The Recursive Algorithms of Yule-Walker Equation in Generalized Stationary Prediction

2013 ◽  
Vol 756-759 ◽  
pp. 3070-3073 ◽  
Author(s):  
Er Yan Zhang ◽  
Xiao Feng Zhu
Keyword(s):  
Toeplitz Matrix ◽  
Gaussian Elimination ◽  
Recursive Algorithm ◽  
Toeplitz Matrices ◽  
Recursive Formula ◽  
Coefficient Matrix ◽  
Recursive Algorithms ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Order Of Magnitude

Toeplitz matrix arises in a remarkable variety of applications such as signal processing, time series analysis, image processing. Yule-Walker equation in generalized stationary prediction is linear algebraic equations that use Toeplitz matrix as coefficient matrix. Making better use of the structure of Toeplitz matrix, we present a recursive algorithm of linear algebraic equations from by using Toeplitz matrix as coefficient matrix , and also offer the proof of the recursive formula. The algorithm, making better use of the structure of Toeplitz matrices, effectively reduces calculation cost. For n-order Toeplitz coefficient matrix, the computational complexity of usual Gaussian elimination is about , while this algorithm is about , decreasing of one order of magnitude.

scholarly journals Direct Methods for Sparse Matrices

1980 ◽  
Vol 9 (123) ◽  
Author(s):  
Ole Østerby ◽  
Zahari Zlatev
Keyword(s):  
Sparse Matrices ◽  
Gaussian Elimination ◽  
Computation Time ◽  
Coefficient Matrix ◽  
Iterative Refinement ◽  
Computational Scheme ◽  
Algebraic Equations ◽  
Large Drop ◽  
Linear Algebraic Equations ◽  
Sparse Problems

<p>The mathematical models of many practical problems lead to systems of linear algebraic equations where the coefficient matrix is large and sparse. Typical examples are the solutions of partial differential equations by finite difference or finite element methods but many other applications could be mentioned.</p><p>When there is a large proportion of zeros in the coefficient matrix then it is fairly obvious that we do not want to store all those zeros in the computer, but it might not be quite so obvious how to get around it. We first describe storage techniques which are convenient to use with direct solution methods, and we then show how a very efficient computational scheme can be based on Gaussian elimination and iterative refinement.</p><p>A serious problem in the storage and handling of sparse matrices is the appearance of fill-ins, i.e. new elements which are created in the process of generating zeros below the diagonal. Many of these new elements tend to be smaller than the original matrix elements, and if they are smaller than a quantity which we shall call the drop tolerance we simply ignore them. In this way we may preserve the sparsity quite well but we probably introduce rather large errors in the LU decomposition to the effect that the solution becomes unacceptable. In order to retrieve the accuracy we use iterative refinement and we show theoreticaly and with practical experiments that it is ideal for the purpose.</p><p>Altogether, the combination of Gaussian elimination, a large drop tolerance, and iterative refinement gives a very efficient and competitive computational scheme for sparse problems. For dense matrices iterative refinement will always require more storage and computation time, and the extra accuracy it yields may not be enough to justify it. For sparse problems, however, iterative refinement combined with a large drop tolerance will in most cases give very accurate results and reliable error estimates with less storage and computation time.</p>

scholarly journals A Simple Algorithm for Finding a Non-negative Basic Solution of a System of Linear Algebraic Equations

2021 ◽  
Vol 28 (3) ◽  
pp. 234-237
Author(s):  
Gleb D. Stepanov
Keyword(s):  
Simplex Method ◽  
Gaussian Elimination ◽  
Simple Algorithm ◽  
Basic Solution ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Linear Programming Problems ◽  
Two Stages ◽  
Computational Resources

