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Journal articles on the topic 'Matrice de Seidel'

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Author: Grafiati

Published: 18 May 2024

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1

Manjhi, Pankaj Kumar, and Ninian Nauneet Kujur. "On Goethals and Seidel Array." Indian Journal Of Science And Technology 17, no. 16 (April 19, 2024): 1643–46. http://dx.doi.org/10.17485/ijst/v17i16.2937.

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Objectives: In this article, we aim to find a series of Hadamard matrices by suitable selection of the special class of matrices given in the Goethals and Seidel array and study the pattern obtained. Methods: In the presented work, the search technique of Hadamard matrices has been done by selecting special class of (0,1) negacyclic matrices instead of the back diagonal identity matrix given in Geothals and Seidel arrays and the possible existence of negacyclic matrices for the corresponding four matrices have been explored in each case. Findings: Corresponding to the special class of (0,1) negacyclic matrices, no sets of four negacyclic matrices have been obtained in the Goethal Seidel array, for even orders. For odd orders, except in the case when all four matrices are equal and the case when there is a pair of equal matrices, many outputs have been obtained for the remaining cases, the search domain being upto 11,9 and 7 for the case of two different, three different and four different matrices respectively, in the Goethal Seidel array. Novelty: The selection of special class of negacyclic matrices instead of the back diagonal identity matrix and finding the corresponding four negacyclic matrices in Goethals and Seidel arrays for constructing Hadamard matrices provides a new approach to the construction of Hadamard matrices. Keywords: Hadamard matrix, Circulant matrix, Williamson matrices, Orthogonal array, Goethals and Seidel array

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2

Askari, Jalal. "A note on the Seidel and Seidel Laplacian matrices." Boletim da Sociedade Paranaense de Matemática 41 (December 23, 2022): 1–6. http://dx.doi.org/10.5269/bspm.51593.

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In this paper we investigate the spectrum of the Seidel and Seidel Laplacian matrix of a graph. We generalized the concept of Seidel Laplacian matrix which denoted by Seidel matrix and obtained some results related to them. This can be intuitively understood as a consequence of the relationship between the Seidel and Seidel Laplacian matrix in the graph by Zagreb index. In closing, we mention some alternatives to and generalization of the Seidel and Seidel Laplacian matrices. Also, we obtain relation between Seidel and Seidel Laplacian energy, related to all graphs with order n.

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3

Enyew, Tesfaye Kebede, Gurju Awgichew, Eshetu Haile, and Gashaye Dessalew Abie. "Second-refinement of Gauss-Seidel iterative method for solving linear system of equations." Ethiopian Journal of Science and Technology 13, no. 1 (April 30, 2020): 1–15. http://dx.doi.org/10.4314/ejst.v13i1.1.

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Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

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4

Zhang, Cheng-yi, Dan Ye, Cong-Lei Zhong, and SHUANGHUA SHUANGHUA. "Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices." Electronic Journal of Linear Algebra 30 (February 8, 2015): 843–70. http://dx.doi.org/10.13001/1081-3810.1972.

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It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible Hâmatrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general Hâmatrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general Hâmatrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general Hâmatrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

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5

Dil, Ayhan, and Mirac Cetin Firengiz. "Shifted Euler-Seidel matrices." Miskolc Mathematical Notes 18, no. 1 (2017): 173. http://dx.doi.org/10.18514/mmn.2017.1662.

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6

Szöllősi, Ferenc, and Patric R. J. Östergård. "Enumeration of Seidel matrices." European Journal of Combinatorics 69 (March 2018): 169–84. http://dx.doi.org/10.1016/j.ejc.2017.10.009.

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7

TUTAŞ, Nesrin. "Euler-Seidel matrices over F_p." TURKISH JOURNAL OF MATHEMATICS 38 (2014): 16–24. http://dx.doi.org/10.3906/mat-1209-36.

