**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1202

# Search results for: Jacobi iterative matrix

##### 1202 The Convergence Results between Backward USSOR and Jacobi Iterative Matrices

**Authors:**
Zuan-De Wang,
Hou-biao Li,
Zhong-xi Gao

**Abstract:**

In this paper, the backward Ussor iterative matrix is proposed. The relationship of convergence between the backward Ussor iterative matrix and Jacobi iterative matrix is obtained, which makes the results in the corresponding references be improved and refined.Moreover,numerical examples also illustrate the effectiveness of these conclusions.

**Keywords:**
Backward USSOR iterative matrix,
Jacobi iterative matrix,
convergence,
spectral radius

##### 1201 The Relationship of Eigenvalues between Backward MPSD and Jacobi Iterative Matrices

**Authors:**
Zhuan-de Wang,
Hou-biao Li,
Zhong-xi Gao

**Abstract:**

In this paper, the backward MPSD (Modified Preconditioned Simultaneous Displacement) iterative matrix is firstly proposed. The relationship of eigenvalues between the backward MPSD iterative matrix and backward Jacobi iterative matrix for block p-cyclic case is obtained, which improves and refines the results in the corresponding references.

**Keywords:**
Backward MPSD iterative matrix,
Jacobi iterative matrix,
eigenvalue,
p-cyclic matrix.

##### 1200 Jacobi-Based Methods in Solving Fuzzy Linear Systems

**Authors:**
Lazim Abdullah,
Nurhakimah Ab. Rahman

**Abstract:**

Linear systems are widely used in many fields of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. Therefore it is important to perform numerical algorithms or procedures that would treat general fuzzy linear systems and solve them using iterative methods. This paper aims are to solve fuzzy linear systems using four types of Jacobi based iterative methods. Four iterative methods based on Jacobi are used for solving a general n × n fuzzy system of linear equations of the form Ax = b , where A is a crisp matrix and b an arbitrary fuzzy vector. The Jacobi, Jacobi Over-Relaxation, Refinement of Jacobi and Refinement of Jacobi Over-Relaxation methods was tested to a five by five fuzzy linear system. It is found that all the tested methods were iterated differently. Due to the effect of extrapolation parameters and the refinement, the Refinement of Jacobi Over-Relaxation method was outperformed the other three methods.

**Keywords:**
Fuzzy linear systems,
Jacobi,
Jacobi Over-
Relaxation,
Refinement of Jacobi,
Refinement of Jacobi Over-
Relaxation.

##### 1199 On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations

**Authors:**
H. D. Ibrahim,
H. C. Chinwenyi,
H. N. Ude

**Abstract:**

In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.

**Keywords:**
conjugate gradient,
linear equations,
symmetric and positive definite matrix,
Gauss-Seidel,
Jacobi,
algorithm

##### 1198 Preconditioned Jacobi Method for Fuzzy Linear Systems

**Authors:**
Lina Yan,
Shiheng Wang,
Ke Wang

**Abstract:**

A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp Mmatrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

**Keywords:**
preconditioning,
M-matrix,
Jacobi method,
fuzzy linear
system (FLS).

##### 1197 Numerical Study of Iterative Methods for the Solution of the Dirichlet-Neumann Map for Linear Elliptic PDEs on Regular Polygon Domains

**Authors:**
A. G. Sifalakis,
E. P. Papadopoulou,
Y. G. Saridakis

**Abstract:**

**Keywords:**
Elliptic PDEs,
Dirichlet to Neumann Map,
Global Relation,
Collocation,
Iterative Methods,
Jacobi,
Gauss-Seidel,
GMRES,
Bi-CGSTAB.

##### 1196 Iterative solutions to the linear matrix equation AXB + CXTD = E

**Authors:**
Yongxin Yuan,
Jiashang Jiang

**Abstract:**

**Keywords:**
matrix equation,
iterative algorithm,
parameter estimation,
minimum norm solution.

##### 1195 A New Preconditioned AOR Method for Z-matrices

**Authors:**
Guangbin Wang,
Ning Zhang,
Fuping Tan

**Abstract:**

In this paper, we present a preconditioned AOR-type iterative method for solving the linear systems Ax = b, where A is a Z-matrix. And give some comparison theorems to show that the rate of convergence of the preconditioned AOR-type iterative method is faster than the rate of convergence of the AOR-type iterative method.

