scholarly journals Perron Complementation on Linear Systems Involving M-Matrices

2021 ◽  
Vol 09 (05) ◽  
Author(s):  
Jianhong Xu ◽  
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Scheme ◽  
Regular Splitting

We propose in this paper a generalized Perron complementation method for uncoupling a consistent linear system which involves an irreducible, either singular or nonsingular, M-matrix. We show that this uncoupling arises naturally from a regular splitting, which also leads to an efficient iterative scheme for solving the linear system.

scholarly journals Properweak regular splitting and its application to convergence of alternating iterations

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6563-6573 ◽  
Author(s):  
Debasisha Mishra
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Iterative Method ◽  
Linear Systems ◽  
Iterative Scheme ◽  
Convergence Theory ◽  
System Of Equations ◽  
Comparison Result ◽  
Linear System Of Equations ◽  
Regular Splitting

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

Hybrid multilevel solution of sparse least-squares linear systems

2017 ◽  
Vol 34 (8) ◽  
pp. 2752-2766
Author(s):  
Christos K. Filelis-Papadopoulos ◽  
George A. Gravvanis
Keyword(s):  
Linear System ◽  
Least Squares ◽  
Linear Systems ◽  
Iterative Scheme ◽  
Qr Factorization ◽  
Content Type ◽  
Coupled Columns ◽  
Independent Column ◽  
Pcg Method ◽  
Triangular Factor

Purpose Large sparse least-squares problems arise in different scientific disciplines such as optimization, data analysis, machine learning and simulation. This paper aims to propose a two-level hybrid direct-iterative scheme, based on novel block independent column reordering, for efficiently solving large sparse least-squares linear systems. Design/methodology/approach Herewith, a novel block column independent set reordering scheme is used to separate the columns in two groups: columns that are block independent and columns that are coupled. The permutation scheme leads to a two-level hierarchy. Using this two-level hierarchy, the solution of the least-squares linear system results in the solution of a reduced size Schur complement-type square linear system, using the preconditioned conjugate gradient (PCG) method as well as backward substitution using the upper triangular factor, computed through sparse Q-less QR factorization of the columns that are block independent. To improve the convergence behavior of the PCG method, the upper triangular factor, computed through sparse Q-less QR factorization of the coupled columns, is used as a preconditioner. Moreover, to further reduce the fill-in, then the column approximate minimum degree (COLAMD) algorithm is used to permute the block consisting of the coupled columns. Findings The memory requirements for solving large sparse least-squares linear systems are significantly reduced compared to Q-less QR decomposition of the original as well as the permuted problem with COLAMD. The memory requirements are reduced further by choosing to form larger blocks of independent columns. The convergence behavior of the iterative scheme is improved due to the chosen preconditioning scheme. The proposed scheme is inherently parallel due to the introduction of block independent column reordering. Originality/value The proposed scheme is a hybrid direct-iterative approach for solving sparse least squares linear systems based on the implicit computation of a two-level approximate pseudo-inverse matrix. Numerical results indicating the applicability and effectiveness of the proposed scheme are given.

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.

A Note About Linear Systems on Curves

1984 ◽  
Vol 27 (3) ◽  
pp. 371-374
Author(s):  
Allen Tannenbaum
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Minimal Degree ◽  
Maximal Dimension ◽  
Degree Relative ◽  
Irreducible Curve ◽  
The Given

AbstractInverting the Castelnuovo bound in two ways, we show that for given integers p ≥ 0, d > 1, n > 1, we can find a smooth irreducible curve of genus p which contains a linear system of degree d and of maximal dimension relative to the given data p and d, and a smooth irreducible curve of genus p which contains a linear system of dimension n and of minimal degree relative to the data p and n.

scholarly journals Applications of Symmetric and Nonsymmetric MSSOR Preconditioners to Large-Scale Biot's Consolidation Problems with Nonassociated Plasticity

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xi Chen ◽  
Kok Kwang Phoon
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Iterative Scheme ◽  
Linear Equations ◽  
Pile Group ◽  
Newton Iteration ◽  
Soil Consolidation ◽  
Considerable Potential ◽  
Stiffness Method ◽  
Biot’S Consolidation

Two solution schemes are proposed and compared for large 3D soil consolidation problems with nonassociated plasticity. One solution scheme results in the nonsymmetric linear equations due to the Newton iteration, while the other leads to the symmetric linear systems due to the symmetrized stiffness strategies. To solve the resulting linear systems, the QMR and SQMR solver are employed in conjunction with nonsymmetric and symmetric MSSOR preconditioner, respectively. A simple footing example and a pile-group example are used to assess the performance of the two solution schemes. Numerical results disclose that compared to the Newton iterative scheme, the symmetric stiffness schemes combined with adequate acceleration strategy may lead to a significant reduction in total computer runtime as well as in memory requirement, indicating that the accelerated symmetric stiffness method has considerable potential to be exploited to solve very large problems.

On the Design of Finite Time Settling Control for Linear Systems

1992 ◽  
Vol 114 (3) ◽  
pp. 359-368 ◽  
Author(s):  
S. Choura
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Finite Time ◽  
Linear Time ◽  
Minimum Energy ◽  
Time Varying ◽  
Parameter Uncertainties ◽  
Final State ◽  
Variable Function ◽  
Time Varying Systems

The design of controllers combining feedback and feedforward for the finite time settling control of linear systems, including linear time-varying systems, is considered. The feedforward part transfers the initial state of a linear system to a desired final state in finite time, and the feedback part reduces the effects of uncertainties and disturbances on the system performance. Two methods for determining the feedforward part, without requiring the knowledge of the explicit state solutions, are proposed. In the first method, a numerical procedure for approximating combined controls that drive linear time-varying systems to their final state in finite time is given. The feedforward part is a variable function of time and is selected based on a set of necessary conditions, such as magnitude constraints. In the second method, an analytical procedure for constructing combined controls for linear time-invariant systems is presented, where the feedforward part is accurately determined and it is of the minimum energy control type. It is shown that both methods facilitate the design of the feedforward part of combined controllers for the finite time settling of linear systems. The robustness of driving a linear system to its desired state in finite time is analyzed for three types of uncertainties. The robustness analysis suggests a modification of the feedforward control law to assure the robustness of the control strategy to parameter uncertainties for arbitrary final times.

scholarly journals H-Matrices in Fuzzy Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Linear System ◽  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Coefficient Matrix ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Fuzzy Linear System ◽  
Fuzzy Linear Systems

We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an H-matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.

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