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8

Vinsensia, Desi, Yulia Utami, Fathia Siregar, and Muhammad Arifin. "Improve refinement approach iterative method for solution linear equition of sparse matrices." Jurnal Teknik Informatika C.I.T Medicom 15, no. 6 (January 30, 2024): 306–13. http://dx.doi.org/10.35335/cit.vol15.2024.721.pp306-313.

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In this paper, systems of linear equations on sparse matrices investigated through modified improve method using Gauss-Seidel and successive overrelaxation (SOR) approach. Taking into adapted convergence rate on the Improve refinement Gauss-seidel outperformed the prior two Gauss-Seidel methods in terms of rate of convergence and number of iterations required to solve the problem by applying a modified version of the Gauss-Seidel approach. to observe the effectiveness of this method, the numerical example is given. The main findings in this study, that Gauss seidel improvement refinement gives optimum spectral radius and convergence rate. Similarly, the SOR improved refinement method gives. Considering their performance, using parameters such as time to converge, number of iterations required to converge and spectral radius level of accuracy. However, SOR works with relaxation values so that it greatly affects the convergence rate and spectral radius results if given greater than 1.

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9

Shen, Shuhui, and Xiaojun Zhang. "Constructions of Goethals–Seidel Sequences by Using k-Partition." Mathematics 11, no. 2 (January 6, 2023): 294. http://dx.doi.org/10.3390/math11020294.

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In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will be presented. We first propose a method based on T-matrices directly to obtain a quad of Goethals–Seidel sequences. Next, by introducing the k-partition, we utilize two classes of 8-partitions to obtain a new class of polynomials still remaining the same (anti)symmetrical properties, with which a quad of Goethals–Seidel sequences could be constructed. Moreover, an adoption of the 4-partition together with a quad of four symmetrical sequences can also lead to a quad of Goethals–Seidel sequences.

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10

Dessalew, Gashaye, Tesfaye Kebede, Gurju Awgichew, and Assaye Walelign. "Generalized Refinement of Gauss-Seidel Method for Consistently Ordered 2-Cyclic Matrices." Abstract and Applied Analysis 2021 (May 31, 2021): 1–7. http://dx.doi.org/10.1155/2021/8343207.

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This paper presents generalized refinement of Gauss-Seidel method of solving system of linear equations by considering consistently ordered 2-cyclic matrices. Consistently ordered 2-cyclic matrices are obtained while finite difference method is applied to solve differential equation. Suitable theorems are introduced to verify the convergence of this proposed method. To observe the effectiveness of this method, few numerical examples are given. The study points out that, using the generalized refinement of Gauss-Seidel method, we obtain a solution of a problem with a minimum number of iteration and obtain a greater rate of convergence than other previous methods.

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11

Vostrikov, Anton. "Matrix vitrages and regular Hadamard matrices." Information and Control Systems, no. 5 (October 26, 2021): 2–9. http://dx.doi.org/10.31799/1684-8853-2021-5-2-9.

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Introduction: The Kronecker product of Hadamard matrices when a matrix of order n replaces each element in another matrix of order m, inheriting the sign of the replaced element, is a basis for obtaining orthogonal matrices of order nm. The matrix insertion operation when not only signs but also structural elements (ornamental patterns of matrix portraits) are inherited provides a more general result called a "vitrage". Vitrages based on typical quasi-orthogonal Mersenne (M), Seidel (S) or Euler (E) matrices, in addition to inheriting the sign and pattern, inherit the value of elements other than unity (in amplitude) in a different way, causing the need to revise and systematize the accumulated experience. Purpose: To describe new algorithms for generalized product of matrices, highlighting the constructions that produce regular high-order Hadamard matrices. Results: We have proposed an algorithm for obtaining matrix vitrages by inserting Mersenne matrices into Seidel matrices, which makes it possible to expand the additive chains of matrices of the form M-E-M-E-… and S-E-M-E-…, obtained by doubling the orders and adding an edge. The operation of forming a matrix vitrage allows you to obtain matrices of high orders, keeping the ornamental pattern as an important invariant of the structure. We have shown that the formation of a matrix vitrage inherits the logic of the Scarpi product, but is cannot be reduced to it, since a nonzero distance in order between the multiplicands M and S simplifies the final regular matrix ornamental pattern due to the absence of cyclic displacements. The alternation of M and S matrices allows you to extend the multiplicative chains up to the known gaps in the S matrices. This sheds a new light on the theory of a regular Hadamard matrix as a product of Mersenne and Seidel matrices. Practical relevance: Orthogonal sequences with floating levels and efficient algorithms for finding regular Hadamard matrices with certain useful properties are of direct practical importance for the problems of noise-proof coding, compression and masking of video data.