**Keywords:**
Z-matrix,
AOR-type iterative method,
precondition,
comparison.

##### 1194 Complexity Reduction Approach with Jacobi Iterative Method for Solving Composite Trapezoidal Algebraic Equations

**Authors:**
Mohana Sundaram Muthuvalu,
Jumat Sulaiman

**Abstract:**

In this paper, application of the complexity reduction approach based on half- and quarter-sweep iteration concepts with Jacobi iterative method for solving composite trapezoidal (CT) algebraic equations is discussed. The performances of the methods for CT algebraic equations are comparatively studied by their application in solving linear Fredholm integral equations of the second kind. Furthermore, computational complexity analysis and numerical results for three test problems are also included in order to verify performance of the methods.

**Keywords:**
Complexity reduction approach,
Composite trapezoidal
scheme,
Jacobi method,
Linear Fredholm integral equations

##### 1193 Some New Upper Bounds for the Spectral Radius of Iterative Matrices

**Authors:**
Guangbin Wang,
Xue Li,
Fuping Tan

**Abstract:**

**Keywords:**
doubly α diagonally dominant matrix,
eigenvalue,
iterative matrix,
spectral radius,
upper bound.

##### 1192 An Iterative Method for the Symmetric Arrowhead Solution of Matrix Equation

**Authors:**
Minghui Wang,
Luping Xu,
Juntao Zhang

**Abstract:**

**Keywords:**
Symmetric arrowhead matrix,
iterative method,
like-minimum norm,
minimum norm,
Algorithm LSQR.

##### 1191 Iterative Solutions to Some Linear Matrix Equations

**Authors:**
Jiashang Jiang,
Hao Liu,
Yongxin Yuan

**Abstract:**

In this paper the gradient based iterative algorithms are presented to solve the following four types linear matrix equations: (a) AXB = F; (b) AXB = F, CXD = G; (c) AXB = F s. t. X = XT ; (d) AXB+CYD = F, where X and Y are unknown matrices, A,B,C,D, F,G are the given constant matrices. It is proved that if the equation considered has a solution, then the unique minimum norm solution can be obtained by choosing a special kind of initial matrices. The numerical results show that the proposed method is reliable and attractive.

**Keywords:**
Matrix equation,
iterative algorithm,
parameter estimation,
minimum norm solution.

##### 1190 Two Iterative Algorithms to Compute the Bisymmetric Solution of the Matrix Equation A1X1B1 + A2X2B2 + ... + AlXlBl = C

**Authors:**
A.Tajaddini

**Abstract:**

In this paper, two matrix iterative methods are presented to solve the matrix equation A1X1B1 + A2X2B2 + ... + AlXlBl = C the minimum residual problem l i=1 AiXiBi−CF = minXi∈BRni×ni l i=1 AiXiBi−CF and the matrix nearness problem [X1, X2, ..., Xl] = min[X1,X2,...,Xl]∈SE [X1,X2, ...,Xl] − [X1, X2, ..., Xl]F , where BRni×ni is the set of bisymmetric matrices, and SE is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than former methods. Paige’s algorithms are used as the frame method for deriving these matrix iterative methods. The numerical example is used to illustrate the efficiency of these new methods.

**Keywords:**
Bisymmetric matrices,
Paige’s algorithms,
Least
square.

##### 1189 Preconditioned Mixed-Type Splitting Iterative Method For Z-Matrices

**Authors:**
Li Jiang,
Baoguang Tian

**Abstract:**

**Keywords:**
Z-matrix,
mixed-type splitting iterative method,
precondition,
comparison theorem,
linear system.

##### 1188 Semiconvergence of Alternating Iterative Methods for Singular Linear Systems

**Authors:**
Jing Wu

**Abstract:**

In this paper, we discuss semiconvergence of the alternating iterative methods for solving singular systems. The semiconvergence theories for the alternating methods are established when the coefficient matrix is a singular matrix. Furthermore, the corresponding comparison theorems are obtained.

**Keywords:**
Alternating iterative method,
Semiconvergence,
Singular
matrix.

##### 1187 An Iterative Method for the Least-squares Symmetric Solution of AXB+CYD=F and its Application

**Authors:**
Minghui Wang

**Abstract:**

Based on the classical algorithm LSQR for solving (unconstrained) LS problem, an iterative method is proposed for the least-squares like-minimum-norm symmetric solution of AXB+CYD=E. As the application of this algorithm, an iterative method for the least-squares like-minimum-norm biymmetric solution of AXB=E is also obtained. Numerical results are reported that show the efficiency of the proposed methods.

