A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve

[EN] In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity (m >= 1) is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev-Halley are the special members of our schemes for m=1. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity.Kanwar, V.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Rajput, M.; Behl, R. (2023). A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve. Algorithms. 16(3). https://doi.org/10.3390/a1603015616

A dynamic convergence control scheme for the solution of the radial equilibrium equation in through-flow analyses

One of the most frequently encountered numerical problems in scientific analyses is the solution of non-linear equations. Often the analysis of complex phenomena falls beyond the range of applicability of the numerical methods available in the public domain, and demands the design of dedicated algorithms that will approximate, to a specified precision, the mathematical solution of specific problems. These algorithms can be developed from scratch or through the amalgamation of existing techniques. The accurate solution of the full radial equilibrium equation (REE) in streamline curvature (SLC) through-flow analyses presents such a case. This article discusses the development, validation, and application of an 'intelligent' dynamic convergence control (DCC) algorithm for the fast, accurate, and robust numerical solution of the non-linear equations of motion for two-dimensional flow fields. The algorithm was developed to eliminate the large extent of user intervention, usually required by standard numerical methods. The DCC algorithm was integrated into a turbomachinery design and performance simulation software tool and was tested rigorously, particularly at compressor operating regimes traditionally exhibiting convergence difficulties (i.e. far off-design conditions). Typical error histories and comparisons of simulated results against experimental are presented in this article for a particular case study. For all case studies examined, it was found that the algorithm could successfully 'guide' the solution down to the specified error tolerance, at the expense of a slightly slower iteration process (compared to a conventional Newton-Raphson scheme). This hybrid DCC algorithm can also find use in many other engineering and scientific applications that require the robust solution of mathematical problems by numerical instead of analytical means

Spectrum optimization in multi-user multi-carrier systems with iterative convex and nonconvex approximation methods

Several practical multi-user multi-carrier communication systems are characterized by a multi-carrier interference channel system model where the interference is treated as noise. For these systems, spectrum optimization is a promising means to mitigate interference. This however corresponds to a challenging nonconvex optimization problem. Existing iterative convex approximation (ICA) methods consist in solving a series of improving convex approximations and are typically implemented in a per-user iterative approach. However they do not take this typical iterative implementation into account in their design. This paper proposes a novel class of iterative approximation methods that focuses explicitly on the per-user iterative implementation, which allows to relax the problem significantly, dropping joint convexity and even convexity requirements for the approximations. A systematic design framework is proposed to construct instances of this novel class, where several new iterative approximation methods are developed with improved per-user convex and nonconvex approximations that are both tighter and simpler to solve (in closed-form). As a result, these novel methods display a much faster convergence speed and require a significantly lower computational cost. Furthermore, a majority of the proposed methods can tackle the issue of getting stuck in bad locally optimal solutions, and hence improve solution quality compared to existing ICA methods.Comment: 33 pages, 7 figures. This work has been submitted for possible publicatio