Local convergence of the Gauss-Newton-Kurchatov method under generalized Lipschitz conditions

We investigate the local convergence of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems. This method is a combination of Gauss-Newton and Kurchatov methods and it is used for problems with the decomposition of the operator. The convergence analysis of the method is performed under the generalized Lipshitz conditions. The conditions of convergence, radius and the convergence order of the considered method are established. Given numerical examples confirm the theoretical results.

Perturbation series for Jacobi matrices and the quantum Rabi model

We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.

Solving Prandtl-Blasius Boundary Layer Equation Using Maple

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting. Similarly, the study resolves some boundary layer related problems and provide relevant Maple codes for these.

Solving Prandtl-Blasius boundary layer equation using Maple

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting.

Limit theorems for some classes of power series type distributions

Several classes of distributions of power series type with finite and infinite radii of convergence are considered. For such distributions local limit theorems are obtained as the parameter of distribution tends to the right end of the interval of convergence. For the case when the convergence radius equals to 1, we prove an integral limit theorem on the convergence of distributions of random variables (1 − x)ξxas x → 1− to the gamma-distribution (ξx is a random variable with corresponding distribution of the power series type). The proofs are based on the steepest descent method.

Filter banks from the Hadamard product of the Fibonacci sequence and a geometric sequence

This note investigates a kind of filter banks related to the Fibonacci sequence. The Fibonacci sequence cannot be directly considered as a low-pass filter since the convergence radius of its [Formula: see text]-transform does not contain the unit disc centered at the origin. Adopting the Hadamard product of the Fibonacci sequence and a geometric sequence, a low-pass filter is designed and its corresponding filter banks are constructed.

On the local convergence of the Modified Newton method

The aim of this paper is to investigate the local convergence of the Modified Newton method, i.e. the classical Newton method in which the first derivative is re-evaluated periodically after m steps. The convergence order is shown to be m + 1. A new algorithm is proposed for the estimation the convergence radius of the method. We propose also a threshold for the number of steps after which is recommended to re-evaluate the first derivative in the Modified Newton method.

Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results.

Local Convergence of a Family of Weighted-Newton Methods

This article considers the fourth-order family of weighted-Newton methods. It provides the range of initial guesses that ensure the convergence. The analysis is given for Banach space-valued mappings, and the hypotheses involve the derivative of order one. The convergence radius, error estimations, and results on uniqueness also depend on this derivative. The scope of application of the method is extended, since no derivatives of higher order are required as in previous works. Finally, we demonstrate the applicability of the proposed method in real-life problems and discuss a case where previous studies cannot be adopted.

Approximation of Cluster Integrals for Various Lattice-Gas Models

An approximation for cluster integrals of an arbitrary high order has been proposed for the well-known lattice-gas model with an arbitrary geometry and dimensions. The approximation is based on the recently obtained accurate relations for the convergence radius of the virial power series in the activity parameter for the pressure and density. As compared to the previous studies of the symmetric virial expansions for the gaseous and condensed states of a lattice gas, the proposed approximation substantially approaches the pressure values at the saturation and boiling points. For the Lee–Yang lattice-gas model, the approximation considerably improves the convergence to the known exact solution.