**Keywords:**
Matrix equation,
bisymmetric matrix,
least squares problem,
like-minimum norm,
iterative algorithm.

##### 1186 A New Approach to the Approximate Solutions of Hamilton-Jacobi Equations

**Authors:**
Joe Imae,
Kenjiro Shinagawa,
Tomoaki Kobayashi,
Guisheng Zhai

**Abstract:**

We propose a new approach on how to obtain the approximate solutions of Hamilton-Jacobi (HJ) equations. The process of the approximation consists of two steps. The first step is to transform the HJ equations into the virtual time based HJ equations (VT-HJ) by introducing a new idea of ‘virtual-time’. The second step is to construct the approximate solutions of the HJ equations through a computationally iterative procedure based on the VT-HJ equations. It should be noted that the approximate feedback solutions evolve by themselves as the virtual-time goes by. Finally, we demonstrate the effectiveness of our approximation approach by means of simulations with linear and nonlinear control problems.

**Keywords:**
Nonlinear Control,
Optimal Control,
Hamilton-Jacobi Equation,
Virtual-Time

##### 1185 On Algebraic Structure of Improved Gauss-Seidel Iteration

**Authors:**
O. M. Bamigbola,
A. A. Ibrahim

**Abstract:**

Analysis of real life problems often results in linear systems of equations for which solutions are sought. The method to employ depends, to some extent, on the properties of the coefficient matrix. It is not always feasible to solve linear systems of equations by direct methods, as such the need to use an iterative method becomes imperative. Before an iterative method can be employed to solve a linear system of equations there must be a guaranty that the process of solution will converge. This guaranty, which must be determined apriori, involve the use of some criterion expressible in terms of the entries of the coefficient matrix. It is, therefore, logical that the convergence criterion should depend implicitly on the algebraic structure of such a method. However, in deference to this view is the practice of conducting convergence analysis for Gauss- Seidel iteration on a criterion formulated based on the algebraic structure of Jacobi iteration. To remedy this anomaly, the Gauss- Seidel iteration was studied for its algebraic structure and contrary to the usual assumption, it was discovered that some property of the iteration matrix of Gauss-Seidel method is only diagonally dominant in its first row while the other rows do not satisfy diagonal dominance. With the aid of this structure we herein fashion out an improved version of Gauss-Seidel iteration with the prospect of enhancing convergence and robustness of the method. A numerical section is included to demonstrate the validity of the theoretical results obtained for the improved Gauss-Seidel method.

**Keywords:**
Linear system of equations,
Gauss-Seidel iteration,
algebraic structure,
convergence.

##### 1184 Iterative Methods for An Inverse Problem

**Authors:**
Minghui Wang,
Shanrui Hu

**Abstract:**

An inverse problem of doubly center matrices is discussed. By translating the constrained problem into unconstrained problem, two iterative methods are proposed. A numerical example illustrate our algorithms.

**Keywords:**
doubly center matrix,
electric network theory,
iterative methods,
least-square problem.

##### 1183 An Iterative Algorithm to Compute the Generalized Inverse A(2) T,S Under the Restricted Inner Product

**Authors:**
Xingping Sheng

**Abstract:**

**Keywords:**
Generalized inverse A(2)
T,
S,
Restricted inner product,
Iterative method,
Orthogonal projection.

##### 1182 An Iterative Method for Quaternionic Linear Equations

**Authors:**
Bin Yu,
Minghui Wang,
Juntao Zhang

**Abstract:**

By the real representation of the quaternionic matrix, an iterative method for quaternionic linear equations Ax = b is proposed. Then the convergence conditions are obtained. At last, a numerical example is given to illustrate the efficiency of this method.

**Keywords:**
Quaternionic linear equations,
Real representation,
Iterative algorithm.

##### 1181 On the Solution of Fully Fuzzy Linear Systems

**Authors:**
Hsuan-Ku Liu

**Abstract:**

A linear system is called a fully fuzzy linear system (FFLS) if quantities in this system are all fuzzy numbers. For the FFLS, we investigate its solution and develop a new approximate method for solving the FFLS. Observing the numerical results, we find that our method is accurate than the iterative Jacobi and Gauss- Seidel methods on approximating the solution of FFLS.

**Keywords:**
Fully fuzzy linear equations,
iterative method,
homotopy perturbation method,
approximate solutions.

##### 1180 Approximating Fixed Points by a Two-Step Iterative Algorithm

**Authors:**
Safeer Hussain Khan

**Abstract:**

In this paper, we introduce a two-step iterative algorithm to prove a strong convergence result for approximating common fixed points of three contractive-like operators. Our algorithm basically generalizes an existing algorithm..Our iterative algorithm also contains two famous iterative algorithms: Mann iterative algorithm and Ishikawa iterative algorithm. Thus our result generalizes the corresponding results proved for the above three iterative algorithms to a class of more general operators. At the end, we remark that nothing prevents us to extend our result to the case of the iterative algorithm with error terms.

