The iterative methods for centrosymmetric matrices

This paper presents some iterative methods for solving system of linear equations namely the Jacobi method and the modified Jacobi method. The Jacobi method is an algorithm for solving system of linear equations with largest absolute values in each row and column dominated by the diagonal elements. The modified Jacobi method also known as the Gauss Seidel method or the method of successive displacement is useful for the solution of system of linear equations. The comparative results analysis of the two methods was considered. We also discussed the rate of convergence of the Jacobi method and the modified Jacobi method. Finally, the results showed that the modified Jacobi method is more efficient, accurate and converges faster than its counterpart “the Jacobi Method”

International Journal of Applied Mathematical Research, 2013

In this paper, based on a block splitting of the coefficient matrix, we present a new generalized iterative method for solving the linear system Ax = b. This method is well-defined even when some elements on the diagonal of A are zero. Convergence analysis and comparison theorems of the proposed method are provided. Specially, the results show that our new generalized AOR iterative method also, converges when A is an H-matrix. And for L-matrices, our new generalized Jacobi iterative method is faster than the classical Jacobi. The Numerical examples are also given to illustrate our results.

Linear Algebra and its Applications, 2000

We present a method for the multiplication of an arbitrary vector by a symmetric centrosymmetric matrix, requiring 5 4 n 2 + O(n) floating-point operations, rather than the 2n 2 operations needed in the case of an arbitrary matrix. Combining this method with Trench's algorithm for Toeplitz matrix inversion yields a method for solving Toeplitz systems with the same complexity as Levinson's algorithm.

Journal of Computational and Applied Mathematics, 2008

In Ujević [A new iterative method for solving linear systems, Appl. Math. Comput. 179 (2006) 725-730], the author obtained a new iterative method for solving linear systems, which can be considered as a modification of the Gauss-Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujević's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujević.

Linear and Multilinear Algebra, 2017

Recently, Tang et al. [Numer Algorithms. 2014;66(2):379-397] have offered a cyclic iterative method for determining the unique solution of the coupled matrix equations A i XB i = F i , i = 1, 2,. .. , N. Analogues to the gradient-based algorithm, the proposed algorithm relies on a fixed parameter whereas it has wider convergence region. Nevertheless, the application of the algorithm to find the centrosymmetric solution of the mentioned problem has been left as a project to be investigated and the optimal value for the fixed parameter has not been derived. In this paper, we focus on a more general class of the coupled linear matrix equations that incorporate the mentioned ones in the earlier refereed work. More precisely, we first develop the authors' propounded algorithm to resolve our considered coupled linear matrix equations over centro-symmetric matrices. Afterwards, we disregard the restriction of the existence of the unique (centro-symmetric) solution and also modify the authors' algorithm by applying an oblique projection technique which allows to produce a sequence of approximate solutions which gratify an optimality property. Numerical results are reported to confirm the validity of the established results and to demonstrate the superior performance of the modified version of the cyclic iterative algorithm.

Consider the linear system Ax = b where the coefficient matrix A is an M-matrix. Here, it is proved that the rate of convergence of the Gauss-Seidel method is faster than the mixedtype splitting and AOR (SOR) iterative methods for solving Mmatrix linear systems. Furthermore, we improve the rate of convergence of the mixed-type splitting iterative method by applying a preconditioned matrix. Comparison theorems show that the rate of convergence of the preconditioned Gauss-Seidel method is faster than the preconditioned mixed-type splitting and AOR (SOR) iterative methods. Finally, some numerical examples are presented to illustrate the reality of our results.

BIT, 1988

A method is presented to solve Ax = b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues of A. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.

Acta Mathematica Hungarica, 2001

A possible type of the matrix splitting is introduced. Using this matrix splitting, we introduce a few properties and representations of generalized inverses as well as iterative methods for computing various solutions of singu- lar linear systems. This matrix splitting is a generalization of the known index splitting from [13] and a proper splitting from [4]. Using a generalization of

Circuits, Systems, and Signal Processing, 1989

The structure inherent to the centrosymmetric matrices is exploited to obtain factorization results leading to significant computational savings in many engineering applications. Several interesting properties of the matrices are discussed with a view toward algorithm computational complexity. It is shown that the multiplicative complexity involved in the process of principal component (eigenvalue/eigenvector) extraction, and in the evaluation of the determinant and inverse of such matrices, can be reduced by nearly 75% by employing the results presented here. The theory presented here unifies and generalizes previous computationally efficient results on several specialized generalized centrosymmetric matrices; for example, the class of symmetric centrosymmetric (or doubly symmetric) matrices is shown to be a special case of the class of centrosymmetric matrices, and since the results obtained here are applicable over the field of complex numbers, the factorization results available for centrosymmetric matrices are readily extended to the complex field. The centrosymmetric matrices play an important role in a number of areas such as pattern recognition, antenna theory, mechanical and electrical systems, and quantum physics. Specific examples of pattern recognition feature selection, a uniform linear antenna array, vibration in structures, and the quantummechanical oscillator are discussed to demonstrate that the theory developed here has a wide range of applications. In addition, certain specialized cases of the centrosymmetric matrices have, in the past, proved their indisputable usefulness in the areas of communication theory, digital filters, linear systems, linear prediction, and speech analysis.

2013

We present a class of two stage iterative methods, called two stage mixed-type splitting (TMS) algorithm, and analyze the convergence of this method for solving large nonsingular system of equations Ax = b. Based on this new method, we establish another two stage models such as two stage (SOR, AOR, SSOR, SAOR) algorithms. The convergence of these algorithms is discussed, under the condition that A is Z-matrix. Besides, convergence is further studied for splitting of M-matrix. Numerical example is also reported to confirm the convergence analysis.