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Journal articles on the topic 'Tridiagonal Matrix Algoritm'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Zgirouski, A. A., and N. A. Likhoded. "Modified method of parallel matrix sweep." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 4 (January 7, 2020): 425–34. http://dx.doi.org/10.29235/1561-2430-2019-55-4-425-434.

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The topic of this paper refers to efficient parallel solvers of block-tridiagonal linear systems of equations. Such systems occur in numerous modeling problems and require usage of high-performance multicore computation systems. One of the widely used methods for solving block-tridiagonal linear systems in parallel is the original block-tridiagonal sweep method. We consider the algorithm based on the partitioning idea. Firstly, the initial matrix is split into parts and transformations are applied to each part independently to obtain equations of a reduced block-tridiagonal system. Secondly, the reduced system is solved sequentially using the classic Thomas algorithm. Finally, all the parts are solved in parallel using the solutions of a reduced system. We propose a modification of this method. It was justified that if known stability conditions for the matrix sweep method are satisfied, then the proposed modification is stable as well.

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2

Ran, Rui-sheng, Ting-zhu Huang, Xing-ping Liu, and Tong-xiang Gu. "An inversion algorithm for general tridiagonal matrix." Applied Mathematics and Mechanics 30, no. 2 (February 2009): 247–53. http://dx.doi.org/10.1007/s10483-009-0212-x.

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3

Mattor, Nathan, Timothy J. Williams, and Dennis W. Hewett. "Algorithm for solving tridiagonal matrix problems in parallel." Parallel Computing 21, no. 11 (November 1995): 1769–82. http://dx.doi.org/10.1016/0167-8191(95)00033-0.

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4

Takahira, S., T. Sogabe, and T. S. Usuda. "Bidiagonalization of (k, k + 1)-tridiagonal matrices." Special Matrices 7, no. 1 (January 1, 2019): 20–26. http://dx.doi.org/10.1515/spma-2019-0002.

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Abstract In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].

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Chathalingath, Anishchandran, and Arun Manoharan. "Performance Optimization of Tridiagonal Matrix Algorithm [TDMA] on Multicore Architectures." International Journal of Grid and High Performance Computing 11, no. 4 (October 2019): 1–12. http://dx.doi.org/10.4018/ijghpc.2019100101.

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Fast and efficient tridiagonal solvers are highly appreciated in scientific and engineering domain, but challenging optimization task for computer engineers. The state-of-the-art developments in multi-core computing paves the way to meet this challenge to an extent. The technical advances in multi-core computing provide opportunities to exploit lower levels of parallelism and concurrency for inherently sequential algorithms. In this article, the authors present an optimal performance pipelined parallel variant of the conventional Tridiagonal Matrix Algorithm (TDMA), aka the Thomas algorithm, on a multi-core CPU platform. The implementation, analysis and performance comparison of the proposed pipelined parallel TDMA and the conventional version are performed on an Intel SIMD multi-core architecture. The results are compared in terms of elapsed time, speedup, cache miss rate. For a system of ‘n' linear equations where n = 2^36 in presented pipelined parallel TDMA achieves speedup of 1.294X with a parallel efficiency of 43% initially and inclines towards linear speed up as the system grows.

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6

Duan, Zhi Jian. "Parallel Algorithm for Solving Block-Tridiagonal Linear Systems." Applied Mechanics and Materials 427-429 (September 2013): 2420–23. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.2420.

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Efficient parallel iterative algorithm is investigated for solving block-tridiagonal linear systems on distributed-memory multi-computers. Based on Galerkin theory, the communication only need twice between the adjacent processors per iteration step. Furthermore, the condition for convergence is given when the coefficient matrix A is a symmetric positive definite matrix. Numerical experiments implemented on the cluster verify that our algorithm parallel acceleration rates and efficiency are higher than the multisplitting one, and has the advantages over the multisplitting method of high efficiency and low memory space.

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7

El-Mikkawy, Moawwad, and Faiz Atlan. "A novel algorithm for inverting a generalk-tridiagonal matrix." Applied Mathematics Letters 32 (June 2014): 41–47. http://dx.doi.org/10.1016/j.aml.2014.02.015.

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8

Fujiwara, Yasuhiro, Sekitoshi Kanai, Yasutoshi Ida, Atsutoshi Kumagai, and Naonori Ueda. "Fast algorithm for anchor graph hashing." Proceedings of the VLDB Endowment 14, no. 6 (February 2021): 916–28. http://dx.doi.org/10.14778/3447689.3447696.

