Two-stage Iterative Method Research Articles

Convergence of two-stage iterative methods for singular symmetric positive semidefinite systems*

Convergence of two-stage iterative methods for singular symmetric positive semidefinite (spsd) systems is studied. The main tool we used to derive the iterative methods and to analyze their convergence is the extended diagonally compensated reduction

The two-stage iterative methods for symmetric positive definite matrices

Two-stage iterative methods for the solution of the symmetric positive definite linear system Ax= b are considered. In this paper, in particular: strong conditions and feeble conditions on splittings are given so that the two-stage methods are convergent for any number of inner iterations and for any nonnegative even number of inner iterations, respectively. Some suitable examples of strong P-regular splittings are presented and the convergence of some feasible two-stage methods is discussed, in which the outer splitting is weak P-regular and the inner splitting is strong or weak P-regular splittings.

The monotonicity of two-stage iterative methods

Two-stage iterative methods for the solution of linear systems are analyzed when the coefficient matrix is Hermitian positive definite. Comparison theorems, based on the number of inner iterations performed, are given.

A class of asynchronous multisplitting two-stage iterations for large sparse block systems of weakly nonlinear equations

For the block system of weakly nonlinear equations Ax= G( x), where A∈ R n×n is a large sparse block matrix and G: R n→ R n is a block nonlinear mapping having certain smoothness properties, we present a class of asynchronous parallel multisplitting block two-stage iteration methods in this paper. These methods are actually the block variants and generalizations of the asynchronous multisplitting two-stage iteration methods studied by Bai and Huang (Journal of Computational and Applied Mathematics 93(1) (1998) 13–33), and they can achieve high parallel efficiency of the multiprocessor system, especially, when there is load imbalance. Under quite general conditions that A∈ R n×n is a block H-matrix of different types and G: R n→ R n is a block P-bounded mapping, we establish convergence theories of these asynchronous multisplitting block two-stage iteration methods. Numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment.

Two-stage iterative methods for consistent Hermitian positive semidefinite systems

We study stationary and nonstationary two-stage iterative methods for the solution of consistent and singular linear systems Ax= b, where A is a Hermitian positive semidefinite matrix. When the outer splitting is P-regular and inner splitting is convergent, we prove the convergence of stationary two-stage iterative methods with inner iteration number q being a positive even number; conditions are given to ensure the convergence of stationary iterative methods for any positive inner iteration number q. Also, we present some theoretical results on the convergence of nonstationary two-stage methods. A comparison theorem is obtained for Hermitian positive semidefinite matrix and some applications to it are discussed. Moreover, we obtain a monotonicity result on the stationary two-stage iterative methods.

A class of parallel hybrid two-stage iteration methods for block bordered linear systems

A class of parallel hybrid two-stage iteration method and its accelerated overrelaxation variant are established for solving the large sparse block bordered system of linear equations, and their convergence are proved when the coefficient matrix of this linear system is an M-matrix and an H-matrix, respectively.

Asynchronous multisplitting two-stage iterations for systems of weakly nonlinear equations

For the large sparse systems of weakly nonlinear equations arising in the discretizations of many classical differential and integral equations, this paper presents a class of asynchronous parallel multisplitting two-stage iteration methods for getting their solutions by the high-speed multiprocessor systems. Under suitable assumptions, we study the global convergence properties of these asynchronous multisplitting two-stage iteration methods. Moreover, for this class of new methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some reasonable assumptions when the involved nonlinear mapping is only assumed to be directionally differentiable. Numerical computations show that our new methods are feasible and efficient for parallely solving the system of weakly nonlinear equations.

The convergence of the two-stage iterative method for Hermitian positive definite linear systems

This paper sets up the convergence theory of the two-stage iterative method for solving Hermitian positive definite systems of linear equations, and investigates the influences of the splitting matrices and the inner iteration number on the asymptotic convergence rate of this method.

Convergence of two-stage iterative methods for Hermitian positive definite matrices

Two-stage iterative methods for the solution of linear systems are studied. Convergence of both stationary and nonstationary cases is analyzed when the coefficient matrix is Hermitian positive definite.

The monotone convergence of the two-stage iterative method for solving large sparse systems of linear equations

This paper sets up the monotone convergence theory for the two-stage iterative method proposed by Frommer and Szyld in [1], and investigates the influence of the splitting matrices and the inner iteration number sequence on the monotone convergence rate of this method.

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations.