This article describes an algorithm for obtaining a non-negative basic solution of a system of linear algebraic equations. This problem, which undoubtedly has an independent interest, in particular, is the most time-consuming part of the famous simplex method for solving linear programming problems.Unlike the artificial basis Orden’s method used in the classical simplex method, the proposed algorithm does not attract artificial variables and economically consumes computational resources.The algorithm consists of two stages, each of which is based on Gaussian exceptions. The first stage coincides with the main part of the Gaussian complete exclusion method, in which the matrix of the system is reduced to the form with an identity submatrix. The second stage is an iterative cycle, at each of the iterations of which, according to some rules, a resolving element is selected, and then a Gaussian elimination step is performed, preserving the matrix structure obtained at the first stage. The cycle ends either when the absence of non-negative solutions is established, or when one of them is found.Two rules for choosing a resolving element are given. The more primitive of them allows for ambiguity of choice and does not exclude looping (but in very rare cases). Use of the second rule ensures that there is no looping.

scholarly journals Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs

2019 ◽  
Vol 214 ◽  
pp. 05004
Author(s):  
Milena Veneva ◽  
Alexander Ayriyan
Keyword(s):  
Performance Analysis ◽  
High Performance ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Performance Study ◽  
Band Matrix ◽  
Linear Algebraic Equations ◽  
The Individual ◽  
Performance Computing ◽  
Symbolic Algorithms

This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed.

scholarly journals ROTATION METHOD FOR RECONSTRUCTING PROCESS AND FIELD FROM IMPERFECT DATA

2004 ◽  
Vol 14 (08) ◽  
pp. 2991-2997 ◽  
Author(s):  
PETER C. CHU ◽  
LEONID M. IVANOV ◽  
TATYANA M. MARGOLINA
Keyword(s):  
Condition Number ◽  
A Priori ◽  
Coefficient Matrix ◽  
Minimum Condition ◽  
Lorenz Attractor ◽  
Algebraic Equations ◽  
Signal Ratio ◽  
Imperfect Data ◽  
Linear Algebraic Equations ◽  
Noise Statistics

Reconstruction of processes and fields from noisy data is to solve a set of linear algebraic equations. Three factors affect the accuracy of reconstruction: (a) a large condition number of the coefficient matrix, (b) high noise-to-signal ratio in the source term, and (c) no a priori knowledge of noise statistics. To improve reconstruction accuracy, the set of linear algebraic equations is transformed into a new set with minimum condition number and noise-to-signal ratio using the rotation matrix. The procedure does not require any knowledge of low-order statistics of noises. Several examples including highly distorted Lorenz attractor illustrate the benefit of using this procedure.

scholarly journals A Fast Optimization Algorithm of FEM/BEM Simulation for Periodic Surface Acoustic Wave Structures

Information ◽  
2019 ◽  
Vol 10 (3) ◽  
pp. 90
Author(s):  
Honglang Li ◽  
Zixiao Lu ◽  
Yabing Ke ◽  
Yahui Tian ◽  
Wei Luo
Keyword(s):  
Acoustic Wave ◽  
Surface Acoustic Wave ◽  
Toeplitz Matrix ◽  
Recursive Algorithm ◽  
Coefficient Matrix ◽  
Time Cost ◽  
Accurate Analysis ◽  
Periodic Surface ◽  
Speed Up ◽  
Element Method

The accurate analysis of periodic surface acoustic wave (SAW) structures by combined finite element method and boundary element method (FEM/BEM) is important for SAW design, especially in the extraction of couple-of-mode (COM) parameters. However, the time cost is very large. With the aim to accelerate the calculation of SAW FEM/BEM analysis, some optimization algorithms for the FEM and BEM calculation have been reported, while the optimization for the solution to the final FEM/BEM equations which is also with a large amount of calculation is hardly reported. In this paper, it was observed that the coefficient matrix of the final FEM/BEM equations for the periodic SAW structures was similar to a Toeplitz matrix. A fast algorithm based on the Trench recursive algorithm for the Toeplitz matrix inversion was proposed to speed up the solution of the final FEM/BEM equations. The result showed that both the time and memory cost of FEM/BEM was reduced furtherly.

