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Journal articles on the topic 'Gauss Jacobi Method'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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1

S., Sathya, and Ramesh T. "Comparison of Gauss Jacobi Method and Gauss Seidel Method using Scilab." International Journal of Trend in Scientific Research and Development 3, no. 6 (2019): 1051–53. https://doi.org/10.5281/zenodo.3589302.

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Numerical Method is the important aspects in solving real world problems that are related to Mathematics, science, medicine, business etc. In this paper, We comparing the two methods by using the scilab 6.0.2 software coding to solve the iteration problem. which are Gauss Jacobi and Gauss Seidel methods of linear equations. S. Sathya | T. Ramesh "Comparison of Gauss Jacobi Method and Gauss Seidel Method using Scilab" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29316.pdf

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2

Sukarna, S., Muhammad Abdy, and R. Rahmat. "Perbandingan Metode Iterasi Jacobi dan Metode Iterasi Gauss-Seidel dalam Menyelesaikan Sistem Persamaan Linear Fuzzy." Journal of Mathematics, Computations, and Statistics 2, no. 1 (2020): 1. http://dx.doi.org/10.35580/jmathcos.v2i1.12447.

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Penelitian ini mengkaji tentang menyelesaian Sistem Persamaan Linear Fuzzy dengan Membanding kan Metode Iterasi Jacobi dan Metode Iterasi Gauss-Seidel. Metode iterasi Jacobi merupakan salah satu metode tak langsung, yang bermula dari suatu hampiran Metode iterasi Jacobi ini digunakan untuk menyelesaikan persamaan linier yang proporsi koefisien nol nya besar. Iterasi dapat diartikan sebagai suatu proses atau metode yang digunakan secara berulang-ulang (pengulangan) dalam menyelesaikan suatu permasalahan matematika ditulis dalam bentuk . Pada metode iterasi Gauss-Seidel, nilai-nilai yang paling akhir dihitung digunakan di dalam semua perhitungan. Jelasnya, di dalam iterasi Jacobi, menghitung dalam bentuk . Setelah mendapatkan Hasil iterasi kedua Metode tersebut maka langkah selanjutnya membandingkan kedua metode tersebut dengan melihat jumlah iterasinya dan nilai Galatnya manakah yang lebih baik dalam menyelesaikan Sistem Persamaan Linear Fuzzy.Kata kunci: Sistem Persamaan Linear Fuzzy, Metode Itersi Jacobi, Metode Iterasi Gauss-Seidel. This study examines the completion of the Linear Fuzzy Equation System by Comparing the Jacobi Iteration Method and the Gauss-Seidel Iteration Method. The Jacobi iteration method is one of the indirect methods, which stems from an almost a method of this Jacobi iteration method used to solve linear equations whose proportion of large zero coefficients. Iteration can be interpreted as a process or method used repeatedly (repetition) in solving a mathematical problem written in the form . In the Gauss-Seidel iteration method, the most recently calculated values are used in all calculations. Obviously, inside Jacobi iteration, counting in form After obtaining the result of second iteration of the Method then the next step compare both methods by seeing the number of iteration and the Error value which is better in solving Linear Fuzzy Equation System.Keywords: Linear Fuzzy Equation System, Jacobi Itersi Method, Gauss-Seidel Iteration Method.

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3

Zhou, Bo Jian, Xu Hong Li, and Jie He. "A Comparative Study of Two Alternative Methods for the Path-Based Logit Stochastic User Equilibrium Problem." Advanced Materials Research 756-759 (September 2013): 1433–36. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.1433.

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in this paper, a computational study of two alternative methods for the path-based logit stochastic user equilibrium model is conducted. The two methods under investigation are the Jacobi gradient projection method and the Gauss-Seidel gradient projection method. We compare the two methods on the Sioux Falls network. Numerical results indicate that for the path-based logit SUE problem, Jacobi gradient projection method is more efficient than Gauss-Seidel gradient projection method.

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4

Bhrawy, A. H., M. A. Alghamdi, and D. Baleanu. "Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/513808.

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The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

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Akram, Muhammad, Ghulam Muhammad, Ali N. A. Koam, and Nawab Hussain. "Iterative Methods for Solving a System of Linear Equations in a Bipolar Fuzzy Environment." Mathematics 7, no. 8 (2019): 728. http://dx.doi.org/10.3390/math7080728.