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Anchor graph hashing is used in many applications such as cancer detection, web page classification, and drug discovery. It computes the hash codes from the eigenvectors of the matrix representing the similarities between data points and anchor points; anchors refer to the points representing the data distribution. In performing an approximate nearest neighbor search, the hash codes of a query data point are determined by identifying its closest anchor points. Anchor graph hashing, however, incurs high computation cost since (1) the computation cost of obtaining the eigenvectors is quadratic to the number of anchor points, and (2) the similarities of the query data point to all the anchor points must be computed. Our proposal, Tridiagonal hashing , increases the efficiency of anchor graph hashing because of its two advances: (1) we apply a graph clustering algorithm to compute the eigenvectors from the tridiagonal matrix obtained from the similarities between data points and anchor points, and (2) we detect anchor points closest to the query data point by using a dimensionality reduction approach. Experiments show that our approach is several orders of magnitude faster than the previous approaches. Besides, it yields high search accuracy than the original anchor graph hashing approach.

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9

Kumar, Surendra, Shashi, and Árpad Pethö. "An algorithm for the numerical inversion of a tridiagonal matrix." Communications in Numerical Methods in Engineering 9, no. 4 (April 1993): 353–59. http://dx.doi.org/10.1002/cnm.1640090409.

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10

MILOVANOVIĆ, E. I., M. D. MIHAJLOVIĆ, I. Ž. MILOVANOVIĆ, and M. K. STOJČEV. "SOLVING TRIDIAGONAL LINEAR SYSTEMS ON MIMD COMPUTERS." Parallel Processing Letters 04, no. 01n02 (June 1994): 53–64. http://dx.doi.org/10.1142/s0129626494000077.

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A partitioning algorithm for solving tridiagonal system of linear equations which is suitable for implementation on multiprocessor systems with small to moderate number of processors is described in this paper. The method is based on decomposing a matrix of order n×n into p partitions of size n×k, where n=pk. The fundamental idea of the proposed algorithm is that the elimination is performed using alternatively superdiagonal and subdiagonal elements, contrary to the Gaussian elimination which uses main diagonal elements. Performance results obtained when the proposed algorithm is implemented on the p-processor orthogonal and linear arrays, both of MIMD type, are presented.

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11

Issakhov, Alibek, and Bakytzan Zhumagulov. "Numerical Modelling of the Thermal Process in the Aquatic Environment." Advanced Materials Research 787 (September 2013): 669–74. http://dx.doi.org/10.4028/www.scientific.net/amr.787.669.

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This paper presents the mathematical model of the thermal influence to the aquatic environment of thermal power plant, which is solved by the Navier-Stokes and temperature equations for an incompressible fluid in a stratified medium. Numerical algorithm based on the projection method which solved with fractional step method. Three dimensional Poisson equation solved with Fourier method with combination of tridiagonal matrix method (Thomas algorithm).

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12

Lai, Jiangzhou, and Linzhang Lu. "Real Fast Structure-Preserving Algorithm for Eigenproblem of Complex Hermitian Matrices." Mathematical Problems in Engineering 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/438320.

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It is well known that the flops for complex operations are usually 4 times of real cases. In the paper, using real operations instead of complex, a real fast structure-preserving algorithm for eigenproblem of complex Hermitian matrices is given. We make use of the real symmetric and skew-Hamiltonian structure transformed by Wilkinson's way, focus on symplectic orthogonal similarity transformations and their structure-preserving property, and then reduce it into a two-by-two block tridiagonal symmetric matrix. Finally a real algorithm can be quickly obtained for eigenvalue problems of the original Hermitian matrix. Numerical experiments show that the fast algorithm can solve real complex Hermitian matrix efficiently, stably, and with high precision.

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13

Ferreira, Carla, and Beresford Parlett. "Convergence of LR algorithm for a one-point spectrum tridiagonal matrix." Numerische Mathematik 113, no. 3 (June 3, 2009): 417–31. http://dx.doi.org/10.1007/s00211-009-0238-2.

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14

Parker, J. T., P. A. Hill, D. Dickinson, and B. D. Dudson. "Parallel tridiagonal matrix inversion with a hybrid multigrid-Thomas algorithm method." Journal of Computational and Applied Mathematics 399 (January 2022): 113706. http://dx.doi.org/10.1016/j.cam.2021.113706.

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15

Fu, Xi Lian. "A Fast Algorithm for the Inverse of a Tridiagonal Period Matrices in Signal Processing." Advanced Materials Research 159 (December 2010): 469–76. http://dx.doi.org/10.4028/www.scientific.net/amr.159.469.

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The theory and method of matrix computation, as an important tool, have much important applications such as in computational mathematics, physics, image processing and recognition, missile system design, rotor bearing system, nonlinear kinetics, economics and biology etc. In this paper, Motivated by the references, especially [2], we give the estimates for the lower bounds on the inverse elements of strictly diagonally dominant tridiagonal period matrices.