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12

Topno, Sheet Nihal, and Shyam Saurabh. "Block Structured Hadamard matrices from certain arrays." Journal of Combinatorial Mathematics and Combinatorial Computing 117 (December 31, 2023): 169–75. http://dx.doi.org/10.61091/jcmcc117-15.

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We have constructed Block structured Hadamard matrices in which odd number of blocks are used in a row (column). These matrices are different than those introduced by Agaian. Generalised forms of arrays developed by Goethals-Seidel, Wallis-Whiteman and Seberry-Balonin heve been employed. Such types of matrices are applicable in the constructions of nested group divisible designs.

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13

Álvarez, Víctor, José Andrés Armario, Raúl M. Falcón, María Dolores Frau, Félix Gudiel, María Belén Güemes, and Amparo Osuna. "On Cocyclic Hadamard Matrices over Goethals-Seidel Loops." Mathematics 8, no. 1 (December 20, 2019): 24. http://dx.doi.org/10.3390/math8010024.

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About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. Much more recently, it has been proved that this notion may naturally be extended to define cocyclic Hadamard matrices developed over a loop. This paper delves into this last topic by introducing the concepts of coboundary, pseudocoboundary and pseudococycle over a quasigroup, and also the notion of the pseudococyclic Hadamard matrix. Furthermore, Goethals-Seidel loops are introduced as a family of Moufang loops so that every Hadamard matrix of Goethals-Seidel type (which is known not to be cocyclically developed over any group) is actually pseudococyclically developed over them. Finally, we also prove that, no matter if they are pseudococyclic matrices, the usual cocyclic Hadamard test is unexpectedly applicable.

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14

Duncan, David M., Thomas R. Hoffman, and James P. Solazzo. "Equiangular tight frames and fourth root seidel matrices." Linear Algebra and its Applications 432, no. 11 (June 2010): 2816–23. http://dx.doi.org/10.1016/j.laa.2009.12.017.

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15

Ghorbani, Ebrahim. "On eigenvalues of Seidel matrices and Haemers’ conjecture." Designs, Codes and Cryptography 84, no. 1-2 (July 16, 2016): 189–95. http://dx.doi.org/10.1007/s10623-016-0248-x.

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16

Wang, Yang, Jie Liu, Xiaoxiong Zhu, Qingyang Zhang, Shengguo Li, and Qinglin Wang. "Improving Structured Grid-Based Sparse Matrix-Vector Multiplication and Gauss–Seidel Iteration on GPDSP." Applied Sciences 13, no. 15 (August 3, 2023): 8952. http://dx.doi.org/10.3390/app13158952.

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Structured grid-based sparse matrix-vector multiplication and Gauss–Seidel iterations are very important kernel functions in scientific and engineering computations, both of which are memory intensive and bandwidth-limited. GPDSP is a general purpose digital signal processor, which is a very significant embedded processor that has been introduced into high-performance computing. In this paper, we designed various optimization methods, which included a blocking method to improve data locality and increase memory access efficiency, a multicolor reordering method to develop Gauss–Seidel fine-grained parallelism, a data partitioning method designed for GPDSP memory structures, and a double buffering method to overlap computation and access memory on structured grid-based SpMV and Gauss–Seidel iterations for GPDSP. At last, we combined the above optimization methods to design a multicore vectorization algorithm. We tested the matrices generated with structured grids of different sizes on the GPDSP platform and obtained speedups of up to 41× and 47× compared to the unoptimized SpMV and Gauss–Seidel iterations, with maximum bandwidth efficiencies of 72% and 81%, respectively. The experiment results show that our algorithms could fully utilize the external memory bandwidth. We also implemented the commonly used mixed precision algorithm on the GPDSP and obtained speedups of 1.60× and 1.45× for the SpMV and Gauss–Seidel iterations, respectively.