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We develop the solution procedures to solve the bipolar fuzzy linear system of equations (BFLSEs) with some iterative methods namely Richardson method, extrapolated Richardson (ER) method, Jacobi method, Jacobi over-relaxation (JOR) method, Gauss–Seidel (GS) method, extrapolated Gauss-Seidel (EGS) method and successive over-relaxation (SOR) method. Moreover, we discuss the properties of convergence of these iterative methods. By showing the validity of these methods, an example having exact solution is described. The numerical computation shows that the SOR method with ω = 1 . 25 is more accurate as compared to the other iterative methods.

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Bhrawy, A. H., and W. M. Abd-Elhameed. "New Algorithm for the Numerical Solutions of Nonlinear Third-Order Differential Equations Using Jacobi-Gauss Collocation Method." Mathematical Problems in Engineering 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/837218.

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A new algorithm for solving the general nonlinear third-order differential equation is developed by means of a shifted Jacobi-Gauss collocation spectral method. The shifted Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithm, and some comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

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7

Ahmadi, Afshin, Felice Manganiello, Amin Khademi, and Melissa C. Smith. "A Parallel Jacobi-Embedded Gauss-Seidel Method." IEEE Transactions on Parallel and Distributed Systems 32, no. 6 (2021): 1452–64. http://dx.doi.org/10.1109/tpds.2021.3052091.

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8

Abdelkawy, M. A., and António M. Lopes. "Spectral Solutions for Fractional Black–Scholes Equations." Mathematical Problems in Engineering 2022 (July 20, 2022): 1–9. http://dx.doi.org/10.1155/2022/9365292.

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This paper presents a numerical method to solve accurately the fractional Black–Scholes model of pricing evolution. A fully spectral collocation technique for the two independent variables is derived. The shifted fractional Jacobi–Gauss–Radau and shifted fractional Jacobi–Gauss–Lobatto collocation techniques are utilized. Firstly, the independent variables are interpolated at the shifted fractional Jacobi nodes, and the solution of the model is approximated by means of a sequence of shifted fractional Jacobi orthogonal functions. Then, the residuals at the shifted fractional Jacobi quadrature locations are estimated. As a result, an algebraic system of equations is obtained that can be solved using any appropriate approach. The accuracy of the proposed method is demonstrated using two numerical examples. It is observed that the new technique is more accurate, efficient, and feasible than other approaches reported in the literature. Indeed, the results show the exponential convergence of the method, both for smooth and nonsmooth solutions.

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Menezes, Matheus da Silva, Paulo Henrique Lopes Silva, João Paulo Caraú de Oliveira, Raimundo Leandro Andrade Marques, and Ivan Mezzomo. "A Parallel Iterative hybrid Gauss–Jacobi–Seidel method." Journal of Computational and Applied Mathematics 468 (November 2025): 116629. https://doi.org/10.1016/j.cam.2025.116629.

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10

Boubaker, K. "A Confirmed Model to Polymer Core-Shell Structured Nanofibers Deposited via Coaxial Electrospinning." ISRN Polymer Science 2012 (October 14, 2012): 1–6. http://dx.doi.org/10.5402/2012/603108.

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A model to core-shell structured polymer nanofibers deposited via coaxial electrospinning is presented. Investigations are based on a modified Jacobi-Gauss collocation spectral method, proposed along with the Boubaker Polynomials Expansion Scheme (BPES), for providing solution to a nonlinear Lane-Emden-type equation. The spatial approximation has been based on shifted Jacobi polynomials with was n the polynomial degree. The Boubaker Polynomials Expansion Scheme (BPES) main features, concerning the embedded boundary conditions, have been outlined. The modified Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique, and a comparison is made with existing results. It has been revealed that both methods are easy to implement and yield very accurate results.

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Agboola, Sunday O., Semiu A. Ayinde, Olajide Ibikunle, and Abiodun D. Obaromi. "Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain." Dutse Journal of Pure and Applied Sciences 9, no. 1a (2023): 127–38. http://dx.doi.org/10.4314/dujopas.v9i1a.13.