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16

Swarztrauber, Paul N. "A parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix." Mathematics of Computation 60, no. 202 (May 1, 1993): 651. http://dx.doi.org/10.1090/s0025-5718-1993-1164126-4.

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17

Li, Kuiyuan. "A fully parallel method for tridiagonal eigenvalue problem." International Journal of Mathematics and Mathematical Sciences 17, no. 4 (1994): 741–52. http://dx.doi.org/10.1155/s0161171294001043.

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In this paper, a fully parallel method for finding all eigenvalues of a real matrix pencil(A,B)is given, whereAandBare real symmetric tridiagonal andBis positive definite. The method is based on the homotopy continuation coupled with the strategy ?Divide-Conquer? and Laguerre iterations. The numerical results obtained from implementation of this method on both single and multiprocessor computers are presented. It appears that our method is strongly competitive with other methods. The natural parallelism of our algorithm makes it an excellent candidate for a variety of advanced architectures.

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18

Iannelli, G. S., and A. J. Baker. "Accuracy and Efficiency Assessments for a Weak Statement CFD Algorithm for High-Speed Aerodynamics." Journal of Engineering for Gas Turbines and Power 116, no. 3 (July 1, 1994): 468–73. http://dx.doi.org/10.1115/1.2906844.

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A bilinear finite element, implicit Runge-Kutta space-time discretization has been established for an aerodynamics weak statement CFD algorithm. The algorithm admits real-gas effect simulation, for reliable hypersonic flow characterization, via an equilibrium reacting air model. The terminal algebraic system is solved using an efficient block-tridiagonal quasi-Newton linear algebra procedure that employs tensor matrix product factorizations within a lexicographic mesh-sweeping protocol. A block solution-adaptive remeshing, for totally arbitrary convex elements, is also utilized to facilitate accurate shock and/or boundary layer flow resolution. Numerical validations are presented for representative benchmark supersonic-hypersonic aerodynamics problem statements.

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19

Fan, Yan-Hong, Ling-Hui Wang, You Jia, Xing-Guo Li, Xue-Xia Yang, and Chih-Cheng Chen. "Investigation of High-Efficiency Iterative ILU Preconditioner Algorithm for Partial-Differential Equation Systems." Symmetry 11, no. 12 (November 28, 2019): 1461. http://dx.doi.org/10.3390/sym11121461.

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In this paper, we investigate an iterative incomplete lower and upper (ILU) factorization preconditioner for partial-differential equation systems. We discretize the partial-differential equations into linear equation systems. An iterative scheme of linear systems is used. The ILU preconditioners of linear systems are performed on the different computation nodes of multi-central processing unit (CPU) cores. Firstly, the preconditioner of general tridiagonal matrix equations is tested on supercomputers. Then, the effects of partial-differential equation systems on the speedup of parallel multiprocessors are examined. The numerical results estimate that the parallel efficiency is higher than in other algorithms.

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20

Krishna, H. "An asymptotically superior algorithm for computing the characteristic polynomial of a tridiagonal matrix." Proceedings of the IEEE 76, no. 10 (1988): 1393–94. http://dx.doi.org/10.1109/5.16343.

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21

Hadj, Ahmed Driss Aiat, and Mohamed Elouafi. "A fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix." Applied Mathematics and Computation 202, no. 2 (August 2008): 441–45. http://dx.doi.org/10.1016/j.amc.2008.02.026.

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22

Gärtner, Bernd, and Markus Sprecher. "A polynomial-time algorithm for the tridiagonal and Hessenberg P-matrix linear complementarity problem." Operations Research Letters 40, no. 6 (November 2012): 484–86. http://dx.doi.org/10.1016/j.orl.2012.08.013.

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23

Dall'Osso, Aldo. "An iterative back substitution algorithm for the solution of tridiagonal matrix systems with fringes." Journal of Computational and Applied Mathematics 169, no. 1 (August 2004): 87–99. http://dx.doi.org/10.1016/j.cam.2003.11.004.

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24

Ivanov, M. A. "An economic algorithm for solving systems of difference equations with a block-tridiagonal matrix." USSR Computational Mathematics and Mathematical Physics 26, no. 2 (January 1986): 84–86. http://dx.doi.org/10.1016/0041-5553(86)90013-3.

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25

A. Ale, Georges Adjibola, Emmanuel E. T. Olodo, Valery Doko, and Antoine Vianou. "Development of a Numerical Model for Vibration Analysis of Low Folded Shells under Dynamic Actions." Modern Applied Science 14, no. 4 (March 30, 2020): 111. http://dx.doi.org/10.5539/mas.v14n4p111.