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17

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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18

Abuzin, Leonid, Nikolai Unknown, Dragomir Ðoković, and Ilias Kotsireas. "Hadamard matrices from Goethals — Seidel difference families with a repeated block." Information and Control Systems, no. 5 (October 16, 2019): 2–9. http://dx.doi.org/10.31799/1684-8853-2019-5-2-9.

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Purpose: To construct Hadamard matrices by using Goethals — Seidel difference families having a repeated block, generalizingthe so called propus construction. In particular we construct the first examples of symmetric Hadamard matrices of order 236.Methods: The main ingredient of the propus construction is a difference family in a finite abelian group of order v consisting offour blocks (X1, X2, X3, X4) where X1 is symmetric and X2 X3. The parameters (v; k1, k2, k3, k4; λ) of such family must satisfythe additional condition ki  λ  v. We modify this construction by imposing different symmetry conditions on some of theblocks and construct many examples of Hadamard matrices of this kind. In this paper we work with the cyclic group Zv of order v.For larger values of v we build the blocks Xi by using the orbits of a suitable small cyclic subgroup of the automorphism groupof Zv. Results: We continue the systematic search for symmetric Hadamard matrices of order 4v by using the propus construction.Such searches were carried out previously for odd v  51. We extend it to cover the case v53. Moreover we construct thefirst examples of symmetric Hadamard matrices of order 236. A wide collection of symmetric and skew-symmetric Hadamardmatrices was obtained and the corresponding difference families tabulated by using the symmetry properties of their blocks.Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking ofvideo information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in themathematical network Internet together with executable on line algorithms.

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19

Liu, Zhongyun, Xiaorong Qin, Nianci Wu, and Yulin Zhang. "The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices." Canadian Mathematical Bulletin 60, no. 4 (December 1, 2017): 807–15. http://dx.doi.org/10.4153/cmb-2016-077-5.

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AbstractIt is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted CSCS) i.e., T = C−S with C a circulantmatrix and S a skew circulantmatrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive deûnite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss–Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant α such that the shifted CSCS iteration converges much faster than the Gauss–Seidel iteration, no matter whether the CSCS itself is convergent or not.

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20

Xiong, Zhuang, and Yaoping Hou. "Eigenvalue-free interval for Seidel matrices of threshold graphs." Applied Mathematics and Computation 427 (August 2022): 127177. http://dx.doi.org/10.1016/j.amc.2022.127177.

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21

Kohno, T. "Improving the Modified Gauss-Seidel Method for Z-Matrices." Linear Algebra and its Applications 267, no. 1-3 (December 1997): 113–23. http://dx.doi.org/10.1016/s0024-3795(97)00063-3.

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22

Kohno, Toshiyuki, Hisashi Kotakemori, Hiroshi Niki, and Masataka Usui. "Improving the modified Gauss-Seidel method for Z-matrices." Linear Algebra and its Applications 267 (December 1997): 113–23. http://dx.doi.org/10.1016/s0024-3795(97)80045-6.

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23

Rizzolo, Douglas. "Determinants of Seidel matrices and a conjecture of Ghorbani." Linear Algebra and its Applications 579 (October 2019): 51–54. http://dx.doi.org/10.1016/j.laa.2019.05.025.

