scholarly journals Convergence theory of iterative methods based on proper splittings and proper multisplittings for rectangular linear systems

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1835-1851
Author(s):  
Vaibhav Shekhar ◽  
Chinmay Giri ◽  
Debasisha Mishra
Keyword(s):  
Linear Systems ◽  
Iterative Scheme ◽  
Iterative Solution ◽  
Differential Algebraic Equations ◽  
Convergence Theory ◽  
Algebraic Equations ◽  
Linear System Of Equations ◽  
Convergence Results ◽  
Comparison Results ◽  
Singular Linear System

Multisplitting methods are useful to solve differential-algebraic equations. In this connection, we discuss the theory of matrix splittings and multisplittings, which can be used for finding the iterative solution of a large class of rectangular (singular) linear system of equations of the form Ax = b. In this direction, many convergence results are proposed for different subclasses of proper splittings in the literature. But, in some practical cases, the convergence speed of the iterative scheme is very slow. To overcome this issue, several comparison results are obtained for different subclasses of proper splittings. This paper also presents a few such results. However, this idea fails to accelerate the speed of the iterative scheme in finding the iterative solution. In this regard, Climent and Perea [J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper multisplittings. With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results. Some of the results obtained in this paper are even new for the iterative theory of nonsingular linear systems.

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.

Control Constraint Realization for Multibody Systems

2009 ◽  
Author(s):  
Alessandro Fumagalli ◽  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Paolo Mantegazza
Keyword(s):  
Linear Systems ◽  
Multibody Systems ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Practical Observation ◽  
Alternative Condition ◽  
Deformable Bodies

This paper discusses the problem of control constraint realization applied to generic under-actuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that Differential-Algebraic Equations (DAE) need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

scholarly journals Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Closed Form ◽  
Lie Group ◽  
Iterative Scheme ◽  
Differential Algebraic Equations ◽  
Group Method ◽  
Computational Results ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Lie Group Method

We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.

Convergence of Linear Multistep Methods and One-Leg Methods for Index-2 Differential-Algebraic Equations with a Variable Delay

2012 ◽  
Vol 4 (5) ◽  
pp. 636-646 ◽  
Author(s):  
Hongliang Liu ◽  
Aiguo Xiao
Keyword(s):  
Multistep Methods ◽  
Differential Algebraic Equations ◽  
Linear Multistep Methods ◽  
Variable Delay ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Convergence Results ◽  
Index 2

AbstractLinear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay. The corresponding convergence results are obtained and successfully confirmed by some numerical examples. The results obtained in this work extend the corresponding ones in literature.

PERIODIC SOLUTIONS OF DIFFERENTIAL ALGEBRAIC EQUATIONS WITH TIME DELAYS: COMPUTATION AND STABILITY ANALYSIS

2006 ◽  
Vol 16 (01) ◽  
pp. 67-84 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
DIRK ROOSE
Keyword(s):  
Stability Analysis ◽  
Periodic Solutions ◽  
Time Delays ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Floquet Multipliers ◽  
Infinite Dimensional ◽  
Local Asymptotic Stability ◽  
Convergence Results

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

scholarly journals Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Hongliang Liu ◽  
Yayun Fu ◽  
Bailing Li
Keyword(s):  
Time Lag ◽  
Relaxation Method ◽  
Differential Algebraic Equations ◽  
Waveform Relaxation ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Convergence Results ◽  
Waveform Relaxation Method ◽  
Fractional Delay ◽  
Delay Differential

Fractional order delay differential-algebraic equations have the characteristics of time lag and memory and constraint limit. These yield some difficulties in the theoretical analysis and numerical computation. In this paper, we are devoted to solving them by the waveform relaxation method. The corresponding convergence results are obtained, and some numerical examples show the efficiency of the method.

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