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In this work a numerical model is developed for vibration analysis of low folded shells under dynamic actions. At first it is done to describe the used finite element discrete model based on Lagrange variational principles. To solve the eigen value problem of these structures a numerical algorithm is proposed using Householder's QR-iteration transformations. This method provides a tridiagonal matrix whose eigen values coincide with those of the initial matrix and significantly reduces the iteration number compared to the Lanczos method. Implementation of the method is carried out on seven folded shell mathematical models. Obtained results show that accuracy can be improved and computational time can be significantly reduced compared to the methods available in the technical literature for this class of problems.

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26

Reuter, Matthew G., and Judith C. Hill. "An efficient, block-by-block algorithm for inverting a block tridiagonal, nearly block Toeplitz matrix." Computational Science & Discovery 5, no. 1 (July 31, 2012): 014009. http://dx.doi.org/10.1088/1749-4699/5/1/014009.

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27

Zhou, Daming, Fei Gao, Elena Breaz, Alexandre Ravey, and Abdellatif Miraoui. "Tridiagonal Matrix Algorithm for Real-Time Simulation of a Two-Dimensional PEM Fuel Cell Model." IEEE Transactions on Industrial Electronics 65, no. 9 (September 2018): 7106–18. http://dx.doi.org/10.1109/tie.2017.2787598.

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28

Climent, Joan-Josep, Leandro Tortosa, and Antonio Zamora. "A BSP Recursive Divide and Conquer Algorithm to Compute the Inverse of a Tridiagonal Matrix." Journal of Parallel and Distributed Computing 59, no. 3 (December 1999): 423–44. http://dx.doi.org/10.1006/jpdc.1999.1586.

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29

Maeda, Kazuki, and Satoshi Tsujimoto. "A generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system." Journal of Computational and Applied Mathematics 300 (July 2016): 134–54. http://dx.doi.org/10.1016/j.cam.2015.12.032.

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30

Zhai, Shuying, Xinlong Feng, and Zhifeng Weng. "New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/246025.

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Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with ordersO(τmin(1+α,2−α)+h4)andO(τ2−α+h4)are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.

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31

Zhumagulov, Bakhytzhan, Alibek Issakhov, and Askar Khikmetov. "Numerical Simulation of Unstable Stratified Turbulent Flow in Open Channels." International Journal of Nonlinear Sciences and Numerical Simulation 16, no. 5 (August 1, 2015): 221–27. http://dx.doi.org/10.1515/ijnsns-2015-0074.

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AbstractThis paper considers the unstable stratified turbulent flow in an open channel. A mathematical model of unstable stratified turbulent flow is introduced, which allows to assess the mean and fluctuation characteristics of the turbulent flow. The numerical algorithm is developed for solving this problem. A numerical method is based on the projection method, which divides the problem into three stages. At the first stage, it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by fractional steps method. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). Finally, at the third stage, it is expected that the transfer is only due to the pressure gradient. The simulation results are in satisfactory agreement with the experimental data.

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32

Najim, Monssif, M’barek Feddaoui, Abderrahman Nait Alla, and Adil Charef. "Comparative numerical study of single and two-phase models of nanofluid liquid film evaporation in a vertical channel." MATEC Web of Conferences 307 (2020): 01034. http://dx.doi.org/10.1051/matecconf/202030701034.

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The main purpose of this study is to survey numerically comparison of two-phase and single-phase models of heat and mass transfer of Al2O3-water nanofluid liquid film flowing downward a vertical channel. A finite difference method is developed to produce the computational predictions for heat and mass transfer during the evaporation of the liquid film approached by the single-phase and two-phase models. The model solves the coupled governing equations in both nanofluid and gas phases together with the boundary and interfacial conditions. The systems of equations obtained by using an implicit finite difference method are solved by Tridiagonal Matrix Algorithm. The results show that the two-phase model is more realistic since it takes into account the thermophoresis and Brownian effects.

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33

Kudryavtsev, Oleg. "Finite Difference Methods for Option Pricing under Lévy Processes: Wiener-Hopf Factorization Approach." Scientific World Journal 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/963625.

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In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.

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34

Monroy-Loperena, Rosendo. "Simulation of Multicomponent Multistage Vapor−Liquid Separations. An Improved Algorithm Using the Wang−Henke Tridiagonal Matrix Method." Industrial & Engineering Chemistry Research 42, no. 1 (January 2003): 175–82. http://dx.doi.org/10.1021/ie0108898.