**Keywords:**
Contractive-like operator,
iterative algorithm,
fixed point,
strong convergence.

##### 1179 Conjugate Gradient Algorithm for the Symmetric Arrowhead Solution of Matrix Equation AXB=C

**Authors:**
Minghui Wang,
Luping Xu,
Juntao Zhang

**Abstract:**

*AXB=C*and the associate optimal approximation problem are considered for the symmetric arrowhead matrix solutions in the premise of consistency. The convergence results of the method are presented. At last, a numerical example is given to illustrate the efficiency of this method.

**Keywords:**
Iterative method,
symmetric arrowhead matrix,
conjugate gradient algorithm.

##### 1178 On the Positive Definite Solutions of Nonlinear Matrix Equation

**Authors:**
Tian Baoguang,
Liang Chunyan,
Chen Nan

**Abstract:**

In this paper, the nonlinear matrix equation is investigated. Based on the fixed-point theory, the boundary and the existence of the solution with the case r>-δ_{i} are discussed. An algorithm that avoids matrix inversion with the case -1<-δ_{i}<0 is proposed.

**Keywords:**
Nonlinear matrix equation,
Positive definite solution,
The maximal-minimal solution,
Iterative method,
Free-inversion

##### 1177 Parallel Multisplitting Methods for Singular Linear Systems

**Authors:**
Guangbin Wang,
Fuping Tan

**Abstract:**

In this paper, we discuss convergence of the extrapolated iterative methods for linear systems with the coefficient matrices are singular H-matrices. And we present the sufficient and necessary conditions for convergence of the extrapolated iterative methods. Moreover, we apply the results to the GMAOR methods. Finally, we give one numerical example.

**Keywords:**
Singular H-matrix,
linear systems,
extrapolated iterative method,
GMAOR method,
convergence.

##### 1176 A Note on the Convergence of the Generalized AOR Iterative Method for Linear Systems

**Authors:**
Zhong-xi Gao,
Hou-biao Li

**Abstract:**

Recently, some convergent results of the generalized AOR iterative (GAOR) method for solving linear systems with strictly diagonally dominant matrices are presented in [Darvishi, M.T., Hessari, P.: On convergence of the generalized AOR method for linear systems with diagonally dominant cofficient matrices. Appl. Math. Comput. 176, 128-133 (2006)] and [Tian, G.X., Huang, T.Z., Cui, S.Y.: Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant cofficient matrices. J. Comp. Appl. Math. 213, 240-247 (2008)]. In this paper, we give the convergence of the GAOR method for linear systems with strictly doubly diagonally dominant matrix, which improves these corresponding results.

**Keywords:**
Diagonally dominant matrix,
GAOR method,
Linear
system,
Convergence

##### 1175 Robust Iterative PID Controller Based on Linear Matrix Inequality for a Sample Power System

**Authors:**
Ahmed Bensenouci

**Abstract:**

**Keywords:**
Linear matrix inequality,
power system,
robust
iterative PID,
robust output feedback control

##### 1174 New Explicit Group Newton's Iterative Methods for the Solutions of Burger's Equation

**Authors:**
Tan K. B.,
Norhashidah Hj. M. Ali

**Abstract:**

In this article, we aim to discuss the formulation of two explicit group iterative finite difference methods for time-dependent two dimensional Burger-s problem on a variable mesh. For the non-linear problems, the discretization leads to a non-linear system whose Jacobian is a tridiagonal matrix. We discuss the Newton-s explicit group iterative methods for a general Burger-s equation. The proposed explicit group methods are derived from the standard point and rotated point Crank-Nicolson finite difference schemes. Their computational complexity analysis is discussed. Numerical results are given to justify the feasibility of these two proposed iterative methods.

**Keywords:**
Standard point Crank-Nicolson (CN),
Rotated point Crank-Nicolson (RCN),
Explicit Group (EG),
Explicit Decoupled Group (EDG).

##### 1173 Extending Global Full Orthogonalization method for Solving the Matrix Equation AXB=F

**Authors:**
Fatemeh Panjeh Ali Beik

**Abstract:**

**Keywords:**
Matrix equations,
Iterative methods,
Block Krylovsubspace methods.