scholarly journals Optimal Backward Perturbation Analysis for the Block Skew Circulant Linear Systems with Skew Circulant Blocks

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaolin Jiang ◽  
Juan Li ◽  
Jianwei Zhou
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Perturbation Analysis ◽  
Spectral Decomposition ◽  
Circulant Matrix ◽  
Singular Value ◽  
Backward Perturbation ◽  
Skew Circulant

We first give the block style spectral decomposition of arbitrary block skew circulant matrix with skew circulant blocks. Secondly, we obtain the singular value of block skew circulant matrix with skew circulant blocks as well. Finally, based on the block style spectral decomposition, we deal with the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks.

scholarly journals On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Juan Li ◽  
Zhaolin Jiang ◽  
Nuo Shen ◽  
Jianwei Zhou
Keyword(s):  
Linear System ◽  
Perturbation Analysis ◽  
Spectral Decomposition ◽  
Coefficient Matrix ◽  
Circulant Matrix ◽  
Singular Value ◽  
Kronecker Products ◽  
Backward Perturbation ◽  
Skew Circulant

We first give the style spectral decomposition of a special skew circulant matrixCand then get the style decomposition of arbitrary skew circulant matrix by making use of the Kronecker products between the elements of first row in skew circulant and the special skew circulantC. Besides that, we obtain the singular value of skew circulant matrix as well. Finally, we deal with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix on the base of its style spectral decomposition.

scholarly journals Analysis of the Structured Perturbation for the BCSCB Linear System

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Xia Tang ◽  
Zhaolin Jiang
Keyword(s):  
Linear System ◽  
Relative Error ◽  
Condition Number ◽  
Spectral Decomposition ◽  
Circulant Matrix ◽  
Networked Systems ◽  
Perturbation Bound ◽  
Numerical Example ◽  
Backward Perturbation ◽  
Skew Circulant

Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm.

The Linear System Solver of Programs Named CORTH and KYLIN2

2017 ◽  
Author(s):  
Pingzhou Ming ◽  
Junjie Pan ◽  
Xiaolan Tu ◽  
Dong Liu ◽  
Hongxing Yu
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Nuclear Power ◽  
Experimental Results ◽  
Gmres Method ◽  
Thermal Hydraulics ◽  
Lattice Calculation ◽  
Minimal Residual

Sub-channel thermal-hydraulics program named CORTH and assembly lattice calculation program named KYLIN2 have been developed in Nuclear Power Institute of China (NPIC). For the sake of promoting the computing efficiency of these programs and achieving the better description on fined parameters of reactor, the programs’ structure and details are interpreted. Then the characteristics of linear systems of these programs are analyzed. Based on the Generalized Minimal Residual (GMRES) method, different parallel schemes and implementations are considered. The experimental results show that calculation efficiencies of them are improved greatly compared with the serial situation.

scholarly journals (M,β)-Stability of Positive Linear Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Octavian Pastravanu ◽  
Mihaela-Hanako Matcovschi
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Transient Behavior ◽  
Short Term ◽  
Time Dynamics ◽  
The Right ◽  
Long Term Behavior ◽  
Stability Concept

The main purpose of this work is to show that the Perron-Frobenius eigenstructure of a positive linear system is involved not only in the characterization of long-term behavior (for which well-known results are available) but also in the characterization of short-term or transient behavior. We address the analysis of the short-term behavior by the help of the “(M,β)-stability” concept introduced in literature for general classes of dynamics. Our paper exploits this concept relative to Hölder vectorp-norms,1≤p≤∞, adequately weighted by scaling operators, focusing on positive linear systems. Given an asymptotically stable positive linear system, for each1≤p≤∞, we prove the existence of a scaling operator (built from the right and left Perron-Frobenius eigenvectors, with concrete expressions depending onp) that ensures the best possible values for the parametersMandβ, corresponding to an “ideal” short-term (transient) behavior. We provide results that cover both discrete- and continuous-time dynamics. Our analysis also captures the differences between the cases where the system dynamics is defined by matrices irreducible and reducible, respectively. The theoretical developments are applied to the practical study of the short-term behavior for two positive linear systems already discussed in literature by other authors.

scholarly journals A Modification of Minimal Residual Iterative Method to Solve Linear Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Xingping Sheng ◽  
Youfeng Su ◽  
Guoliang Chen
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Linear Systems ◽  
Residual Error ◽  
Projection Technique ◽  
Numerical Example ◽  
Minimal Residual

We give a modification of minimal residual iteration (MR), which is 1V-DSMR to solve the linear systemAx=b. By analyzing, we find the modifiable iteration to be a projection technique; moreover, the modification of which gives a better (at least the same) reduction of the residual error than MR. In the end, a numerical example is given to demonstrate the reduction of the residual error between the 1V-DSMR and MR.

Quantum CFD Simulations for Heat Transfer Applications

2020 ◽  
Author(s):  
Chao Lu ◽  
Zhao Hu ◽  
Bei Xie ◽  
Ning Zhang
Keyword(s):  
Heat Transfer ◽  
Quantum Computing ◽  
Linear System ◽  
Linear Equation ◽  
Linear Systems ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Phase ◽  
Cfd Simulations ◽  
Analytical Result

Abstract In this paper, computational heat transfer (CHT) equations were solved using the state-of-art quantum computing (QC) technology. The CHT equations can be discretized into a linear equation set, which can be possibly solved by a QC system. The linear system can be characterized by Ax = b. The A matrix in this linear system is a Hermitian matrix. The linear system is then solved by using the HHL algorithm, which is a quantum algorithm to solve a linear system. The quantum circuit requires an Ancilla qubit, clock qubits, qubits for b and a classical bit to record the result. The process of the HHL algorithm can be described as follows. Firstly, the qubit for b is initialized into the phase as desire. Secondly, the quantum phase estimation (QPE) is used to determine the eigenvalues of A and the eigenvalues are stored in clock qubits. Thirdly, a Rotation gate is used to rotate the inversion of eigenvalues and information is passed to the Ancilla bit to do Pauli Y-rotation operation. Fourthly, revert the whole processes to untangle qubits and measure all of the qubits to output the final results for x. From the existing literature, a few 2 × 2 matrices were successfully solved with QC technology, proving the possibility of QC on linear systems [1]. In this paper, a quantum circuit is designed to solve a CHT problem. A simple 2 by 2 linear equation is modeled for the CHT problem and is solved by using the quantum computing. The result is compared with the analytical result. This result could initiate future studies on determining the quantum phase parameters for more complicated QC linear systems for CHT applications.

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