A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection

The researchers aimed to study the nonlinear fractional order model of malaria infection based on the Caputo-Fabrizio fractional derivative. The homotopy analysis transform method (HATM) is applied based on the floating-point arithmetic (FPA) and the discrete stochastic arithmetic (DSA). In the FPA, to show the accuracy of the method we use the absolute error which depends on the exact solution and a positive value ε. Because in real life problems we do not have the exact solution and the optimal value of ε, we need to introduce a new condition and arithmetic to show the efficiency of the method. Thus the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. The CESTAC method is based on the DSA. Also, a new termination criterion is used which is based on two successive approximations. Using the CESTAC method we can find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorem of the CESTAC method is proved to show that the number of common significant digits (NCSDs) between two successive approximations are almost equal to the NCSDs of the exact and approximate solutions. Plotting several graphs, the regions of convergence are demonstrated for different number of iterations k = 5, 10. The numerical results based on the simulated data show the advantages of the DSA in comparison with the FPA.

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The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method

Mathematics ◽

10.3390/math9030260 ◽

2021 ◽

Vol 9 (3) ◽

pp. 260

Author(s):

Samad Noeiaghdam ◽

Denis Sidorov ◽

Abdul-Majid Wazwaz ◽

Nikolai Sidorov ◽

Valery Sizikov

Keyword(s):

Decomposition Method ◽

Adomian Decomposition Method ◽

Absolute Error ◽

Floating Point ◽

Cestac Method ◽

Stochastic Arithmetic ◽

Floating Point Arithmetic ◽

Adomian Decomposition ◽

Discontinuous Kernels ◽

Point Arithmetic

The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.

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A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library

Mathematics ◽

10.3390/math9121321 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1321

Author(s):

Samad Noeiaghdam ◽

Sanda Micula ◽

Juan J. Nieto

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Fractional Order ◽

Point Theorem ◽

Successive Approximations ◽

Banach Fixed Point Theorem ◽

Order Model ◽

Existence Of A Solution ◽

Cestac Method ◽

Fractional Order Model

In this paper, a nonlinear fractional order model of COVID-19 is approximated. For this aim, at first we apply the Caputo–Fabrizio fractional derivative to model the usual form of the phenomenon. In order to show the existence of a solution, the Banach fixed point theorem and the Picard–Lindelof approach are used. Additionally, the stability analysis is discussed using the fixed point theorem. The model is approximated based on Indian data and using the homotopy analysis transform method (HATM), which is among the most famous, flexible and applicable semi-analytical methods. After that, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, which are based on discrete stochastic arithmetic (DSA), are applied to validate the numerical results of the HATM. Additionally, the stopping condition in the numerical algorithm is based on two successive approximations and the main theorem of the CESTAC method can aid us analytically to apply the new terminations criterion instead of the usual absolute error that we use in the floating-point arithmetic (FPA). Finding the optimal approximations and the optimal iteration of the HATM to solve the nonlinear fractional order model of COVID-19 are the main novelties of this study.

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Specification for binary floating point arithmetic for microprocessor systems

10.3403/00218955u ◽

2015 ◽

Keyword(s):

Floating Point ◽

Floating Point Arithmetic ◽

Point Arithmetic

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Numerical algorithms for high-performance computational science

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2019.0066 ◽

2020 ◽

Vol 378 (2166) ◽

pp. 20190066 ◽

Cited By ~ 2

Author(s):

Jack Dongarra ◽

Laura Grigori ◽

Nicholas J. Higham

Keyword(s):

High Performance ◽

Numerical Algorithms ◽

Computational Science ◽

Floating Point ◽

Important Criterion ◽

Data Movement ◽

Floating Point Arithmetic ◽

High Performance Computers ◽

Point Arithmetic ◽

Speed And Accuracy

A number of features of today’s high-performance computers make it challenging to exploit these machines fully for computational science. These include increasing core counts but stagnant clock frequencies; the high cost of data movement; use of accelerators (GPUs, FPGAs, coprocessors), making architectures increasingly heterogeneous; and multi- ple precisions of floating-point arithmetic, including half-precision. Moreover, as well as maximizing speed and accuracy, minimizing energy consumption is an important criterion. New generations of algorithms are needed to tackle these challenges. We discuss some approaches that we can take to develop numerical algorithms for high-performance computational science, with a view to exploiting the next generation of supercomputers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

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