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The Physical or Mathematical behaviour of this model may be represented by describing all the different states it may occupy and by indicating how it moves among these states. In this study, the stationary distribution of Markov chains was solved using iterative methods that begin with an initial estimate of the solution vector and then modified it in a way that brings it closer and closer to the real solution with each step or iteration. These methods also involved matrix operations like multiplication with one or more vectors, which preserves the transition matrices while speeding up the process. We computed the solutions using Jacobi iterative method and Gauss-Seidel iterative method in order to shed more light on the solutions of stationary distribution in Markov chain. This was done with the aid of several already-existing laws, theorems, and formulas of Markov chain and the application of normalization principle and matrix operations such as lower, upper, and diagonal matrices. The stationary distribution vector’s 𝜋𝑖,𝑖=1,2,…,4 are obtained for the illustrative example one as 𝜋(3) = (0.078125,0.109375,0.21875,0.59375) as well as the four eigenvalues of the matrix as 𝜆1=1.0, 𝜆2=−0.7718, 𝜆3,4=−0.1141±0.5576𝑖 using Jacobi iterative technique, and for illustrative example two using Gauss-Siedel method as 𝝅(𝟑) = (0.090909, 0.181818, 0.363636, 0.363636). The research shown that Gauss Siedel method converged faster than Jacobi method.

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12

Amin, A. Z., A. K. Amin, M. A. Abdelkawy, A. A. Alluhaybi, and I. Hashim. "Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral equation." PLOS ONE 18, no. 5 (2023): e0283746. http://dx.doi.org/10.1371/journal.pone.0283746.

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A numerical approach based on shifted Jacobi-Gauss collocation method for solving mixed Volterra-Fredholm integral equations is introduced. The novel technique with shifted Jacobi-Gauss nodes is applied to reduce the mixed Volterra-Fredholm integral equations to a system of algebraic equations that has an easy solved. The present algorithm is extended to solve the one and two-dimensional mixed Volterra-Fredholm integral equations. Convergence analysis for the present method is discussed and confirmed the exponential convergence of the spectral algorithm. Various numerical examples are approached to demonstrate the powerful and accuracy of the technique.

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Sun, Shi, Ziping Huang, Cheng Wang, and Liming Guo. "The Cascadic Multigrid Method of the Weak Galerkin Method for Second-Order Elliptic Equation." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/7912845.

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This paper is devoted to the analysis of the cascadic multigrid algorithm for solving the linear system arising from the weak Galerkin finite element method. The proposed cascadic multigrid method is optimal for conjugate gradient iteration and quasi-optimal for Jacobi, Gauss-Seidel, and Richardson iterations. Numerical results are also provided to validate our theoretical analysis.

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Li, Zonghai, Yujie Duan, and Junji Jia. "Deflection of charged massive particles by a four-dimensional charged Einstein–Gauss–Bonnet black hole." Classical and Quantum Gravity 39, no. 1 (2021): 015002. http://dx.doi.org/10.1088/1361-6382/ac38d0.

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Abstract Based on the Jacobi metric method, this paper studies the deflection of a charged massive particle by a novel four-dimensional charged Einstein–Gauss–Bonnet black hole. We focus on the weak field approximation and consider the deflection angle with finite distance effects. To this end, we use a geometric and topological method, which is to apply the Gauss–Bonnet theorem to the Jacobi space to calculate the deflection angle. We find that the deflection angle contains a pure gravitational contribution δ g, a pure electrostatic δ c and a gravitational–electrostatic coupling term δ gc. We find that the deflection angle increases (decreases) if the Gauss–Bonnet coupling constant α is negative (positive). Furthermore, the effects of the BH charge, the particle charge-to-mass ratio and the particle velocity on the deflection angle are analyzed.

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15

Abdelkawy, Mohamed A., Ahmed Z. M. Amin, António M. Lopes, Ishak Hashim, and Mohammed M. Babatin. "Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel." Fractal and Fractional 6, no. 1 (2021): 19. http://dx.doi.org/10.3390/fractalfract6010019.

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We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

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Peykrayegan, Narges, Mehdi Ghovatmand, Mohammad Hadi Noori Skandari, and Dumitru Baleanu. "An approximate approach for fractional singular delay integro-differential equations." AIMS Mathematics 7, no. 5 (2022): 9156–71. http://dx.doi.org/10.3934/math.2022507.