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24

Shen, Hailong, Xinhui Shao, Zhenxing Huang, and Chunji Li. "PRECONDITIONED GAUSS-SEIDEL ITERATIVE METHOD FOR Z-MATRICES LINEAR SYSTEMS." Bulletin of the Korean Mathematical Society 48, no. 2 (March 31, 2011): 303–14. http://dx.doi.org/10.4134/bkms.2011.48.2.303.

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25

Evans, D. J., C. Li, and Y. Xue. "The extrapolated gauss-seidel methods and generally consistently ordered matrices." International Journal of Computer Mathematics 23, no. 1 (January 1987): 77–97. http://dx.doi.org/10.1080/00207168708803609.

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26

Kotakemori, Hisashi, Hiroshi Niki, and Naotaka Okamoto. "A generalization of the adaptive gauss-seidel method forZ-matrices." International Journal of Computer Mathematics 64, no. 3-4 (January 1997): 317–26. http://dx.doi.org/10.1080/00207169708804594.

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27

Doković, Dragomir ſ. "Ten hadamard matrices of order 1852 of Goethals-Seidel type." European Journal of Combinatorics 13, no. 4 (July 1992): 245–48. http://dx.doi.org/10.1016/s0195-6698(05)80030-7.

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28

Đoković, Dragomir Ž. "Two Hadamard matrices of order 956 of Goethals-Seidel type." Combinatorica 14, no. 3 (September 1994): 375–77. http://dx.doi.org/10.1007/bf01212983.

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29

Yuan, J. Y., and D. D. Zontini. "Comparison theorems of preconditioned Gauss–Seidel methods for M-matrices." Applied Mathematics and Computation 219, no. 4 (November 2012): 1947–57. http://dx.doi.org/10.1016/j.amc.2012.08.037.

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30

Crnković, Dean, Ronan Egan, and Andrea Švob. "Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs." Advances in Mathematics of Communications 14, no. 4 (2020): 591–602. http://dx.doi.org/10.3934/amc.2020032.

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31

Liu, Qingbing, Guoliang Chen, and Jing Cai. "Convergence analysis of the preconditioned Gauss–Seidel method for H-matrices." Computers & Mathematics with Applications 56, no. 8 (October 2008): 2048–53. http://dx.doi.org/10.1016/j.camwa.2008.03.033.

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32

Jha, Navnit, R. K. Mohanty, and Vinod Chauhan. "Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System." Advances in Numerical Analysis 2013 (October 24, 2013): 1–10. http://dx.doi.org/10.1155/2013/614508.

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Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

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33

Wen, Chun, Ting-Zhu Huang, Xian-Ming Gu, Zhao-Li Shen, Hong-Fan Zhang, and Chen Liu. "Multipreconditioned GMRES for simulating stochastic automata networks." Open Mathematics 16, no. 1 (August 24, 2018): 986–98. http://dx.doi.org/10.1515/math-2018-0083.

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AbstractStochastic Automata Networks (SANs) have a large amount of applications in modelling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seidel are inefficient due to the huge size of the generator matrices. In this paper, the multipreconditioned GMRES (MPGMRES) is considered by using two or more preconditioners simultaneously. Meanwhile, a selective version of the MPGMRES is presented to overcome the rapid increase of the storage requirements and make it practical. Numerical results on two models of SANs are reported to illustrate the effectiveness of these proposed methods.

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34

Li, Wen, and Weiwei Sun. "Modified Gauss–Seidel type methods and Jacobi type methods for Z-matrices." Linear Algebra and its Applications 317, no. 1-3 (September 2000): 227–40. http://dx.doi.org/10.1016/s0024-3795(00)00140-3.

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35

Bodmann, Bernhard G., Vern I. Paulsen, and Mark Tomforde. "Equiangular tight frames from complex Seidel matrices containing cube roots of unity." Linear Algebra and its Applications 430, no. 1 (January 2009): 396–417. http://dx.doi.org/10.1016/j.laa.2008.08.002.