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35

Azim, M. A. "Isothermal free jets in high-temperature surroundings." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225, no. 8 (May 16, 2011): 1913–18. http://dx.doi.org/10.1177/0954406211401488.

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Two types of isothermal free jets, named positively and negatively buoyant, have been studied numerically to discern the effect of surrounding temperatures on their flow dynamics. Turbulence closure in those jets was achieved by standard k - ε model. The governing equations were solved using Implicit θ-Scheme and Tridiagonal Matrix Algorithm. Calculations were made for the jets having constant temperature at 20 °C and by varying surrounding temperatures from 20°C to 1000°C. It is clear that negatively buoyant jets but not the positively buoyant jets are nearly invariant to the change in surrounding temperatures compared to non-buoyant jet. Change in fluid dynamical behaviour of positively buoyant jets due to surrounding temperature change seems promising as it may offer the advantages of fuel jets in high-temperature air combustion.

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36

KARLSSON, HANS O. "LANCZOS ALGORITHMS AND CROSS CORRELATION FUNCTIONS Cif(E)." Journal of Theoretical and Computational Chemistry 02, no. 04 (December 2003): 523–35. http://dx.doi.org/10.1142/s0219633603000719.

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Cross correlation (CC) functions Cif(E) play an important role in chemical physics. They appear in the description of reactive scattering, photo-dissociation, photo-electron spectroscopy and electron transfer to mention a few. In this paper, we discuss two methods based on the Lanczos algorithm to compute the CC function for several initial and final states at the same time, without diagonalization. The property of the coupled two-term recursions variant of the Lanczos algorithm that yields a decomposition [Formula: see text] of the tridiagonal Lanczos matrix is crucial. The first method, the quasi minimal-recursive residue generation method (QM-RRGM) is based on solving a set of linear equations whereas the second method is based on a band-Lanczos method. The computational cost is of the same order of magnitude for both methods and is given by the number of matrix-vector multiplications in the underlying Lanczos method. Only a small set of scalars needs to be updated each recursion. The methods are compared for a model problem, the continuum resonance Raman cross section for a collinear model of CH2IBr . Both methods shows similar convergence properties. By adding a pre-conditioner, the rate of convergence can be increased dramatically.

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Hartman, David, and Milan Hladik. "REGULARITY RADIUS: PROPERTIES, APPROXIMATION AND A NOT A PRIORI EXPONENTIAL ALGORITHM." Electronic Journal of Linear Algebra 33 (May 16, 2018): 122–36. http://dx.doi.org/10.13001/1081-3810.3749.

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The radius of regularity, sometimes spelled as the radius of nonsingularity, is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. There are basically two approaches to handle this situation. Firstly, approximation algorithms are applied and secondly, tighter bounds for radius of regularity are considered. Improvements of both approaches have been recently shown by Hartman and Hlad\'{i}k (doi:10.1007/978-3-319-31769-4\_9) utilizing relaxation of the radius computation to semidefinite programming. An estimation of the regularity radius using any of the above mentioned approaches is usually applied to general matrices considering none or just weak assumptions about the original matrix. Surprisingly less explored area is represented by utilization of properties of special classes of matrices as well as utilization of classical algorithms extended to be used to compute the considered radius. This work explores a process of regularity radius analysis and identifies useful properties enabling easier estimation of the corresponding radius values. At first, checking finiteness of this characteristic is shown to be a polynomial problem along with determining a sharp upper bound on the number of nonzero elements of the matrix to obtain infinite radius. Further, relationship between maximum (Chebyshev) norm and spectral norm is used to construct new bounds for the radius of regularity. Considering situations where the known bounds are not tight enough, a new method based on Jansson-Rohn algorithm for testing regularity of an interval matrix is presented which is not a priory exponential along with numerical experiments. For a situation where an input matrix has a special form, several corresponding results are provided such as exact formulas for several special classes of matrices, e.g., for totally positive and inverse non-negative, or approximation algorithms, e.g., rank-one radius matrices. For tridiagonal matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design a polynomial algorithm to compute the radius of regularity.

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38

Issakhov, Alibek. "Numerical Modeling of Influence of the Thermal Power Plant with Considering the Hydrometeorological Condition at the Reservoir – Cooler." International Journal of Energy Optimization and Engineering 3, no. 2 (April 2014): 1–16. http://dx.doi.org/10.4018/ijeoe.2014040101.

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This paper presents the mathematical model of the thermal power plant in cooling pond under different hydrometeorological conditions, which is solved by three dimensional Navier - Stokes equations and temperature equation for an incompressible fluid in a stratified medium. A numerical method based on the projection method, which divides the problem into three stages. At the first stage it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by method of fractional steps. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). Finally at the third stage it is expected that the transfer is only due to the pressure gradient. To increase the order of approximation compact scheme was used. Then qualitatively and quantitatively approximate the basic laws of the hydrothermal processes depending on different hydrometeorological conditions are determined.