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<abstract><p>In this article, we present Jacobi-Gauss collocation method to numerically solve the fractional singular delay integro-differential equations, because such methods have better superiority, capability and applicability than other methods. We first apply a technique to replace the delay function in the considered equation and suggest an equivalent system. We then propose a Jacobi-Gauss collocation approach to discretize the obtained system and to achieve an algebraic system. Having solved the algebraic system, an approximate solution is gained for the original equation. Three numerical examples are solved to show the applicability of presented approximate approach. Obtaining the approximations of the solution and its fractional derivative simultaneously and an acceptable approximation by selecting a small number of collocation points are advantages of the suggested method.</p></abstract>

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17

Cai, Haotao. "Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels." Journal of Function Spaces 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/4751357.

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We develop a generalized Jacobi-Galerkin method for second kind Volterra integral equations with weakly singular kernels. In this method, we first introduce some known singular nonpolynomial functions in the approximation space of the conventional Jacobi-Galerkin method. Secondly, we use the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation so as to obtain high-order accuracy for the approximation. Then, we establish that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order. One numerical example is presented to demonstrate the effectiveness of the proposed method.

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Doha, E. H., A. H. Bhrawy, and R. M. Hafez. "A Jacobi Dual-Petrov-Galerkin Method for Solving Some Odd-Order Ordinary Differential Equations." Abstract and Applied Analysis 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/947230.

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A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for theJth order ODE involvesn-fold indefinite integrals forn=1,…,J. Extension of the JDPG for ODEs with polynomial coefficients is treated using the Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs.

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Hamani, Fatima, and Azedine Rahmoune. "Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Method." Tatra Mountains Mathematical Publications 80, no. 3 (2021): 35–52. http://dx.doi.org/10.2478/tmmp-2021-0030.

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Abstract In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L ∞ and weighted L 2 norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.

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SEBRO, AMRACH, HAILU MULETA, and SOLOMON GEBREGIORGIS. "Extrapolated refinement of generalized gauss- seidel scheme for solving system of linear equations." Berhan International Research Journal of Science and Humanities 4 (March 22, 2020): 12–23. http://dx.doi.org/10.61593/dbu.birjsh.01.01.55.

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In this paper, Extrapolated Refinement of Generalized Gauss-Seidel scheme for solving system of linear equations has been presented. In order to accelerate the rate of convergence of the scheme, the one-parameter family of splitting procedure has been introduced and the convergence of the method is well established. To validate the proposed method, three numerical examples were considered. Comparisons were made among Refinement of Generalized Jacobi, Generalized Gauss-Seidel, Refinement of Generalized Gauss-Seidel and Extrapolated Refinement of Generalized Gauss-Seidel schemes with respect to the number of iterations to converge, computational running time and storage capacity. The numerical results presented in tables show that the Extrapolated Refinement of Generalized Gauss-Seidel scheme is more efficient than the other three schemes considered for comparison.

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Abdulbaset S Alkrbash, Alhadi A Abosbaia, and Abdulhafid M Elfaghi. "Comparison of Basic Iterative Methods Used to Solve of Heat and Fluid Flow Problems." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 101, no. 1 (2023): 186–91. http://dx.doi.org/10.37934/arfmts.101.1.186191.

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In this research, a comparison of the convergence rate of different basic methods is made in order to solve the Poisson equation, which is similar to some of the resulting equations in computational fluid mechanics. The Jacobi method, the point Gauss-Seidel method, the successive over-relaxation method (SOR), the line Gauss method (TDMA), the ADI method and Strongly Implicit (SIP) method are among the iterative approaches investigated, which are then compared to find the most optimal method. The selection criteria included the number of iterations and the time needed to reach convergence. In both selection criteria, the SIP approach has been shown to be the most efficient.

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Audu, Khadeejah James, James Nkereuwem Essien, Abraham Baba Zhiri, and Aliyu Rasheed Taiwo. "A THIRD REFINEMENT OF JACOBI METHOD FOR SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS." FUDMA JOURNAL OF SCIENCES 7, no. 5 (2023): 234–39. http://dx.doi.org/10.33003/fjs-2023-0705-1955.

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Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.