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36

Zheng, Bing, and Shu-Xin Miao. "Two new modified Gauss–Seidel methods for linear system with M-matrices." Journal of Computational and Applied Mathematics 233, no. 4 (December 2009): 922–30. http://dx.doi.org/10.1016/j.cam.2009.08.056.

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37

Sun, Li-Ying. "Some extensions of the improved modified Gauss–Seidel iterative method forH-matrices." Numerical Linear Algebra with Applications 13, no. 10 (2006): 869–76. http://dx.doi.org/10.1002/nla.498.

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38

Reyes, Victor, and Ignacio Araya. "Non-Convex Optimization: Using Preconditioning Matrices for Optimally Improving Variable Bounds in Linear Relaxations." Mathematics 11, no. 16 (August 17, 2023): 3549. http://dx.doi.org/10.3390/math11163549.

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The performance of branch-and-bound algorithms for solving non-convex optimization problems greatly depends on convex relaxation techniques. They generate convex regions which are used for improving the bounds of variable domains. In particular, convex polyhedral regions can be represented by a linear system A.x=b. Then, bounds of variable domains can be improved by minimizing and maximizing variables in the linear system. Reducing or contracting optimally variable domains in linear systems, however, is an expensive task. It requires solving up to two linear programs for each variable (one for each variable bound). Suboptimal strategies, such as preconditioning, may offer satisfactory approximations of the optimal reduction at a lower cost. In non-square linear systems, a preconditioner P can be chosen such that P.A is close to a diagonal matrix. Thus, the projection of the equivalent system P.A.x=P.b over x, by using an iterative method such as Gauss–Seidel, can significantly improve the contraction. In this paper, we show how to generate an optimal preconditioner, i.e., a preconditioner that helps the Gauss–Seidel method to optimally reduce the variable domains. Despite the cost of generating the preconditioner, it can be re-used in sub-regions of the search space without losing too much effectiveness. Experimental results show that, when used for reducing domains in non-square linear systems, the approach is significantly more effective than Gauss-based elimination techniques. Finally, the approach also shows promising results when used as a component of a solver for non-convex optimization problems.

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39

Akbari, S., S. M. Cioabă, S. Goudarzi, A. Niaparast, and A. Tajdini. "On a question of Haemers regarding vectors in the nullspace of Seidel matrices." Linear Algebra and its Applications 615 (April 2021): 194–206. http://dx.doi.org/10.1016/j.laa.2021.01.003.

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40

Bodmann, Bernhard G., and Helen J. Elwood. "Complex equiangular Parseval frames and Seidel matrices containing $p$th roots of unity." Proceedings of the American Mathematical Society 138, no. 12 (December 1, 2010): 4387. http://dx.doi.org/10.1090/s0002-9939-2010-10435-5.

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41

Evans, D. J., and C. Li. "The extrapolated gauss-seidel plus semi-iterative method for generalized consistently ordered matrices." International Journal of Computer Mathematics 25, no. 1 (January 1988): 55–66. http://dx.doi.org/10.1080/00207168808803660.

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42

Natale Primero, Gino, Jesús Rafael García Núñez, and Alvaro Correa Arroyave. "Utilización de matrices dispersas en el Método de los Elementos Finitos." Ingeniería e Investigación, no. 27 (May 1, 1992): 18–37. http://dx.doi.org/10.15446/ing.investig.n27.20759.

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El presente es el primero de una serie de artículos relacionados con el tema de los Elementos Finitos y su aplicación en el campo de la ingeniería, y en particular de la Ingeniería Geotécnica, que publicaremos en esta sección. En el primero de ellos estamos presentando el método ilustrándolo con un ejemplo muy sencillo en el que se determinan los esfuerzos de una viga constituida por un material CHILE bajo el supuesto de deformaciones planas y esfuerzos biaxiales. La finalidad de este primer artículo es mostrar las ventajas que ofrecen las matrices dispersas para el manejo de la matriz banda que genera un tipo de problemas como el planteado, frente a los sistemas tradicionales de Choleski-Jordan y Gauss-Seidel, entre otros. Posteriores artículos ilustraran el método para materiales CHOLE (Continuos, Homogéneos, Ortotrópicos y linealmente elásticos) y otros tipos de materiales en donde el comportamiento elástico no cumple con la linearidad.

