Improving the Computational Efficiency of a Variant of Steffensen’s Method for Nonlinear Equations

Khdhr, Fuad W.; Saeed, Rostam K.; Soleymani, Fazlollah

doi:10.3390/math7030306

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…This procedure should be done for all loops in the network separately (in our case for I, II, III, IV and V). However, in order to simplify calculations, derivative-free methods can be used [28,29].…”

Section: The Multi-point Iterative Hardy Cross Methodsmentioning

confidence: 99%

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

Brkić

Praks

2019

Applied Sciences

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P.P.) Dejan Brkić: https://orcid.org/0000-0002-2502-0601, Pavel Praks https://orcid.org/0000-0002-3913-7800 #Dejan Brkić was on short visit funded by IT4Innovations, VŠB -Technical Abstract: Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method which considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton-Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modelling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton-Raphson iterative method.

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Section: The Multi-point Iterative Hardy Cross Methodsmentioning

confidence: 99%

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

Brkić

Praks

2019

Applied Sciences

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“…An example of an approximation of the Colebrook equation based on internal cycles is given by Serghides [51]. Serghides' methods belong to Steffensen types of methods, which do not require derivatives (other Steffensen types of methods can be used in addition [52,53]).…”

Section: Approximations By Multi-point Methods With Internal Cyclesmentioning

confidence: 99%

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

Brkić

Praks

2019

Fluids

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Even a relatively simple equation such as Colebrook’s offers a lot of possibilities to students to increase their computational skills. The Colebrook’s equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton–Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Padé polynomials, special functions such as Lambert W, artificial intelligence such as neural networks, etc. In addition, to develop explicit approximations or to improve their accuracy, regression analysis, genetic algorithms, and curve fitting techniques can be used too. In this learning numerical exercise, a few numerical examples will be shown along with the explanation of the estimated pedagogical impact for university students. Students can see what the difference is between the classical vs. floating-point algebra used in computers.

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“…There are only a few papers that have been concerned with the one-step iterative schemes with memory [9][10][11]. In this paper, we develop several memory-dependent one-step iterative methods for a high-performance solution of nonlinear equations.…”

Section: Introductionmentioning

confidence: 99%

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Liu,

Chang,

Kuo

2024

Symmetry

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In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

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Improving the Computational Efficiency of a Variant of Steffensen’s Method for Nonlinear Equations

Cited by 6 publications

References 20 publications

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

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