Wang Deren

On the convergence of the parallel multisplitting AOR algorithm

Abstract We present a class of relaxed parallel multisplitting algorithms, called the parallel multisplitting AOR algorithm, for solving large nonsingular systems of equations Ax = b . This new algorithm is a generalization and improvement of the relaxed parallel multisplitting method [ Linear Algebra Appl. 119:141–152 (1989)]. Based on the new algorithm model, we establish another algorithm called the relaxed parallel multisplitting AOR algorithm. The convergence of these algorithms is discussed; under the condition that A is a monotone matrix, we obtain corresponding convergence results. These convergence conditions are convenient to verify.

The theory of Smale's point estimation and its applications

Abstract The main result of this paper is that we exact Smales point estimation theory, i.e., without assuming γ k = ‖P′(z) −1 P (k) (z) k! ‖ (k ⩾ 2) being bounded by γ, the point estimation convergence theorem of the Ne method is set up by making use of the majorizing method. The proof of the theorem is simple and precise, while the required point estimation conditions are weaker than all those of known point estimation convergence theorems. Another result of this paper is an application of the above new theory to the Durand-Kerner method. We compare the point estimation conditions for the Durand-Kerner method with other known point estimation conditions. Numerical results show that our results have evident advantages.

Models of asynchronous parallel matrix multisplitting relaxed iterations

Abstract By making use of the principle of sufficiently using the delayed information, we propose two models of asynchronous parallel matrix multisplitting accelerated overrelaxation iterative methods for solving systems of linear equations, which have the merits of convenient computations, flexible and free communications, and can cover all the known synchronous as well as asynchronous parallel matrix multisplitting relaxation methods and their special cases. When the coefficient matrix is an H-matrix, we prove the convergence and estimate the convergence rates of these models in a detailed manner.

On the monotone convergence of multisplitting method for a class of systems of weakly nonlinear equations

In this paper, we set up a parallel matrix multisplitting iterative method for a class of system of weakly nonlinear equations, Au = G(u), A∊L(R n), G:R n →R n , which is generally resulted from the discretization of many classical differential equations. For the new method, the two-sided approximation property is deliberately shown, and the comparison theorems between the convergence rates of different multisplit-tings as well as multisplitting and single splittings of the coefficient matrix A∊L(R n ) are given in detail in the sense of monotonicity. Therefore, the monotone convergence theory about this method is thoroghly established. Finally, we apply the built conclusions to several special but very important and practical multisplittings to confirm the correctness and effectiveness of our theory.

Matrix multi-splitting multi-parameter relaxation methods

In this paper, we propose a class of matrix multisplitting multiparameter relaxation methods, for solving large nonsingular systems of equations. This new class of method includes the well known matrix multisplitting relaxation methods such as the matrix multisplitting SOR, methods as well as the extrapolated matrix multisplitting AOR method as its special cases, as well as the matrix multisplitting SSOR and SAOR methods. It therefore forms a series of relaxation methods in the sense of matrix multisplitting which affords more flexible choices for practical application and also makes the parallel computation of serial relaxation methods become possible. The convergence theory of this new class of methods is established under the condition that the coefficient matrix of the system of equations is an H-matrix.

PARALLEL MULTILEVEL ITERATIVE METHODS

Abstract For large-scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self-adjoint elliptic boundary-value problem, by making use of blocked multilevel iteration we construct preconditioning matrices for the coefficient matrix and set up a class of parallel multilevel iterative methods for solving such system. Theoretical analysis shows that besides lending themselves to strongly parallel computation these new methods have convergence rates independent of both the sizes and the level numbers of the grids, and their computational work loads are also bounded by linear functions about the step sizes of the finest grids.

On the determination of the safe initial approximation for the Durand-Kerner algorithm

Abstract In this paper, applying the majorant function method, we present a new proof of the convergence of the Durand-Kerner method, and obtain a greater computable radius estimation of safe initial discs. This result is an improvement of all results now available. Furthermore, combining the Kuhn algorithm for solving all zeros of polynomials, we obtain a discriminant to get safe initial discs. Finally, we compare the complexity between the Durand-Kerner algorithm and several known results, and the numerical results show the superiority of our result.

A CLASS OF FACTORIZATION UPDATE ALGORITHM FOR SOLVING SYSTEMS OF SPARSE NONLINEAR EQUATIONS

In this paper, we establish a class of sparse update algorithm based on matrix triangular factorizations for solving a system of sparse equations. The localQ-superlinear convergence of the algorithm is proved without introducing anm-step refactorization. We compare the numerical results of the new algorithm with those of the known algorithms, The comparison implies that the new algorithm is satisfactory.

Asynchronous multisplitting relaxed iterations for weakly nonlinear systems

In this paper, we propose a class of asynchronous parallel multisplitting accelerated overrelaxation methods for solving the system of weakly nonlinear equations Aφ(x) + Bψ(X) = G(x) with A, BeL(Rn), φ, ψ:Rn →Rn being diagonal mappings andG:Rn→Rn a general mapping, which is constantly resulted from the discretization of many classical differential equations.Under suitable conditions of both the coefficient matrices and the nonlinear mappings, as well as reasonable constraints of the multiple splittings and the relaxation parameters, the global convergence theories of these new methods are set up thoroughly.

On the optimal properties of the krawczyk-type interval operator∗

In this paper we discuss a kind of Krawczyk-type interval operator for solving a system of the nonlinear equations, and obtain that: i) The existence test condition presented in [4] without the interval operations is further studied. Combining the interval operator B(X A), we obtain an existence test which is easier to apply. ii) Some important properties of the interval operator B(X A) are discussed. Particularly we prove that the above existence test and the condition are equivalent. iii) Optimal properties in the same sense of the interval operator B(X A) are discussed, and the function relationship between the eigenvalues of the matrix P = ∣I − AL∣ and the matrix A is given. They provide a basis for the optimal choice of the matrix A. For the Krawczyk-type interval operator, these are new results. Of all these facts, some are an improvement and extension of previous results; some provide useful conditions for constructing more efficient interval algorithms.