scholarly journals Lanczos Bidiagonalization With Partial Reorthogonalization

1998 ◽  
Vol 27 (537) ◽  
Author(s):  
Rasmus Munk Larsen
Keyword(s):  
Eigenvalue Problems ◽  
Sparse Matrix ◽  
Singular Values ◽  
High Accuracy ◽  
Symmetric System ◽  
Lanczos Algorithm ◽  
Lanczos Bidiagonalization ◽  
Left And Right ◽  
Test Matrices ◽  
Matlab Package

A partial reorthogonalization procedure (BPRO) for maintaining semi-orthogonality among the left and right Lanczos vectors in the Lanczos bidiagonalization (LBD) is presented. The resulting algorithm is mathematically equivalent to the symmetric Lanczos algorithm with partial reorthogonalization (PRO) developed by Simon but works directly on the Lanczos bidiagonalization of A. For computing the singular values and vectors of a large sparse matrix with high accuracy, the BPRO algorithm uses only half the amount of storage and a factor of 3-4 less work compared to methods based on PRO applied to an equivalent symmetric system. Like PRO the algorithm presented here is based on simple recurrences which enable it to monitor the loss of orthogonality among the Lanczos vectors directly without forming inner products. These recurrences are used to develop a Lanczos bidiagonalization algorithm with partial reorthogonalization which has been implemented in a MATLAB package for sparse SVD and eigenvalue problems called PROPACK. Numerical experiments with the routines from PROPACK are conducted using a test problem from inverse helioseismology to illustrate the properties of the method. In addition a number of test matrices from the Harwell-Boeing collection are used to compare the accuracy and efficiency of the MATLAB implementations of BPRO and PRO with the svds routine in MATLAB 5.1, which uses an implicitly restarted Lanczos algorithm.

scholarly journals Almost Exact Computation of Eigenvalues in Approximate Differential Problems

2019 ◽  
Vol 24 (4) ◽  
pp. 96 ◽  
Author(s):  
José M. A. Matos ◽  
Maria João Rodrigues
Keyword(s):  
Eigenvalue Problems ◽  
High Accuracy ◽  
Second Step ◽  
Matlab Toolbox ◽  
Differential Problem ◽  
Algebraic Problem ◽  
Polynomial Expression ◽  
Computation Of Eigenvalues ◽  
Potential Use ◽  
Mathematics And Physics

Differential eigenvalue problems arise in many fields of Mathematics and Physics, often arriving, as auxiliary problems, when solving partial differential equations. In this work, we present a method for eigenvalues computation following the Tau method philosophy and using Tau Toolbox tools. This Matlab toolbox was recently presented and here we explore its potential use and suitability for this problem. The first step is to translate the eigenvalue differential problem into an algebraic approximated eigenvalues problem. In a second step, making use of symbolic computations, we arrive at the exact polynomial expression of the determinant of the algebraic problem matrix, allowing us to get high accuracy approximations of differential eigenvalues.

Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations

2015 ◽  
Vol 7 (4) ◽  
pp. 430-440 ◽  
Author(s):  
Xueying Zhang ◽  
Xin An ◽  
C. S. Chen
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Stokes Equations ◽  
Sparse Matrix ◽  
High Accuracy ◽  
Collocation Methods ◽  
Navier Stokes ◽  
Sparse Linear Systems ◽  
Navier Stokes Equations ◽  
Weight Coefficients

AbstractThe local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

A Sparse Matrix-based Method for Rapid Solving the Reynolds Equation

2021 ◽  
pp. 1-14
Author(s):  
Ke He ◽  
Shi Chen ◽  
Zhinan Zhang
Keyword(s):  
Computational Efficiency ◽  
Degrees Of Freedom ◽  
Reynolds Equation ◽  
Sparse Matrix ◽  
Elastohydrodynamic Lubrication ◽  
High Accuracy ◽  
Rapid Analysis ◽  
Long Time ◽  
The Finite Difference Method ◽  
The Difference

Abstract Due to the repeated iteration, the numerical method represented by the finite-difference method has the disadvantages of low computational efficiency and long time-consuming in solving the Reynolds equation. This paper proposed a new sparse matrix-based method to solve the difference Reynolds equation by replacing the pressure iterative process with the sparse matrix solver. Compared with the traditional iterative methods, the computational efficiency of this new method is about two orders of magnitude higher, and it shows high accuracy in different degrees of freedom. Two cases of aerostatic lubrication and elastohydrodynamic lubrication are used to illustrate the effectiveness of this method. This method can support the rapid analysis of fluid lubrication problems and lay the foundation for developing the lubrication calculation library.

Modification of the Craig-Bampton CMS Procedure for General Systems

1997 ◽  
Author(s):  
Bram de Kraker ◽  
Dick H. van Campen
Keyword(s):  
Physical Meaning ◽  
Cross Coupling ◽  
Rotor System ◽  
Transformation Matrix ◽  
Symmetric System ◽  
Beam System ◽  
General Systems ◽  
Left And Right ◽  
Nonsymmetric Matrices

Abstract In this paper the Craig-Bampton CMS procedure for the reduction and successive coupling of undamped structural subsystems with symmetric system matrices will be modified for the case of general damping and nonsymmetric matrices. This leads to a Ritz-transformation matrix based on left- and right static and dynamic modes (complex vectors). The physical meaning of these modes will be illustrated and two examples (a damped beam system and a rotor-system with gyroscopy and a cross-coupling bearing model) will be presented and discussed showing the potential of this extension of the Craig-Bampton procedure.

Stereo Vision Based Distance Measurement and its Application

2014 ◽  
Vol 926-930 ◽  
pp. 3258-3261
Author(s):  
Gan Le Hu ◽  
Yan Chen ◽  
Zheng Guo Gu
Keyword(s):  
Computer Graphics ◽  
Stereo Vision ◽  
Stereo Matching ◽  
High Accuracy ◽  
Good Combination ◽  
Practical Applications ◽  
Time Performance ◽  
Speed Up ◽  
Left And Right ◽  
Made In

In computer graphics, Stereo Vision has been a research hotspot for many years, and it is been widely used in many areas. Stereo Vision is a technique which utilizes computer to simulate human eye system. In order to achieve this simulation, two main problems need to be solved: camera calibration and stereo matching. We focus on stereo matching in this paper. After years of development some achievements have been made in stereo matching, but some problems remain unsolved. The two most important things concerned in stereo vision trend to be contradictory: accuracy and efficiency. This paper presents a method called Advanced-Census which is a good combination of SAD and Census. When finding corresponding pixel in left and right image, SAD can get high accuracy with low speed and Census have the opposite result. Advanced-Census has advantages of both SAD and Census. It retains the speed of Census while having the accuracy of SAD. Although it has the speed of Census but not enough for practical applications, so we speed up Advanced-Census using multi-thread technique and edge detection. After speeding up, Advanced-Census gets nearly real-time performance.

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