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Todorov, Venelin, Slavi Georgiev, and Stoyan Apostolov. "New “Walk on Equations” Monte Carlo Algorithm for Linear Systems." Journal of Physics: Conference Series 2675, no. 1 (December 1, 2023): 012037. http://dx.doi.org/10.1088/1742-6596/2675/1/012037.

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Abstract A novel version of Monte Carlo algorithm for solving systems of linear algebraic equations is presented and studied. The algorithm is similar to the “Walk on Equations” Monte Carlo method recently developed by Ivan Dimov, Sylvain Maire and Jean Michel Sellier. It is done a comparison with the Gauss-Seidel method for matrices up to size of 212. The algorithm could be drastically improved by choosing appropriate values for the relaxation parameters, which in turn leads to dramatic reduction in time and lower relative errors for a given number of iterations. What is more, a sequential Monte Carlo method of John Halton based on an iterative use of the control variate method has been applied. Some of the most important numerical applications are the large system, coming from a finite element approximation of problems, describing a beam structure in constructive mechanics, and the block-diagonal matrices, which come from discretization of models in a regime-switching economy.

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Chang, Shih Yu, Hsiao-Chun Wu, and Yifan Wang. "New Efficient Approach to Solve Big Data Systems Using Parallel Gauss–Seidel Algorithms." Big Data and Cognitive Computing 6, no. 2 (April 19, 2022): 43. http://dx.doi.org/10.3390/bdcc6020043.

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In order to perform big-data analytics, regression involving large matrices is often necessary. In particular, large scale regression problems are encountered when one wishes to extract semantic patterns for knowledge discovery and data mining. When a large matrix can be processed in its factorized form, advantages arise in terms of computation, implementation, and data-compression. In this work, we propose two new parallel iterative algorithms as extensions of the Gauss–Seidel algorithm (GSA) to solve regression problems involving many variables. The convergence study in terms of error-bounds of the proposed iterative algorithms is also performed, and the required computation resources, namely time- and memory-complexities, are evaluated to benchmark the efficiency of the proposed new algorithms. Finally, the numerical results from both Monte Carlo simulations and real-world datasets are presented to demonstrate the striking effectiveness of our proposed new methods.

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Herrera, Maria Camila, Oscar Danilo Montoya, Alexander Molina-Cabrera, Luis Fernando Grisales-Noreña, and Diego Armando Giral-Ramirez. "Convergence analysis of the triangular-based power ﬂow method for AC distribution grids." International Journal of Electrical and Computer Engineering (IJECE) 12, no. 1 (February 1, 2022): 41. http://dx.doi.org/10.11591/ijece.v12i1.pp41-49.

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<p>This paper addresses the convergence analysis of the triangular-based power flow (PF) method in alternating current radial distribution networks. The PF formulation is made via upper-triangular matrices, which enables finding a general iterative PF formula that does not require admittance matrix calculations. The convergence analysis of this iterative formula is carried out by applying the Banach fixed-point theorem (BFPT), which allows demonstrating that under an adequate voltage profile the triangular-based PF always converges. Numerical validations are made, on the well-known 33 and 69 distribution networks test systems. Gauss-seidel, newton-raphson, and backward/forward PF methods are considered for the sake of comparison. All the simulations are carried out in MATLAB software.</p>

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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 (March 30, 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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Estrada, Ernesto. "The many facets of the Estrada indices of graphs and networks." SeMA Journal 79, no. 1 (October 25, 2021): 57–125. http://dx.doi.org/10.1007/s40324-021-00275-w.

