Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric.

Disturbance estimator-based switching function for discrete-time sliding mode control systems with control saturation

In this paper, a new disturbance estimator-based switching function (SF) is presented dedicated to discrete-time sliding mode control (DSMC) systems with disturbances and control saturation (CS). This SF is featured by an ( n+1)th-order disturbance estimator, which utilizes n+1 past disturbance terms to attain accurate disturbance rejection. Moreover, an auxiliary state is integrated into the SF to address the CS problem. One uniqueness of the developed method is that it is capable of achieving the ideal quasi-sliding mode and produces a zero-convergence error of the SF under the influence of disturbances and CS, which is rarely realized in DSMC methods. In addition, system states dynamics is rigorously analysed adopting the Lyapunov technique. The feasibility and virtue of the designed results are verified by a numerical example.

A Family of Fifth and Sixth Convergence Order Methods for Nonlinear Models

We study the local convergence of a family of fifth and sixth convergence order derivative free methods for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the seventh derivative to show convergence. However, we only use the first divided difference of order one as well as the first derivative in our analysis. We also provide computable radius of convergence, error estimates, and uniqueness of the solution results not given in earlier studies. Hence, we expand the applicability of these methods. The dynamical analysis of the discussed family is also presented. Numerical experiments complete this article.

The Equivalence between Successive Approximations and Matricial Load Flow Formulations

This paper shows the equivalence of the matricial form of the classical backward/forward load flow formulation for distribution networks with the recently developed successive approximations (SA) load flow approach. Both formulations allow solving the load flow problem in meshed and radial distribution grids even if these are operated with alternating current (AC) or direct current (DC) technologies. Both load flow methods are completely described in this research to make a fair comparison between them and demonstrate their equivalence. Numerical comparisons in the 33- and 69-bus test feeder with radial topology show that both methods have the same number of iterations to find the solution with a convergence error defined as 1×10−10.

Application of discretization error estimators in stepped column buckling problems

In order to reduce the discretization error, in this paper, Richardson’s Extrapolation and Convergence Error Estimator were used to investigating the buckling problem convergence. The main objective was to verify the convergence order of the stepped column problem and to define a consistent moment of inertia at the point of variation of the cross-section. The variable of interest was the critical buckling load obtained by the Finite Difference Method. The convergent solution obtained errors less than 10-8, and this work showed that the best solution is not defined by excessive mesh refinement, but by the solution convergence analysis.

A Joint Detection and Decoding Scheme for PC-SCMA System Based on Pruning Iteration

Polar coding and sparse code multiple access (SCMA) are key technologies for 5G mobile communication, the joint design of them has a great significance to improve the overall performance of the transmitter-receiver symmetric wireless communication system. In this paper, we firstly propose a pruning iterative joint detection and decoding algorithm (PI-JDD) based on the confidence stability of resource nodes. Branches to be updated are dynamically pruned to avoid redundant iterative, which is able to reduce 24~50% complexity while achieving the approximate error performance of traditional serial joint iterative detection and decoding algorithm S-JIDD. Then, to further reduce the bit error rate (BER) of the receiver, a cyclic redundancy check (CRC) termination mechanism is added at the end of each joint iteration to avoid the convergence error caused by decoding deviation. Simulation results show that the addition of an early stopping criterion can achieve a remarkable performance gain compared with the S-JIDD algorithm. More importantly, the combined algorithm of the two proposed schemes can reduce the computational complexity while achieving better error performance.

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

In present paper, we deal with a backward diffusion problem for a time-fractional diffusion problem with a nonlinear source in a strip domain. We all know this nonlinear problem is severely ill-posed, i.e., the solution does not depend continuously on the measurable data. Therefore, we use the Fourier truncation regularization method to solve this problem. Under an a priori hypothesis and an a priori regularization parameter selection rule, we obtain the convergence error estimates between the regular solution and the exact solution at 0 ≤ x < 1 .

Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, the hypotheses of the present study are based on the first Fréchet-derivative only, thereby the application of methods is expanded. New analysis also provides the radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of derivatives of higher order. Moreover, the order of convergence for the methods is verified by using computational order of convergence or approximate computational order of convergence without using higher order derivatives. Numerical examples are provided to verify the theoretical results and to show the good convergence behavior.