Search

Recently, Zhou et al. [Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math. 224 (2009), pp. 242–249] have proposed the preconditioned generalized accelerated over relaxation (GAOR) methods for solving generalized least squares problems and studied their convergence rates. In this paper, we propose a new type of preconditioners and study the convergence rates of the new preconditioned GAOR methods for solving generalized least squares problems. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Zhou et al. whenever these methods are convergent. Lastly, numerical experiments are provided in order to confirm the theoretical results studied in this paper. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the nonstationary iterative scheme for solving consistent singular linear system with index one. We utilize the group inverse to present a sufficient condition for the convergence of the nonstationary iterative method. Our result extends the known results of the stationary iterative scheme. Finally, we present a sufficient condition for the multisplitting algorithm and provide numerical examples to illustrate the advantages of nonstationary method. [ABSTRACT FROM PUBLISHER]

In this paper, we study the convergence ofP-regular splitting iterative methods for singular and non-singular non-Hermitian positive semidefinite linear systems, which generalize the known results. Some numerical experiments are performed to illustrate the convergence results. [ABSTRACT FROM PUBLISHER]

In this paper, a new iterative refinement of the solution of an ill-conditioned linear system of equations are given. The convergence properties of the method are studied. Some numerical experiments of the method are given and compared with that of two of the available methods. [ABSTRACT FROM AUTHOR]

The paper presents a type of tridiagonal preconditioners for solving linear system Ax=b with nonsingular M-matrix A, and obtains some important convergent theorems about preconditioned Jacobi and Gauss-Seidel type iterative methods. The main results theoretically prove that the tridiagonal preconditioners cannot only accelerate the convergence of iterations, but also generalize some known results. [ABSTRACT FROM AUTHOR]

Despite the significant progress on iterative learning control (ILC) in the past decade, ILC algorithms for non-linear systems are still under development. Monotonic convergence is one of the desirable properties of non-linear ILC. In this paper, by introducing a relaxation index, a Newton method based ILC is shown to converge monotonically. Simulations verify the effectiveness of the new algorithm. Potential improvements are discussed for future work. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give comparison results for a class of row preconditioners. Convergence and monotone properties for the classical iterative methods associated with these preconditioners are analysed. In the final section, numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up toand. [ABSTRACT FROM AUTHOR]

The convergence of numerical approximations to the solutions of differential equations is a key aspect of numerical analysis and scientific computing. Iterative solution methods for the systems of linear(ized) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often paid to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a 'right' norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer. [ABSTRACT FROM AUTHOR]

Large linear systems arising in CFD problems are often solved using iterative methods which typically combine an accelerator such as GMRES and a preconditioner. This paper presents results using several preconditioning techniques, which are not standard in CFD. Numerical tests are carried out for solving three-dimensional incompressible, compressible and magnetohydrodynamic (MHD) problems. A selection of numerical results is presented showing in particular that the flexible GMRES algorithm preconditioned with ILUT factorization provides a fairly robust iterative solver. [ABSTRACT FROM AUTHOR]

The method of moments for the simplified scattering and radiation on thin wire models can lead to the large-scale linear systems with non-Hermitian Toeplitz-like coefficient matrices. Since such reduced linear systems are typically dense and ill conditioned, iterative solvers are rarely considered to deal with the discretized linear systems in these electromagnetic model simulations. However, utilizing the Toeplitz-like structure of coefficient matrices, we can use the fast Toeplitz matrix-vector product (MVP) to refresh iterative solvers for the discretized linear systems. Compared with the traditional solvers, the fast numerical method can reduce the computational cost fromtoand the storage fromtowithout using any lossy compression. Due to the Toeplitz-like structure of coefficient matrices, circulant preconditioners are considered to further accelerate the convergence of iterative solvers. Since the circulant matrices can be diagonalized via fast Fourier transforms (FFTs), the fast Toeplitz MVP is utilized to retain the total computational complexity of the proposed methods at, whereis the number of spatial grid nodes. Theoretical analyses of the spectra of the preconditioned linear systems are also preliminarily investigated. Finally, extensive numerical experiments involving both scattering and radiation issues are employed to demonstrate the effectiveness of the proposed methods in aspects of the number of iterations and CPU time elapsed. [ABSTRACT FROM PUBLISHER]