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Baloch, Muhammad Shakeel Rind, Zubair Ahmed Kalhoro, Mir Sarfraz Khalil, and Prof Abdul Wasim Shaikh. "A New Improved Classical Iterative Algorithm for Solving System of Linear Equations." Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences 58, no. 4 (2022): 69–81. http://dx.doi.org/10.53560/ppasa(58-4)638.

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The fundamental problem of linear algebra is to solve the system of linear equations (SOLE’s). To solve SOLE’s, is one of the most crucial topics in iterative methods. The SOLE’s occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE’s. In this research, an improved iterative scheme namely, ‘’a new improved classical iterative algorithm (NICA)’’ has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE’s is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE’s does arise usually from ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

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Bela Amelia. "Sistem Persamaan Linear dengan Metode Gauss Seidel." Jurnal Pustaka Cendekia Pendidikan 2, no. 2 (2024): 132–36. https://doi.org/10.70292/jpcp.v2i2.22.

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A linear equation is an algebraic equation in which each term contains a constant or multiplication of a constant with a single variable. Systems of linear equations arise directly from real problems that require a solution process. Systems of linear equations can be solved by two methods. The first method is direct, which is usually called the exact method. These methods include inverse, elimination, substitution, LU decomposition, Cholesky decomposition, QR decomposition, Crout decomposition, and ST decomposition. The second method is usually known as the indirect method or iteration method, including the Jacobi iteration method, the Newton method, and the Gauss Seidel method. The Gauss-Seidel method is a method of solving simultaneous equations through an iteration process so that the actual value is obtained by using the initial value in the next process using a previously known value.

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Bello, Samaila, Muhammad Dikko Amadi, and Aminu Haruna Rawayau. "INTERNET OF THINGS-BASED WIRELESS SENSOR NETWORK SYSTEM FOR EARLY DETECTION AND PREVENTION OF VANDALISM/LEAKAGE ON PIPELINE INSTALLATIONS IN THE OIL AND GAS INDUSTRY IN NIGERIA." FUDMA JOURNAL OF SCIENCES 7, no. 5 (2023): 240–46. http://dx.doi.org/10.33003/fjs-2023-0705-1927.

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Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.

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HarpinderKaur, HarpinderKaur. "Convergence of Jacobi and Gauss-Seidel Method and Error Reduction Factor." IOSR Journal of Mathematics 2, no. 2 (2012): 20–23. http://dx.doi.org/10.9790/5728-0222023.

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Doha, E. H., D. Baleanu, A. H. Bhrawy, and M. A. Abdelkawy. "A Jacobi Collocation Method for Solving Nonlinear Burgers-Type Equations." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/760542.

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We solve three versions of nonlinear time-dependent Burgers-type equations. The Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parametersαandβ. In addition, the problem is reduced to the solution of the system of ordinary differential equations (SODEs) in time. This system may be solved by any standard numerical techniques. Numerical solutions obtained by this method when compared with the exact solutions reveal that the obtained solutions produce high-accurate results. Numerical results show that the proposed method is of high accuracy and is efficient to solve the Burgers-type equation. Also the results demonstrate that the proposed method is a powerful algorithm to solve the nonlinear partial differential equations.

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Atendra, Singh Yadav* Ashish Kumar. "NUMERICAL SOLUTION OF SYSTEM OF LINEAR EQUATIONS BY ITERATIVE METHODS." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 4 (2017): 203–8. https://doi.org/10.5281/zenodo.546307.

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Numerical method is the important aspects in solving real world problems that are related to mathematics, science, medicine, business are very few examples. Numerical method is the area related to mathematics and computer science which create, analysis and implements algorithm to numerically solve the system of linear equations. Numerical methods commonly involve an iterative method (as to find roots). They are now mostly used as preconditions for the popular iterative solvers. While it is difficult task solve as it takes a lot of time but it is an interesting part of Mathematics. In this paper the main emphasis on the beginners that how to iterate the solution of numerical to get appropriate results.

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Kang, Sanggoo, Yin Chao Wu, and Suyun Ham. "Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack." Applied Sciences 10, no. 9 (2020): 3129. http://dx.doi.org/10.3390/app10093129.