An efficient two-stage iterative method for the steady-state analysis of Markov regenerative stochastic Petri net models

Abstract To enhance the modeling power of stochastic Petri nets (SPNs), new steady-state analysis methods have been proposed for nets that include non-exponential transitions. The underlying stochastic process is a Markov regenerative process (MRP) when at most one non-exponential transition is enabled in each marking. Time-efficient algorithms for constructing and solving the MRP have been developed. However, the space required to solve such models is often extremely large. This largeness is due to the large number of transitions in the MRP. Traditional analysis methods require that all these transitions be stored in primary memory for efficient computation. If the size of available memory is smaller than that needed to store these transitions, a time-efficient computation is impossible using these methods. To use this class of SPNs to model realistic systems, the space complexity of MRP analysis algorithms must be reduced. In this paper, we propose a new steady-state analysis method that is both time and space efficient. The new method takes advantage of the structure of the underlying process to reduce both computation time and required memory. The performance of the proposed method is compared to existing methods using several SPN examples.

An efficient two-stage iterative method for the steady-state analysis of Markov regenerative stochastic Petri net models

To enhance the modeling power of stochastic Petri nets (SPNs), new steady-state analysis methods have been proposed for nets that include non-exponential transitions. The underlying stochastic process is a Markov regenerative process (MRP) when at most one non-exponential transition is enabled in each marking. Time-efficient algorithms for constructing and solving the MRP have been developed. However, the space required to solve such models is often extremely large. This largeness is due to the large number of transitions in the MRP. Traditional analysis methods require that all these transitions be stored in primary memory for efficient computation. If the size of available memory is smaller than that needed to store these transitions, a time-efficient computation is impossible using these methods. To use this class of SPNs to model realistic systems, the space complexity of MRP analysis algorithms must be reduced. In this paper, we propose a new steady-state analysis method that is both time and space efficient. The new method takes advantage of the structure of the underlying process to reduce both computation time and required memory. The performance of the proposed method is compared to existing methods using several SPN examples.

Asynchronous two-stage iterative methods

Parallel block two-stage iterative methods for the solution of linear systems of algebraic equations are studied. Convergence is shown for monotone matrices and for H-matrices. Two different asynchronous versions of these methods are considered and their convergence investigated.

H-Splittings and two-stage iterative methods

Convergence of two-stage iterative methods for the solution of linear systems is studied. Convergence of the non-stationary method is shown if the number of inner iterations becomes sufficiently large. TheR 1-factor of the two-stage method is related to the spectral radius of the iteration matrix of the outer splitting. Convergence is further studied for splittings ofH-matrices. These matrices are not necessarily monotone. Conditions on the splittings are given so that the two-stage method is convergent for any number of inner iterations.

Convergence of nested classical iterative methods for linear systems

Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.

Two-stage parallel iterative methods for the symmetric linear complementarity problem

In this paper, we propose a two-stage parallel iterative method for solving the symmetric linear complementarity problem. When implemented in a parallel computing environment, the method decomposes the problem into subproblems which are solved by certain iterative procedures concurrently on separate processors. Convergence of the overall method is established under some mild assumptions on how the inner iterations are terminated. Applications of the proposed method to solve strictly convex quadratic programs are discused and numerical results on both a sequential computer (IBM 4381) and a super-computer (CRAYX-MP/24) are reported.

Toward a Fundamental Theory of Optimal Feature Selection: Part II-Implementation and Computational Complexit

Certain algorithms and their computational complexity are examined for use in a VLSI implementation of the real-time pattern classifier described in Part I of this work. The most computationally intensive processing is found in the classifier training mode wherein subsets of the largest and smallest eigenvalues and associated eigenvectors of the input data covariance pair must be computed. It is shown that if the matrix of interest is centrosymmetric and the method for eigensystem decomposition is operator-based, the problem architecture assumes a parallel form. Such a matrix structure is found in a wide variety of pattern recognition and speech and signal processing applications. Each of the parallel channels requires only two specialized matrix-arithmetic modules. These modules may be implemented as linear arrays of processing elements having at most O(N) elements where N is the input data vector dimension. The computations may be done in O(N) time steps. This compares favorably to O(N3) operations for a conventional, or general, rotation-based eigensystem solver and even the O(2N2) operations using an approach incorporating the fast Levinson algorithm for a matrix of Toeplitz structure since the underlying matrix in this work does not possess a Toeplitz structure. Some examples are provided on the convergence of a conventional iterative approach and a novel two-stage iterative method for eigensystem decomposition.

Fifth order implicit multipoint method for solving equations

We have derived a two-stage implicit fifth order multipoint iterative method for solving equations. Further, we have also obtained an implicit third order and a semiexplicit fourth order multipoint method. Comparison of computational results are made with other well-known methods on a number of difficult problems. The implicit multipoint methods are accurate and robust.

On the two-stage iterative method of Douglas for mildly nonlinear elliptic difference equations

On the two-stage iterative method of Douglas for mildly nonlinear elliptic difference equations