Iterative methods with memory for solving nonlinear systems have been designed. For approximating the accelerating parameters the Kurchatov’s divided difference is used as an approximation of the ...derivative of second order. The convergence of the proposed schemes is analyzed by means of Taylor expansions. Numerical examples are shown to compare the performance of the proposed schemes with other known ones, confirming the theoretical results.
Derivative-free iterative methods are useful to approximate the numerical solutions when the given function lacks explicit derivative information or when the derivatives are too expensive to compute. ...Exploring the convergence properties of such methods is crucial in their development. The convergence behavior of such approaches and determining their practical applicability require conducting local as well as semi-local convergence analysis. In this study, we explore the convergence properties of a sixth-order derivative-free method. Previous local convergence studies assumed the existence of derivatives of high order even when the method itself was not utilizing any derivatives. These assumptions imposed limitations on its applicability. In this paper, we extend the local analysis by providing estimates for the error bounds of the method. Consequently, its applicability expands across a broader range of problems. Moreover, the more important and challenging semi-local convergence not investigated in earlier studies is also developed. Additionally, we survey recent advancements in this field. The outcomes presented in this paper can be proved valuable to practitioners and researchers engaged in the development and analysis of derivative-free numerical algorithms. Numerical tests illuminate and validate further the theoretical results.
Clinical researchers use prognostic modeling techniques to identify a-prior patient health status and characterize progression patterns. It is highly desirable to predict future health condition ...especially to implement preventive and intervention strategies in pre-diabetic individuals. Hidden Markov Model (HMM) and its variants are a class of models that provide predictions concerning future condition by exploiting sequences of clinical measurements obtained from a longitudinal sample of patients. Despite the advantages of using these models for prognostic modeling, it still face barriers and significant challenges, to effectively learn dynamic interactions, when using irregularly sampled longitudinal Electronic Medical Records (EMRs) data. Newton's divide difference method (NDDM) is a classical approach for handling irregular data in terms of divided difference. However, as it is polynomial approximation technique, it suffers with Runge Phenomenon. The problem can be even more severe when the interval is a bit extended. Therefore, to tackle this problem, we proposed a novel approximation method based on NDDM as a component with HMM in order to estimate the 8 years risk of developing Type 2 Diabetes Mellitus (T2DM) in a particular individual. The proposed method is evaluated on real world clinical data obtained from CPCSSN. The results demonstrated that our proposed technique has the ability to exploit the available irregularly sampled EMRs data for effective approximation and improved prediction accuracy.
A method without memory as well as a method with memory are developed free of derivatives for solving equations in Banach spaces. The convergence order of these methods is established in the scalar ...case using Taylor expansions and hypotheses on higher-order derivatives which do not appear in these methods. But this way, their applicability is limited. That is why, in this paper, their local and semi-local convergence analyses (which have not been given previously) are provided using only the divided differences of order one, which actually appears in these methods. Moreover, we provide computable error distances and uniqueness of the solution results, which have not been given before. Since our technique is very general, it can be used to extend the applicability of other methods using linear operators with inverses along the same lines. Numerical experiments are also provided in this article to illustrate the theoretical results.
In this article we extend the notions of G-metric and b-metric and define a new metric called Gb-metric with coefficient b≥1. A fixed point theorem is proved in this metric space. We obtain parallel ...results of several existing fixed point theorems such as that of Banach, Geraghty and Boyd–Wong in Gb-metric space using our theorem. As an application of our fixed point theorem we provide a fixed point iteration to solve a class of nonlinear matrix equations of the form Xs+A∗G(X)A=Q, where s≥1, A is an n×n matrix, G is a continuous function from the set of all Hermitian positive definite matrices to the set of all Hermitian positive semi-definite matrices and Q is an n×n Hermitian positive definite matrix. It is noted that the error in estimated solution we get by following our method is lesser than the error we get with Ćirić’s fixed point iteration.
On the way to autonomous driving more and more advanced driver assistance systems (ADASs) are developed to increase safety and comfort. These functions require additional quantified information about ...the environment, for example information about the current road condition, which a human driver would perceive intuitively. In this paper, a road condition estimation, described by a road friction coefficient, is developed using a second order divided differences filter in combination with estimates of the time varying parameters, namely the dynamic wheel radius and the tire stiffness. Using real measurement data of commercial vehicles, the estimation algorithm is compared to a static and less complex longitudinal friction coefficient estimator. The comparison shows good results for both estimators with differences at low excitation and during cornering maneuvers.
Power Systems Dynamic State Estimation (PSDSE) using hybrid measurements from Phasor Measurement Units (PMUs) and Remote Terminal Units (RTUs) in the presence of partially characterized and ...non-stationary Gaussian as well as non-Gaussian measurement noise has been addressed in this work. For Gaussian noise, Adaptive Divided Difference Filter (ADDF) with anomaly detection has been tested in comparison to the adaptive versions of Unscented Kalman Filter (AUKF) and Cubature Kalman Filter (ACKF) for IEEE 30 bus test system and Northern Regional Power Grid (NRPG) 246 bus Indian practical system. In addition, a novel technique termed Adaptive Student's-t Divided Difference Filter (AST-DDF), has been proposed which deals with the presence of outliers in real-time data in the context of synchronous and asynchronous reporting rate of the RTUs. The non-Gaussian measurement noise is assumed to follow student's-t distribution, which intrinsically accommodates outliers. In view of partially characterized time-varying noise statistics, AST-DDF has been validated in IEEE 30 bus and IEEE 118 bus benchmark systems in the presence of bad data and transients caused during the faults in power systems. Results reveal that the adaptive versions of DDF for both the Gaussian and non-Gaussian noise substantiate satisfactory estimation accuracy at a reasonable computational burden and hence recommended for the future Energy Management Systems (EMS).
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