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39

Sabinin, Vladimir. "ON INCOMPLETE FACTORIZATION IMPLICIT TECHNIQUE FOR 2D ELLIPTIC FD EQUATIONS." Mathematical Modelling and Analysis 25, no. 1 (January 13, 2020): 37–52. http://dx.doi.org/10.3846/mma.2020.8485.

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A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions.

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40

Artichowicz, Wojciech, and Dariusz Gąsiorowski. "Computationally Efficient Solution of a 2D Diffusive Wave Equation Used for Flood Inundation Problems." Water 11, no. 10 (October 22, 2019): 2195. http://dx.doi.org/10.3390/w11102195.

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This paper presents a study dealing with increasing the computational efficiency in modeling floodplain inundation using a two-dimensional diffusive wave equation. To this end, the domain decomposition technique was used. The resulting one-dimensional diffusion equations were approximated in space with the modified finite element scheme, whereas time integration was carried out using the implicit two-level scheme. The proposed algorithm of the solution minimizes the numerical errors and is unconditionally stable. Consequently, it is possible to perform computations with a significantly greater time step than in the case of the explicit scheme. An additional efficiency improvement was achieved using the symmetry of the tridiagonal matrix of the arising system of nonlinear equations, due to the application of the parallelization strategy. The computational experiments showed that the proposed parallel implementation of the implicit scheme is very effective, at about two orders of magnitude with regard to computational time, in comparison with the explicit one.

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Sari, Murat. "Solution of the Porous Media Equation by a Compact Finite Difference Method." Mathematical Problems in Engineering 2009 (2009): 1–13. http://dx.doi.org/10.1155/2009/912541.

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Accurate solutions of the porous media equation that usually occurs in nonlinear problems of heat and mass transfer and in biological systems are obtained using a compact finite difference method in space and a low-storage total variation diminishing third-order Runge-Kutta scheme in time. In the calculation of the numerical derivatives, only a tridiagonal band matrix algorithm is encountered. Therefore, this scheme causes to less accumulation of numerical errors and less use of storage space. The computed results obtained by this way have been compared with the exact solutions to show the accuracy of the method. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. This method is seen to be a very good alternative method to some existing techniques for such realistic problems.

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Xu, Yongze, Yu Liu, and Lei Zhang. "Effects of the Solder Phase Transformation on the Optimization of Reflow Soldering Parameters and Temperature Profiles." Discrete Dynamics in Nature and Society 2021 (June 16, 2021): 1–19. http://dx.doi.org/10.1155/2021/9955967.

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In this study, a heat convection model of the reflow oven and a heat conduction model of the soldering area are proposed based on heat transfer theory, and a dynamic Thomas algorithm is developed for solving linear equations with coefficient matrix evolving over time in the tridiagonal system, which is derived from a heat transfer problem with moving boundaries in the solder phase transition process. We have also carried out numerical simulations for investigating the accuracy of the mathematical model, in which the temperature profiles are calculated and compared for different cases with considering or ignoring phase transformations, respectively. Parameters of reflow soldering, such as the conveyor speed, the set temperature in each zone, and a part of the heating factor, are optimized by the use of the nondominated sorting genetic algorithm II. By comparing the temperature profile and optimal solutions in the two cases, numerical results show that phase transitions of the solder have great impacts on optimal parameters and the slope of temperature profiles. Moreover, the phenomenon that the heating factor varies with the maximum set temperature in a banded distribution is investigated and analyzed, which is an important part of this work.

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BOLSTAD, JOHN H. "FOLLOWING PATHS OF SYMMETRY-BREAKING BIFURCATION POINTS." International Journal of Bifurcation and Chaos 02, no. 03 (September 1992): 559–76. http://dx.doi.org/10.1142/s0218127492000707.