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

2013 ◽  
Vol 11 (8) ◽  
Author(s):  
Zahari Zlatev ◽  
Krassimir Georgiev
Keyword(s):  
Iterative Methods ◽  
Original Problem ◽  
Krylov Subspace ◽  
Gaussian Elimination ◽  
Algebraic Equations ◽  
Computational Process ◽  
Science And Engineering ◽  
Linear Algebraic Equations ◽  
Subspace Algorithms ◽  
Reliable Methods

AbstractMany problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.

Calculation of the Blade-to-Blade Compressible Flow Field in Turbo Impellers Using the Finite-Element Method

1977 ◽  
Vol 19 (3) ◽  
pp. 108-112 ◽  
Author(s):  
D. Adler ◽  
Y. Krimerman
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Approach ◽  
Iterative Procedure ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Variational Functional ◽  
Linear Algebraic Equations ◽  
Computer Storage

No variational principle can be found for Wu's blade-to-blade equation and therefore no appropriate variational functional associated with the problem can be derived. This difficulty is overcome by using a Poisson equation as the basis for an iterative procedure. Thus the method retains the advantage of the variational approach in which the coefficient matrix of the linear algebraic equations is always symmetric. The symmetry of the coefficient matrix allows reduction of computer storage.

Laminar Throughflow of Varying-Quality Steam Between Corotating Disks

1978 ◽  
Vol 100 (2) ◽  
pp. 194-200 ◽  
Author(s):  
C. R. Truman ◽  
W. Rice ◽  
D. F. Jankowski
Keyword(s):  
Saturated Vapor ◽  
Nonlinear Partial Differential Equations ◽  
Practical Interest ◽  
Mass Conservation ◽  
Local Analysis ◽  
Liquid Droplets ◽  
Algebraic Equations ◽  
Two Phase ◽  
Linear Algebraic Equations ◽  
Order Of Magnitude

A local analysis was made of the laminar throughflow between corotating disks of a Newtonian vapor containing liquid droplets. Such a flow is of practical interest in multiple-disk turbomachinery, and specifically is a model of a two-phase, single component flow (saturated vapor - saturated liquid steam) which would arise in geothermal applications of a multiple-disk turbine. Local mass conservation and momentum equations for the vapor and droplet fields were used, as well as mass conservation and energy equations for a single droplet. The vapor-droplet interaction was modeled by a drag force based on the well-known drag coefficients for flow past spheres. The governing equations were reduced to parabolic form by order-of-magnitude arguments to allow a marching-type solution. The nonlinear partial differential equations were replaced by nonlinear finite-difference equations, which were linearized. The resulting system of linear algebraic equations was solved directly. Various results are presented, including results for a set of flow parameters which are close to those applicable for multiple-disk turbines operating on geothermal steam.

Numerical Analysis of 2D Cloaking Problems Using Homogeneus Materials

2016 ◽  
Vol 685 ◽  
pp. 56-59 ◽  
Author(s):  
Gennady Alekseev ◽  
Alexey Lobanov ◽  
Yuliya Spivak
Keyword(s):  
Numerical Analysis ◽  
Singular Value Decomposition ◽  
Numerical Experiments ◽  
Scattering Problem ◽  
Coefficient Matrix ◽  
Singular Value ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Matrix Properties ◽  
Value Decomposition

We consider the problem of decreasing the scattering from arbitrary 2D object by surrounding it the shell composed of M layers of homogeneous anisotropic materials. The solution of the scattering problem under study is obtained by solving corresponding 2D Helmholtz equation using cylindrical functions expansion. The coefficients of these expansions are determined by solving system of 4M+3 linear algebraic equations. Efficient numerical algorithm of cloaking problem is developed which is based on singular value decomposition of the coefficient matrix. Properties of the algorithm are studied and the results of numerical experiments are discussed.

Sign in / Sign up

Export Citation Format

Share Document