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. Based on Paige's algorithms LSQR and residual-reducing version of LSQR proposed in Paige and Saunders [LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Softw. 8(1) (1982), pp. 43–71], we provide two matrix iterative algorithms for finding solution with the least norm to the QLS problem by making use of structure of real representation matrices. Numerical experiments are presented to illustrate the efficiency of our algorithms. [ABSTRACT FROM PUBLISHER]

A new splitting iteration method is presented for the system of linear equations when the coefficient matrix is a non-Hermitian positive-definite matrix. The spectral radius, the optimal parameter, and some norm properties of the iteration matrix for the new method are discussed in detail. Based on these results, the new method is convergent under reasonable conditions for any non-Hermitian positive-definite linear system. Finally, the numerical examples show that the new method is more effective than the Hermitian and skew-Hermitian splitting iterative (or positive-definite and skew-Hermitian splitting iterative) method in central processing unit time. [ABSTRACT FROM PUBLISHER]

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver. [ABSTRACT FROM AUTHOR]

In this article, He's variational iteration method has been applied to find the optimal control of linear systems, approximately. Numerical results are given for several test examples involving scalar and second-order systems to demonstrate the applicability and efficiency of the method. [ABSTRACT FROM AUTHOR]

The SOR-like method with two real parameters ω and α is considered for solving the augmented system. The new method is called the modified SOR-like method (MSOR-like method). The MSOR-like method becomes the SOR-like method when α = 0. The functional equation relating the parameters and eigenvalues of the iteration matrix of the MSOR-like method is obtained. Hence the necessary and sufficient condition for the convergence of the GSOR-like method is derived. It is shown that when α is negative, the convergence domain for the parameter ω for the MSOR-like method is larger than that for the SOR-like method. Finally, a numerical computation based on a particular linear system is given which clearly shows that the MSOR-like method outperforms the SOR-like method. [ABSTRACT FROM AUTHOR]

A second-order finite difference scheme derived from rotated discretisation formula is employed in conjunction with a preconditioner to obtain highly accurate and fast numerical solution of the two-dimensional elliptic partial differential equation. The use of a 'splitting' preconditioning strategy will be shown to improve the spectral properties of the matrix of the linear system resulting from this discretisation by minimising the eigenvalue spectrum of the transformed matrix. The application of this technique to several acceleration iterative methods, such as Simultaneous displacement, Richardson's and Chebyshev accelerated methods, are presented and discussed. † E-mail: dj.evans@ntu.ac.uk [ABSTRACT FROM AUTHOR]

In [5] a new iterative method is given for the linear system of equations Au=b , where A is large, sparse and nonsymmetrical and A^{\rm T}+A is symmetric and positive definite (SPD) or equivalently A is positive real. The new iterative method is based on a mixed-type splitting of the matrix A and is called the mixed-type splitting iterative method. The iterative method contains an auxiliary matrix D_1 that is restricted to be symmetric. In this note, the auxiliary matrix is allowed to be more general and it is shown that by proper choice of D 1 , the new iterative method is still convergent. It is also shown that by special choice of D_{1} , the new iterative method becomes the well-known (point) accelerated overrelaxation (AOR) [1] method. Hence, it is shown that the (point) AOR method applied to the positive real system is convergent under the proper choice of the overrelaxation parameters ω and τ. [ABSTRACT FROM AUTHOR]

Two classes of asynchronous relaxed parallel two-stage multisplitting methods based on extrapolated and AOR methods are studied for the solution of nonsingular linear systems, which are called asynchronous outer relaxed or inner relaxed parallel two-stage multisplitting methods. Convergence of these methods is studied for H-matrix. Almost all methods seen in literatures can be viewed as special cases of our methods. [ABSTRACT FROM AUTHOR]