A family of Chebyshev-Halley type methods in Banach spaces

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes

Relaxing convergence conditions for an inverse-free Jarratt type approximation

We consider an inverse-free Jarratt-type approximation of order four in a Banach space (Argyros et al., 1996). We establish a convergence theorem by using recurrence relations. The purpose of this note is to relax convergence conditions and give an example where our convergence theorem can be applied but not the other ones

Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series

AbstractThe close relationship between discrete Sturm–Liouville problems belonging to the so-called limit-circle case, the indeterminate Hamburger moment problem and the search of self-adjoint extensions of the associated semi-infinite Jacobi matrix is well known. In this paper, all these important topics are also related with associated sampling expansions involving analytic Lagrange-type interpolation series

Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors. © 2014 Elsevier Inc. All rights reserved

On the correct choice of equivalent circuit for fitting bulk impedance data of ionic/electronic conductors

Bulk conductivity data of ionically and electronically conducting solid electrolytes and electronic ceramics invariably show a frequency dependence that cannot be modelled by a single-valued resistor. To model this, common practice is to add a constant phase element (CPE) in parallel with the bulk resistance. To fit experimental data on a wide variety of materials, however, it is also essential to include the limiting, high frequency permittivity of the material in the equivalent circuit. Failure to do so can lead to incorrect values for the sample resistance and CPE parameters and to an inappropriate circuit for materials that are electrically heterogeneous

Majorizing sequences for Newton’s method from initial value problems

AbstractThe most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems

An extension of Gander’s result for quadratic equations

AbstractIn the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with R-order of convergence at least four for quadratic equations

A construction procedure of iterative methods with cubical convergence II: Another convergence approach

We extend the analysis of convergence of the iterations considered in Ezquerro et al. [Appl. Math. Comput. 85 (1997) 181] for solving nonlinear operator equations in Banach spaces. We establish a different Kantorovich-type convergence theorem for this family and give some error estimates in terms of a real parameter [-5, 1). © 1998 Elsevier Science Inc. All rights reserved

A note on a modification of Moser's method

We use a recurrence technique to obtain semilocal convergence results for Ulm's iterative method to approximate a solution of a nonlinear equation F (x) = 0fenced((x n + 1 = x n - B n F (x n),, n 0,; B n + 1 = 2 B n - B n F (x n + 1) B n,, n 0 .))This method does not contain inverse operators in its expression and we prove it converges with the Newton rate. We also use this method to approximate a solution of integral equations of Fredholm-type. © 2007 Elsevier Inc. All rights reserved

Dosimetry of large electron beam fields and treatment protocol for mycosis fungoides using the total skin irradiation technique