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AbstractThe Estrada index of a graph/network is defined as the trace of the adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such as the Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many of these extensions, including new ones, such as Gaussian, Mittag–Leffler and Onsager ones. More importantly, we contextualize all of these indices in physico-mathematical frameworks which allow their interpretations and facilitate their extensions and further studies. We also describe several of the bounds and estimations of these indices reported in the literature and analyze many of them computationally for small graphs as well as large complex networks. This article is intended to formalize many of the Estrada indices proposed and studied in the mathematical literature serving as a guide for their further studies.

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Bylina, Beata, and Jarosław Bylina. "Influence of Preconditioning and Blocking on Accuracy in Solving Markovian Models." International Journal of Applied Mathematics and Computer Science 19, no. 2 (June 1, 2009): 207–17. http://dx.doi.org/10.2478/v10006-009-0017-3.

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Influence of Preconditioning and Blocking on Accuracy in Solving Markovian ModelsThe article considers the effectiveness of various methods used to solve systems of linear equations (which emerge while modeling computer networks and systems with Markov chains) and the practical influence of the methods applied on accuracy. The paper considers some hybrids of both direct and iterative methods. Two varieties of the Gauss elimination will be considered as an example of direct methods: the LU factorization method and the WZ factorization method. The Gauss-Seidel iterative method will be discussed. The paper also shows preconditioning (with the use of incomplete Gauss elimination) and dividing the matrix into blocks where blocks are solved applying direct methods. The motivation for such hybrids is a very high condition number (which is bad) for coefficient matrices occuring in Markov chains and, thus, slow convergence of traditional iterative methods. Also, the blocking, preconditioning and merging of both are analysed. The paper presents the impact of linked methods on both the time and accuracy of finding vector probability. The results of an experiment are given for two groups of matrices: those derived from some very abstract Markovian models, and those from a general 2D Markov chain.

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SEGURA UGALDE, ESTEBAN, and EDUARDO PIZA VOLIO. "SECUENCIAS TIPO TURYN." Revista de Matemática: Teoría y Aplicaciones 26, no. 2 (July 12, 2019): 253–79. http://dx.doi.org/10.15517/rmta.v26i2.38317.

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En este artículo estudiamos fundamentalmente las denominadas secuencias tipo Turyn y algunos algoritmos heurísticos para generarlas. La importancia de estas secuencias estriba, al menos, en el hecho de quepueden ser empleadas en la construcción de algunas matrices de Hadamard de órdenes 4(3m - 1), donde m es el largo de la secuencia tipo Turyn a través del uso del teorema de Goethals-Seidal. Simplificamos la demostración del teorema de Turyn (ver Teorema 3). Además, hallamos algunos resultados teóricos interesantes (ver Teorema 5). Finalmente, desarrollamos varios algoritmos heurísticos eficientes, comparables a los algoritmos ya conocidos, que generan secuencias tipo Turyn de tamañosmenores o iguales a 40.

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Oladeinde, M. H., and John A. Akpobi. "Modeling Analysis of the Wire-Drawing Operation Using the Weighted-Residual Finite Element Method." Advanced Materials Research 367 (October 2011): 677–84. http://dx.doi.org/10.4028/www.scientific.net/amr.367.677.

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Numerical analysis of a wire drawing operation to compute the stress distribution along the blank cross-section is presented. The governing equation describing the wire drawing equation is weakened using the Bubnov-Galerkin finite element method to obtain the finite element model. The blank is descritized into a mesh of C0 quadratic isoparametric and C0 cubic finite elements. Stiffness matrices for all elements are obtained using the finite element model which were subsequently assembled by enforcing continuity of the nodal stress. Boundary conditions are applied and the resulting condensed system of equation solved for unknown nodal stresses using Gauss Seidel method. The relative performance of the C0 quadratic and C0 cubic elements are assessed. Parametric analysis is carried out to show the influence of drawing parameters on the stresses generated and drawing load. The analysis was carried out using a Visual Basic.Net program developed by the authors.The results are presented in both graphical and tabular forms.

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