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In this study, singular integral solutions were studied to investigate scattering of Rayleigh waves by subsurface cracks. Defining a wave scattering model by objects, such as cracks, still can be quite a challenge. The model’s analytical solution uses five different numerical integration methods: (1) the Gauss–Legendre quadrature, (2) the Gauss–Chebyshev quadrature, (3) the Gauss–Jacobi quadrature, (4) the Gauss–Hermite quadrature and (5) the Gauss–Laguerre quadrature. The study also provides an efficient dynamic finite element analysis to demonstrate the viability of the wave scattering model with an optimized model configuration for wave separation. The obtained analytical solutions are verified with displacement variation curves from the computational simulation by defining the correlation of the results. A novel, verified model, is proposed to provide variations in the backward and forward scattered surface wave displacements calculated by different frequencies and geometrical crack parameters. The analytical model can be solved by the Gauss–Legendre quadrature method, which shows the significantly correlated displacement variation with the FE simulation result. Ultimately, the reliable analytic model can provide an efficient approach to solving the parametric relationship of wave scattering.

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Doha, Eid H., Ali H. Bhrawy, Ramy M. Hafez, and Robert A. Van Gorder. "Jacobi rational–Gauss collocation method for Lane–Emden equations of astrophysical significance." Nonlinear Analysis: Modelling and Control 19, no. 4 (2014): 537–50. http://dx.doi.org/10.15388/na.2014.4.1.

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Bhrawy, A. H., and A. S. Alofi. "A Jacobi–Gauss collocation method for solving nonlinear Lane–Emden type equations." Communications in Nonlinear Science and Numerical Simulation 17, no. 1 (2012): 62–70. http://dx.doi.org/10.1016/j.cnsns.2011.04.025.

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GHOREISHI, F., and P. MOKHTARY. "SPECTRAL COLLOCATION METHOD FOR MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATIONS." International Journal of Computational Methods 11, no. 05 (2014): 1350072. http://dx.doi.org/10.1142/s0219876213500722.

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In this paper, the spectral collocation method is investigated for the numerical solution of multi-order Fractional Differential Equations (FDEs). We choose the orthogonal Jacobi polynomials and set of Jacobi Gauss–Lobatto quadrature points as basis functions and grid points respectively. This solution strategy is an application of the matrix-vector-product approach in spectral approximation of FDEs. The fractional derivatives are described in the Caputo type. Numerical solvability and an efficient convergence analysis of the method have also been discussed. Due to the fact that the solutions of fractional differential equations usually have a weak singularity at origin, we use a variable transformation method to change some classes of the original equation into a new equation with a unique smooth solution such that, the spectral collocation method can be applied conveniently. We prove that after this regularization technique, numerical solution of the new equation has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method.

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Matiko, Fedir, Vitalii Roman, Halyna Matiko, and Dmytro Yalinskyi. "Investigation of ultrasonic flowmeter error in distorted flow using two-peak Salami functions." Energy engineering and control systems 7, no. 2 (2021): 144–51. http://dx.doi.org/10.23939/jeecs2021.02.144.

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Results of investigating the additional error of ultrasonic flowmeters caused by the distortion of the flow are presented in the article. The location coordinates of acoustic paths were calculated for their number from 1 to 6 according to the different numerical integrating methods: Gauss (Gauss-Legendre, Gauss-Jacobi), Chebyshev (equidistant location of acoustic paths), Westinghouse method, method of OWICS (Optimal Weighted Integration for Circular Sections). This made it possible to realize the flowrate equation for multi-path ultrasonic flowmeters and to determine their additional error for different location of the acoustic paths. The average flow velocity along each path is calculated based on the flow velocity profile in the pipe cross section. Four two-peak Salami functions of velocity are used to calculate the velocity profile of the distorted flow caused by typical local resistances. According to the research results the recommendations were developed for choosing the number of the acoustic paths of the ultrasonic flowmeters and for using the methods for determining the location coordinates of the acoustic paths.

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Li, Cui-Xia, Qun-Fa Cui, and Shi-Liang Wu. "Comparison Theorems for Single and Double Splittings of Matrices." Journal of Applied Mathematics 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/827826.

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Some comparison theorems for the spectral radius of double splittings of different matrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper by Miao and Zheng (2009). Some comparison theorems between the spectral radius of single and double splittings of matrices are established and are applied to the Jacobi and Gauss-Seidel double SOR method.

