Iterative methods for the numerical solution of linear systems

نویسندگان

  • Maria Louka
  • Nikolaos Missirlis
چکیده

The objective of this dissertation is the design and analysis of iterative methods for the numerical solution of large, sparse linear systems. This type of systems emerges from the discretization of Partial Differential Equations. Two special types of linear systems are studied. The first type deals with systems whose coefficient matrix is two cyclic whereas the second type studies the augmented linear systems. Initially, the Preconditioned Simultaneous Displacement (PSD) method, which is a generalized version of the Symmetric SOR (SSOR) method, is studied when the Jacobi iteration matrix is weakly cyclic and its eigenvalues are all real “real case” or all imaginary “imaginary case”. The first result is that the PSD method has better convergence rate than the SSOR method. In particular, in the “imaginary case” its convergence is increased by an order of magnitude compared to the SSOR method. In an attempt to further increase the convergence rate of the PSD method, more parameters were introduced. The new method is called the Modified PSD (MPSD) method. Under the same assumptions the convergence of the MPSD method is studied. It is shown that the optimum MPSD method is equivalent to the optimum MSOR method. Furthermore, the convergence analysis of the Generalized Modified Extrapolated SOR (GMESOR) and Generalized Modified Preconditioned Simultaneous Displacement (GMPSD) methods is studied for the numerical solution of the augmented linear systems. The main result of our analysis is that both methods possess the same rate of convergence and less complexity than the Preconditioned Conjugate Gradient (PCG) method. The last result is important since it proves that the addition of parameters in an iterative method has the same effect in the increase of the rate of convergence as that of the Conjugate Gradient (CG) method which belongs to the Krylov subspace methods.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

On the modified iterative methods for $M$-matrix linear systems

This paper deals with scrutinizing the convergence properties of iterative methods to solve linear system of equations. Recently, several types of the preconditioners have been applied for ameliorating the rate of convergence of the Accelerated Overrelaxation (AOR) method. In this paper, we study the applicability of a general class of the preconditioned iterative methods under certain conditio...

متن کامل

Comparison results on the preconditioned mixed-type splitting iterative method for M-matrix linear systems

Consider the linear system Ax=b where the coefficient matrix A is an M-matrix. In the present work, it is proved that the rate of convergence of the Gauss-Seidel method is faster than the mixed-type splitting and AOR (SOR) iterative methods for solving M-matrix linear systems. Furthermore, we improve the rate of convergence of the mixed-type splitting iterative method by applying a preconditio...

متن کامل

Preconditioned Generalized Minimal Residual Method for Solving Fractional Advection-Diffusion Equation

Introduction Fractional differential equations (FDEs)  have  attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or numerical scheme  may be a good approach, particularly, the schemes in numerical linear algebra for solving ...

متن کامل

‎Finite iterative methods for solving systems of linear matrix equations over reflexive and anti-reflexive matrices

A matrix $Pintextmd{C}^{ntimes n}$ is called a generalized reflection matrix if $P^{H}=P$ and $P^{2}=I$‎. ‎An $ntimes n$‎ ‎complex matrix $A$ is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix $P$ if $A=PAP$ ($A=-PAP$)‎. ‎In this paper‎, ‎we introduce two iterative methods for solving the pair of matrix equations $AXB=C$ and $DXE=F$ over reflexiv...

متن کامل

A New Two-stage Iterative Method for Linear Systems and Its Application in Solving Poisson's Equation

In the current study we investigate the two-stage iterative method for solving linear systems. Our new results shows which splitting generates convergence fast in iterative methods. Finally, we solve the Poisson-Block tridiagonal matrix from Poisson's equation which arises in mechanical engineering and theoretical physics. Numerical computations are presented based on a particular linear system...

متن کامل

Improvements of two preconditioned AOR iterative methods for Z-matrices

‎In this paper‎, ‎we propose two preconditioned AOR iterative methods to solve systems of linear equations whose coefficient matrices are Z-matrix‎. ‎These methods can be considered as improvements of two previously presented ones in the literature‎. ‎Finally some numerical experiments are given to show the effectiveness of the proposed preconditioners‎.‎

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2012