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We propose a pseudo-arclength continuation algorithm for computing paths of Z 2-symmetry-breaking bifurcation points for two-parameter nonlinear elliptic problems. The algorithm consists of an Euler predictor step and a solution step composed of a sequence of Newton iterations. This work generalizes the algorithm of Werner and Spence for locating a one-parameter symmetry-breaking bifurcation point by using the approach of Keller and Fier for following a (two-parameter) path of limit points (a "fold"). By repeated use of the bordering algorithm, we solve linear systems whose matrix is the "symmetric" Jacobian or "antisymmetric" Jacobian, thus fully exploiting any (block tridiagonal) structures present. We give numerical results for the steady, axisymmetric flow between rotating coaxial cylinders (Taylor–Couette flow). For finite cylinders, we compute the fold curve and path of symmetry-breaking bifurcation points for small aspect ratios, and illustrate a new method to accurately locale the two Z 2-symmetric codimension one singularities. For infinite cylinders, we show the projections on the (aspect ratio, Reynolds number) plane of the folds and bifurcation point paths in the neighborhood of the two-cell/four-cell neutral curve crossing. We numerically verify a conjecture of Meyer–Spasche and Wagner concerning the connection of two neutral-curve crossings by a path of secondary subharmonic bifurcations.

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Dongarra, J. J., G. A. Geist, and C. H. Romine. "Algorithm 710: FORTRAN subroutines for computing the eigenvalues and eigenvectors of a general matrix by reduction to general tridiagonal form." ACM Transactions on Mathematical Software 18, no. 4 (December 1992): 392–400. http://dx.doi.org/10.1145/138351.138352.

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Issakhov, Alibek. "Mathematical Modelling of Thermal Process to Aquatic Environment with Different Hydrometeorological Conditions." Scientific World Journal 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/678095.

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This paper presents the mathematical model of the thermal process from thermal power plant to aquatic environment of the reservoir-cooler, which is located in the Pavlodar region, 17 Km to the north-east of Ekibastuz town. The thermal process in reservoir-cooler with different hydrometeorological conditions is considered, which is solved by three-dimensional Navier-Stokes equations and temperature equation for an incompressible flow in a stratified medium. A numerical method based on the projection method, divides the problem into three stages. At the first stage, it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by fractional steps method. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). Finally, at the third stage, it is expected that the transfer is only due to the pressure gradient. Numerical method determines the basic laws of the hydrothermal processes that qualitatively and quantitatively are approximated depending on different hydrometeorological conditions.

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Юлдашев, А. В., Н. В. Репин, and В. В. Спеле. "A parallel preconditioner based on the approximation of an inverse matrix by power series for solving sparse linear systems on graphics processors." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 4 (September 10, 2019): 444–56. http://dx.doi.org/10.26089/nummet.v20r439.

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Рассмотрена применимость метода AIPS, аппроксимирующего обратную матрицу на основе степенного разложения в ряд Неймана, в рамках двухступенчатого предобусловливателя CPR. Предложен ориентированный на архитектуру CUDA параллельный алгоритм решения линейных систем с трехдиагональной матрицей, состоящей из независимых блоков различного размера. Показано, что реализация предложенного алгоритма может более чем в 2 раза превосходить по быстродействию функции решения трехдиагональных систем из библиотеки cuSPARSE. Проведено тестирование метода BiCGStab с предобусловливателем CPRAIPS на современных GPU, в том числе на гибридной вычислительной системе с 4 GPU NVIDIA Tesla V100, показавшее приемлемую масштабируемость данного предобусловливателя, а также возможность ускорить решение линейных систем, характерных для задачи гидродинамического моделирования нефтегазовых месторождений, по сравнению с CPRAMG. The applicability of the AIPS method approximating an inverse matrix using Neumann series is considered in the framework of the CPR two stage preconditioner. A parallel CUDAoriented algorithm is proposed for solving linear systems with tridiagonal matrices consisting of independent blocks of different sizes. It is shown that the implementation of the proposed algorithm can be more than twice the speed of the similar functions from the cuSPARSE library. Experimental evaluation of the BiCGStab method with the CPRAIPS preconditioner on modern GPUs, including a hybrid computing system with 4 GPU NVIDIA Tesla V100, is performed. Numerical experiments show an adequate scalability of this preconditioner as well as the possibility (compared to the CPRAMG) to accelerate the solution of linear systems being typical for the reservoir modeling problems.

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Allet, A., and I. Paraschivoiu. "Viscous Flow and Dynamic Stall Effects on Vertical-Axis Wind Turbines." International Journal of Rotating Machinery 2, no. 1 (1995): 1–14. http://dx.doi.org/10.1155/s1023621x95000157.

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The present paper describes a numerical method, aimed to simulate the flow field of vertical-axis wind turbines, based on the solution of the steady, incompressible, laminar Navier-Stokes equations in cylindrical coordinates. The flow equations, written in conservation law form, are discretized using a control volume approach on a staggered grid. The effect of the spinning blades is simulated by distributing a time-averaged source terms in the ring of control volumes that lie in the path of turbine blades. The numerical procedure used here, based on the control volume approach, is the widely known “SIMPLER” algorithm. The resulting algebraic equations are solved by the TriDiagonal Matrix Algorithm (TDMA) in the r- and z-directions and the Cyclic TDMA in the 0-direction. The indicial model is used to simulate the effect of dynamic stall at low tip-speed ratio values. The viscous model, developed here, is used to predict aerodynamic loads and performance for the Sandia 17-m wind turbine. Predictions of the viscous model are compared with both experimental data and results from the CARDAAV aerodynamic code based on the Double-Multiple Streamtube Model. According to the experimental results, the analysis of local and global performance predictions by the 3D viscous model including dynamic stall effects shows a good improvement with respect to previous 2D models.