Introducción: La terapia total de piel con haz de electrones (TSEBT) es una técnica usada en el tratamiento de enfermedades superficiales de piel tales como linfomas cutáneos. Aunque su efectividad ha sido demostrada a través de varios estudios clínicos en estadios avanzados de la micosis fungoide, no es una técnica disponible en muchos centros de cáncer en México debido a que se requieren campos grandes de electrones para cubrir un cuerpo entero. Objetivo: Para obtener campos grandes de electrones para TSEBT, las distancias convencionales de tratamiento no son suficientes. Esto trae la necesidad de una caracterización y calibración especial del haz de electrones a distancias de 500 cm o más y la necesidad de crear un protocolo de tratamiento para TSEBT en nuestro país. Materiales y métodos: Para la puesta en marcha de esta técnica fue seleccionado el acelerador lineal ELEKTA SYNERGY, con una energía de 6 MeV a una distancia fuente-superficie (SSD) de 500 cm, en modo de alta tasa de dosis (1,000 UM = 100 Gy), usando un cono aplicador de 40 × 40 cm2 y el ángulo de gantry de 90◦, obteniendo un campo grande de electrones de 200 × 100 cm2 (área útil). La calibración de rutina fue realizada a una profundidad de referencia (Zref) de 1.4 cm a una SSD de 100 cm y un cono aplicador de 40 × 40 cm2 usando una cámara de ionización de placas plano-paralelas y un electrómetro (Scandotronix Wellhofer modelo PPC05 FOCUS) y un maniquí de agua. El mismo procedimiento fue realizado para determinar la tasa de dosis absoluta en condiciones de tratamiento (500 cm). Para la caracterización del haz de electrones en términos de porcentaje de dosis a profundidad (PDD) y perfiles de dosis, se utilizó Película Radiocrómica Gafchromic EBT2 (PRC), después de ser calibrada para electrones en un maniquí de agua sólida (Scandotronix Wellhofer) a un Zref = 1.4 cm de profundidad a una Distancia Fuente Isocentro SAD de 100 cm y de 500 cm, para obtener la ecuación de la dosis en respuesta de la densidad óptica. La PDD fue obtenida a 0, 1, 1.2, 1.4, 1.5, 2, 3, 4, 5, 6, 7, 8 y 9 cm de profundidad en el maniquí. La distribución espacial de dosis fue obtenida colocando muestras de PRC de 3 × 3 cm2 sobre la pantalla de acrílico que será situada enfrente del paciente (para así obtener la dosis absorbida máxima en piel). Por último, fue propuesto un protocolo de tratamiento. Resultados: La profundidad de dosis máxima (Zref) para electrones fue de 1.4 ± 0.05 cm, de acuerdo con la distribución de dosis espacial relativa y el porcentaje de dosis en profundidad para una SSD de 500 ± 0.5 cm, sobre un área de 200 × 100 cm2. Se graficaron los perfiles del haz tanto horizontal como vertical, mostrando una simetría horizontal de ±0.35%, aplanado horizontal de ±3.62%, simetría vertical de ±2.1% y aplanado vertical de ±14.2%. Conclusiones: Los resultados de los perfiles horizontal y vertical permiten evaluar la simetría y aplanado del haz de electrones. El PDD fue analizado hasta 9 ± 0.05 cm, estableciendo la profundidad de penetración de los electrones, asegurando un tratamiento superficial a la piel.Introduction: Total Skin Electron Beam Therapy (TSEBT) is a technique used in the treatment of superficial skin diseases, such as cutaneous lymphomas. Although it has demonstrated its effectiveness through several clinical studies of the advanced stages of mycosis fungoides, it is not available in most cancer centers in Mexico, because it requires large electron fields in order to cover the entire body. Objective: In order to obtain large electron fields for TSEBT, conventional treatment distances are not sufficient. This has led to the need for an electron beam with special features and calibration at a distance of 500 cm or more, as well as the need to create a treatment protocol to develop the TSEBT programs in Mexico. Materials and methods: The ELEKTA SYNERGY Linear Accelerator was selected with a 6 MeV energy at a Source Skin Distance (SSD) of 500 cm, in high dose rate mode (1,000 MU=100 Gy), using a cone of 40×40 cm2 and the gantry angled to 90◦, obtaining a large electron field of 200×100 cm2 useful area. Routine calibration was performed at a Zref=1.40 cm and an SSD of 100 cm with a 40×40 cm2 cone using a plane-parallel ionization chamber and electrometer (Scandotronix Wellhofer model PPC05 FOCUS) and a water phantom. The same procedure for the absolute dose rate determination was also performed in treatment conditions (500 cm). For the beam characterization in terms of Percentage Depth Dose (PDD) and beam profiles, Radiochromic Gafchromic® EBT2 film (RCF) was used, after being calibrated for electrons in a solid water phantom (Scanditronix Wellhöfer) at a 1.4 cm depth and Source Axis Distance (SAD) of 100 cm and 500 cm, in order to acquire the equation relating the dose response with optical density. PDD was obtained at 0, 1, 1.2, 1.4, 1.5, 2, 3, 4, 5, 6, 7, 8 and 9 cm depths in the phantom. Spatial dose distribution was obtained by placing 3×3 cm2 samples of RCF on an acrylic screen situated in front of the patient (for the purpose of obtaining maximum absorbed dose on the skin). Lastly, a treatment protocol was proposed. Results: An effective maximum dose depth (Zref) for electrons of 1.4±0.05 cm was obtained according to the relative spatial dose distribution and the percentage depth dose for a SSD of 500±0.5 cm, over an area of 200×100 cm2. Horizontal and vertical beam profiles were plotted, showing a horizontal symmetry of ±0.35%, horizontal flatness of ±3.62%, vertical symmetry of ±2.1%, and vertical flatness of ±14.2%. Conclusions: The resulting horizontal and vertical profiles enabled the electron beam symmetry and flatness to be assessed. PDD was analyzed up to a 9±0.05 cm, establishing the electron depth penetration to ensure treatment of the skin surface