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Bhrawy, A. H., A. S. Alofi, and R. A. Van Gorder. "An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/425648.

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We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomialsJn(α,β)(r)withα,β∈(-1,∞),r∈(0,1)andnthe polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.

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36

Greenberg, Albert G., and Robert J. Vanderbei. "Quicker Convergence for Iterative Numerical Solutions to Stochastic Problems: Probabilistic Interpretations, Ordering Heuristics, and Parallel Processing." Probability in the Engineering and Informational Sciences 4, no. 4 (1990): 493–521. http://dx.doi.org/10.1017/s0269964800001790.

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Gauss-Seidel is a general method for solving a system of equations (possibly nonlinear). It makes repeated sweeps through the variables; within a sweep as each new estimate for a variable is computed, the current estimate for that variable is replaced with the new estimate immediately, instead of on completion of the sweep. The idea is to use new data as soon as it is computed. Gauss- Seidel is often efficient for computing the invariant measure of a Markov chain (especially if the transition matrix is sparse), and for computing the value function in optimal control problems. In many applications the computation can be significantly improved by appropriately ordering the variables within each sweep. A simple heuristic is presented here for computing an ordering that quickens convergence. In parallel processing, several variables must be computed simultaneously, which appears to work against Gauss-Seidel. Simple asynchronous parallel Gauss-Seidel methods are presented here. Experiments indicate that the methods retain the benefit of a good ordering, while further speeding up convergence by a factor of P if P processors participate.In this paper, we focus on the optimal stopping problem. A probabilistic interpretation of the Gauss-Seidel (and the Jacobi) method for computing the value function is given, which motivates our ordering heuristic. However, the ordering heuristic and parallel processing methods apply in a broader context, in particular, to the important problem of computing the invariant measure of a Markov chain.

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Pranowo. "A Novel Space-time Discontinuous Galerkin Method for Solving of One-dimensional Electromagnetic Wave Propagations." TELKOMNIKA Telecommunication, Computing, Electronics and Control 15, no. 3 (2017): 1310–16. https://doi.org/10.12928/TELKOMNIKA.v15i3.5080.

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In this paper we propose a high-order space-time discontinuous Galerkin (STDG) method for solving of one-dimensional electromagnetic wave propagations in homogeneous medium. The STDG method uses finite element Discontinuous Galerkin discretizations in spatial and temporal domain simultaneously with high order piecewise Jacobi polynomial as the basis functions. The algebraic equations are solved using Block Gauss-Seidel iteratively in each time step. The STDG method is unconditionally stable, so the CFL number can be chosen arbitrarily. Numerical examples show that the proposed STDG method is of exponentially accuracy in time.

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Doha, E. H., A. H. Bhrawy, D. Baleanu, and R. M. Hafez. "Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type." Computational Mathematics and Mathematical Physics 53, no. 9 (2013): 1292–302. http://dx.doi.org/10.1134/s0965542513090121.

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Doha, E. H., A. H. Bhrawy, D. Baleanu, and R. H. Hafez. "Efficient Jacobi–Gauss collocation method for solving initial value problems of Brau-type." Журнал вычислительной математики и математической физики 53, no. 9 (2013): 1480. http://dx.doi.org/10.7868/s0044466913090056.

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Doha, E. H., A. H. Bhrawy, M. A. Abdelkawy, and Robert A. Van Gorder. "Jacobi–Gauss–Lobatto collocation method for the numerical solution of nonlinear Schrödinger equations." Journal of Computational Physics 261 (March 2014): 244–55. http://dx.doi.org/10.1016/j.jcp.2014.01.003.

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41

Bhrawy, A. H., M. A. Alghamdi, and Eman S. Alaidarous. "An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions." Abstract and Applied Analysis 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/295936.

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One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

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Antoni, Grégory. "A New Accurate and Efficient Iterative Numerical Method for Solving the Scalar and Vector Nonlinear Equations: Approach Based on Geometric Considerations." International Journal of Engineering Mathematics 2016 (August 7, 2016): 1–18. http://dx.doi.org/10.1155/2016/6390367.

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This paper deals with a new numerical iterative method for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphson method. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.

