadratic polynomials. Different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or complex coefficients. The property of immersion of the basins of attraction in all cases is analyzed.

Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods

In this paper, a family of new fourth-order optimal iterative methods for solving nonlinear equations is proposed. The classical Kings family of fourth-order schemes is obtained as an special case. We also present results for describing the conjugacy classes and dynamics of some of the presented methods for complex polynomials of different degrees.

OOFDM Signal Transmission Using a Single Optical Broadband Source

In this letter, optical OFDM signal transmission is demonstrated by using a broadband optical source over a 10 km fiber link. OFDM signal transmission usually employs lasers to generate the optical carrier since the chromatic fiber dispersion prevents the use of broadband optical sources. However, an interferometric system is introduced to allow signal transmission at a given RF band when a broadband optical source is employed. Optical sources with different optical bandwidths have been employed to explore the limitations of the system. OFDM transmission at 2 Gb/s using both QPSK and 16QAM modulation is demonstrated over 10 km.

Dynamics and Fractal Dimension of Steffensen-Type Methods

In this paper, the dynamical behavior of different optimal iterative schemes for solving nonlinear equations with increasing order, is studied. The tendency of the complexity of the Julia set is analyzed and referred to the fractal dimension. In fact, this fractal dimension can be shown to be a powerful tool to compare iterative schemes that estimate the solution of a nonlinear equation. Based on the box-counting algorithm, several iterative derivative-free methods of different convergence orders are compared.

Broadband Optical Sources for Low-Cost WDM-MB-OFDM Networks

Multiband Orthogonal Frequency Division Multiplexing (MB-OFDM) signal transmission through a 10-km optical fiber link is experimentally demonstrated by using a broadband source. We propose the use of optical broadband sources in multiband OFDM signal transmitters as a low-cost and power fading free solution. The OFDM band is dynamically selected in the receiver by means of an Mach-Zehnder Interferometer (MZI). Therefore, MB-OFDM is demonstrated at 1.5 Gb/s per band in access networks, supporting up to 20 ONUs.

King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

A biparametric family of derivative-free optimal iterative methods of order four, for solving nonlinear equations, is presented. From the error equation of this class, different families of iterative schemes with memory can be designed increasing the order of convergence up to six. The real stability analysis of the biparametric family without memory is made on quadratic polynomials, finding areas in the parametric plane with good performance. Moreover, in order to study the real behavior of the parametric class with memory, we associate it with a discrete multidimensional dynamical system. By analyzing the fixed and critical points of its vectorial rational function, we can select those methods with best stability properties.

Derivative-free high-order methods applied to preliminary orbit determination ☆

Abstract From position and velocity coordinates for several given instants, it is possible to determine the orbital elements of the preliminary orbit, taking only into account mutual gravitational attraction forces between the Earth and the satellite. Nevertheless, it should be refined with later observations from ground stations, whose geographic coordinates are previously known. Different methods developed for this purpose need to find a solution of a nonlinear function. In some classical methods it is usual to employ fixed point or secant methods. The second iterative scheme is often used when it is not possible to obtain the derivative of the nonlinear function. Nowadays, there exist efficient numerical methods that are able to highly improve the results obtained by the classical schemes. We will focus our attention on the method of iteration of the true anomaly, in which the secant method is replaced by more efficient methods, such as the second-order Steffensen’s method, as well as other high-order derivative-free methods.

Software Assisted Energy Efficient and Flexible WDM-PON Access Networks

This paper presents the architecture of a green and flexible WDM-PON access network. We propose the segmentation of a broadband source into smaller frequency windows by means of a wave shaper filter. A central controller commands the configuration of physical network parameters such as the optical bandwidth delivered by the filter and the modulation format of the service to be transported in the network. Different optical bandwidths were defined for the experimental demonstration and two modulation formats were evaluated in our approach. Results found an effective bandwidth of 3 GHz for optical bandwidths below 1 nm. Broader optical pass bands reduce the bandwidth but improve the signal quality measured in the receiver due to a relative higher average optical power available in the photo detector.

Stability and applicability of iterative methods with memory

Based on the third-order Traub’s method, two iterative schemes with memory are introduced. The proper inclusion of accelerating parameters allows the introduction of memory. Therefore, the order of convergence of the iterative methods increases from 3 up to 3.73 without new functional evaluations. One of them includes derivatives and the other one is derivative-free. The stability of the methods with memory is analyzed and their basins of attraction are compared to check the differences between them. The methods are applied to solve two nonlinear problems in Chemistry, such as the fractional conversion of the nitrogen-hydrogen feed that gets converted to ammonia and the Colebrook–White equation.