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Farooq, M. Asif, A. Salahuddin, Asif Mushtaq, and M. Razzaq. "A Simplified Finite Difference Method (SFDM) Solution via Tridiagonal Matrix Algorithm for MHD Radiating Nanofluid Flow over a Slippery Sheet Submerged in a Permeable Medium." Mathematical Problems in Engineering 2021 (January 27, 2021): 1–17. http://dx.doi.org/10.1155/2021/6628009.

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In this paper, we turn our attention to the mathematical model to simulate steady, hydromagnetic, and radiating nanofluid flow past an exponentially stretching sheet. A numerical modeling technique, simplified finite difference method (SFDM), has been applied to the flow model that is based on partial differential equations (PDEs) which is converted to nonlinear ordinary differential equations (ODEs) by using similarity variables. For the resultant algebraic system, the SFDM uses the tridiagonal matrix algorithm (TDMA) in computing the solution. The effectiveness of numerical scheme is verified by comparing it with solution from the literature. However, where reference solution is not available, one can compare its numerical results with the results of MATLAB built-in package bvp4c. The velocity, temperature, and concentration profiles are graphed for a variety of parameters, i.e., Prandtl number, Grashof number, thermal radiation parameter, Darcy number, Eckert number, Lewis number, and Brownian and thermophoresis parameters. The significant effects of the associated emerging thermophysical parameters, i.e., skin friction coefficient, local Nusselt number, and local Sherwood numbers are analyzed and discussed in detail. Numerical results are compared from the available literature and found a close agreement with each other. It is found that the Eckert number upsurges the velocity curve. However, the dimensionless temperature declines with the Grashof number. It is also shown that the SFDM gives good results when compared with the results obtained from bvp4c and results from the literature.

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Koshy, Ranjana, and Ausif Mahmood. "Optimizing Deep CNN Architectures for Face Liveness Detection." Entropy 21, no. 4 (April 20, 2019): 423. http://dx.doi.org/10.3390/e21040423.

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Face recognition is a popular and efficient form of biometric authentication used in many software applications. One drawback of this technique is that it is prone to face spoofing attacks, where an impostor can gain access to the system by presenting a photograph of a valid user to the sensor. Thus, face liveness detection is a necessary step before granting authentication to the user. In this paper, we have developed deep architectures for face liveness detection that use a combination of texture analysis and a convolutional neural network (CNN) to classify the captured image as real or fake. Our development greatly improved upon a recent approach that applies nonlinear diffusion based on an additive operator splitting scheme and a tridiagonal matrix block-solver algorithm to the image, which enhances the edges and surface texture in the real image. We then fed the diffused image to a deep CNN to identify the complex and deep features for classification. We obtained 100% accuracy on the NUAA Photograph Impostor dataset for face liveness detection using one of our enhanced architectures. Further, we gained insight into the enhancement of the face liveness detection architecture by evaluating three different deep architectures, which included deep CNN, residual network, and the inception network version 4. We evaluated the performance of each of these architectures on the NUAA dataset and present here the experimental results showing under what conditions an architecture would be better suited for face liveness detection. While the residual network gave us competitive results, the inception network version 4 produced the optimal accuracy of 100% in liveness detection (with nonlinear anisotropic diffused images with a smoothness parameter of 15). Our approach outperformed all current state-of-the-art methods.

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Frandsen, Gudmund Skovbjerg, and Piotr Sankowski. "Dynamic Normal Forms and Dynamic Characteristic Polynomial." BRICS Report Series 15, no. 2 (April 12, 2008). http://dx.doi.org/10.7146/brics.v15i2.21937.

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We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case our algorithm supports rank-one updates in O(n^2 log n) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n^2 k log n) randomized time, where k is the number of invariant factors of the matrix. The algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the Frobenius normal form or the tridiagonal symmetric form. The algorithm can be extended to solve the matrix eigenproblem with relative error 2^{-b} in additional O(n log^2 n log b) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm the hardness of the problem is studied. For the symmetric case we present an Omega(n^2) lower bound for rank-one updates and an Omega(n) lower bound for element updates.

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