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43

Tang, Shuting, Xiuqin Deng, and Rui Zhan. "The general tensor regular splitting iterative method for multilinear PageRank problem." AIMS Mathematics 9, no. 1 (2023): 1443–71. http://dx.doi.org/10.3934/math.2024071.

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<abstract><p>The paper presents an iterative scheme called the general tensor regular splitting iterative (GTRS) method for solving the multilinear PageRank problem, which is based on a (weak) regular splitting technique and further accelerates the iterative process by introducing a parameter. The method yields familiar iterative schemes through the use of specific splitting strategies, including fixed-point, inner-outer, Jacobi, Gauss-Seidel and successive overrelaxation methods. The paper analyzes the convergence of these solvers in detail. Numerical results are provided to demonstrate the effectiveness of the proposed method in solving the multilinear PageRank problem.</p></abstract>

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44

Yoo, Jeong-Un, and Il-Suek Koh. "Comparison of Linear Iteration Schemes to Improve the Convergence of Iterative Physical Optics for an Impedance Scatterer." Journal of Electromagnetic Engineering and Science 23, no. 1 (2023): 78–80. http://dx.doi.org/10.26866/jees.2023.1.l.12.

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The conventional iterative physical optics (IPO) method updates the surface current based on the Jacobi iteration scheme, which typically diverges for large objects. To control the convergence property of the IPO method, other iteration schemes, such as Gauss–Seidel and successive over-relaxation, can be used. In this study, we compare the convergence properties of three iteration schemes for scatterings by five scatterers comprising electrically perfect or imperfect conductors modeled with an impedance material. The accuracy of the IPO method is compared with that of the multi-level fast multipole method.

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45

Liu, Qirui. "Iterative algorithms and architectures in massive MIMO detection." Applied and Computational Engineering 6, no. 1 (2023): 1143–50. http://dx.doi.org/10.54254/2755-2721/6/20230491.

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Massive multiple-input multiple-output (MIMO) wireless systems play an important role in the 5G networks. The complexity of signal detection in massive MIMO is increasing rapidly due to the growth of the number of transmitting antennas. In this paper, we introduced different iterative algorithms to decrease the computational cost of the approximate minimum mean-square error (MMSE) algorithm, including the Neumann Series algorithm, Jacobi method, Gauss-Seidel method, Successive Over Relaxation method and the Conjugate Gradient method. In addition, the VLSI architecture implementations of algorithms mentioned above are also discussed in the article.

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Ali, Mushtaq Salh, Mostafa Shamsi, Hassan Khosravian-Arab, Delfim F. M. Torres, and Farid Bozorgnia. "A space–time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives." Journal of Vibration and Control 25, no. 5 (2018): 1080–95. http://dx.doi.org/10.1177/1077546318811194.

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We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi–Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer–order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre–Gauss–Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low central processing unit time.

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Mat, Ali Nur Afza, Jumat Sulaiman, Azali Saudi, and Nor Syahida Mohamad. "Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation." Indonesian Journal of Electrical Engineering and Computer Science 23, no. 1 (2021): 471–78. https://doi.org/10.11591/ijeecs.v23.i1.pp471-478.

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In this paper, a similarity finite difference (SFD) solution is addressed for the two-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization scheme to get the second-order SFD approximation equation. We propose a fourpoint similarity explicit group (4-point SEG) iterative method as a numerical solution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To show the 4-point SEG iteration efficiency, two iterative methods, such as Jacobi and Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobi and GS iterations in terms of iteration number and execution time.

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Doha, E. H., A. H. Bhrawy, D. Baleanu, and R. M. Hafez. "A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations." Applied Numerical Mathematics 77 (March 2014): 43–54. http://dx.doi.org/10.1016/j.apnum.2013.11.003.

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La Scala, M., M. Brucoli, F. Torelli, and M. Trovato. "A Gauss-Jacobi-Block-Newton method for parallel transient stability analysis (of power systems)." IEEE Transactions on Power Systems 5, no. 4 (1990): 1168–77. http://dx.doi.org/10.1109/59.99367.

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Peykrayegan, N., M. Ghovatmand, and M. H. Noori Skandari. "On the convergence of Jacobi‐Gauss collocation method for linear fractional delay differential equations." Mathematical Methods in the Applied Sciences 44, no. 2 (2020): 2237–53. http://dx.doi.org/10.1002/mma.6934.

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