scholarly journals ANALYSIS OF NUMERICAL SOLUTION OF POISSON EQUATION BY ILU (0)-CG METHOD

2021 ◽  
Vol 2 (336) ◽  
pp. 74-82
Author(s):  
A. A. Abdurakhimova ◽  
N. M. Kassymbek ◽  
O. Zh. Mamyrbayev
Keyword(s):  
Iterative Methods ◽  
Poisson Equation ◽  
Linear Equations ◽  
Inverse Matrix ◽  
Linear Matrix ◽  
Main Question ◽  
Lu Decomposition ◽  
System Matrix ◽  
The Matrix ◽  
Number Of Iterations

The problem of generalization of the method is the main question that arises when studying the quality of iterative methods. The efficiency of solving systems using iterative methods directly depends on the assumptions about the system of equations to be solved. Prerequisites are used to provide a more efficient solution. Many types of prerequisites are currently known, for example, prerequisites based on the approximation of the system matrix: ILU, IQR, and ILQ; Prerequisites based on the approximation of the inverse matrix: a polynomial, rarely filled approximation of the inverse matrix (for example, AINV), an approximation in the factorized form of the inverse matrix (for example, FSAI, SPAI, etc.). This article analyzes the CG and CG methods with the preconditioner ILU (0) by the example of solving the two-dimensional Poisson equation. The CG method is usually used to solve any system of linear equations. ILU (0) was selected as a prerequisite for the article. The incomplete LU decomposition (ILU (0)) is an efficient precursor and is easily implemented. This suggests a system that can be solved to speed up the accumulation of CG and other iterative methods, that is, to reduce the number of iterations. The ILU (0) preconditioner is very easy to detect using the LU decomposition. Since the linear matrix was rarely filled, the CSR format was used to store the matrix in memory. ILU (0) + CG, i.e. the algorithm with a precondition, was assembled 5-8 times faster than the CG algorithm. Data on the number of iterations of convergence of the method without a preconditioner and with the ILU(0) preconditioner were obtained and analyzed.

scholarly journals Spectral properties of discrete models of multi-dimensional elliptic problems with mixed derivatives

2019 ◽  
Vol 55 (2) ◽  
pp. 207-215
Author(s):  
A. U. Prakonina
Keyword(s):  
Convergence Rate ◽  
Iterative Methods ◽  
Imaginary Part ◽  
Conjugate Gradient ◽  
Discrete Models ◽  
Lu Factorization ◽  
The Matrix ◽  
Mixed Derivatives ◽  
Incomplete Lu Factorization ◽  
Number Of Iterations

The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

scholarly journals Influence of Preconditioning and Blocking on Accuracy in Solving Markovian Models

2009 ◽  
Vol 19 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Beata Bylina ◽  
Jarosław Bylina
Keyword(s):  
Markov Chains ◽  
Iterative Methods ◽  
Linear Equations ◽  
Direct Methods ◽  
Factorization Method ◽  
Markovian Models ◽  
The Matrix ◽  
Gauss Elimination ◽  
Systems Of Linear Equations ◽  
The Impact

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.

scholarly journals On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

2021 ◽  
Vol 31 (4) ◽  
pp. 651-667
Author(s):  
B.Kh. Turmetov ◽  
V.V. Karachik
Keyword(s):  
Integral Representation ◽  
Boundary Value Problems ◽  
Explicit Form ◽  
Poisson Equation ◽  
Green's Function ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Inverse Matrix ◽  
The Matrix ◽  
Neumann Boundary Value Problems

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

scholarly journals Design and Implementation of MIPS Processor for LU Decomposition based on FPGA

2020 ◽  
Vol 4 (3) ◽  
Author(s):  
Rusul Saad Khalil ◽  
Safaa S. Omran
Keyword(s):  
Integrated Circuit ◽  
High Speed ◽  
Linear Equations ◽  
Lu Decomposition ◽  
Gate Arrays ◽  
The Matrix ◽  
Field Programmable ◽  
Programmable Gate Arrays ◽  
Hardware Description ◽  
Lower Triangle

The solution for a set of liner equations require to find the matrix inverse of a square matrix with same number of the linear equations, this operation require many mathematical calculations. To solve this problem, LU decomposition for the matrix is used, which computes two matrices, a lower triangle matrix and an upper triangle matrix. In this, paper a design for 32-bits MIPS (microprocessor without interlocked pipelined stages) processor with the required instructions that used to calculate the LU matrices. The design implemented using VHDL (Very high speed integrated circuit hardware description language) then integrated with FPGA (Field Programmable Gate Arrays) Xilinx Spartan 6. The results for the different parts of the processor are resented in the form of test bench waveform and the architecture of the system is demonstrated and the results was matched with theoretical results.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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