In this paper we describe a multi-grid algorithm for the penalty procedure of Stokes problem. It is proved that the convergence rate of the algorithm is bounded away from 1 independently of the meshsize. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.

We discuss the Hermite-type collocation method for the solution of Volterra integral equation with weakly singular kernel. The constructed approximation is a cubic spline in the continuity class C$^1$. We prove that this method is convergent with order of four.

We analyze the convergence of multigrid methods applied to finite element equations of second order with singularities caused by reentrant angles and abrupt changes in the boundary conditions. Provided much weaker demand of classical multigrid proofs, it is shown in this paper that, for symmetric and positive definite problems in the presence of singularities, multigrid algorithms with even one smoothing step converge at a rate which is independent of the number of levels or unknowns. Furthermore, we extend this result to the nonsymmetric and indefinite problems.

Matrix analysis on additive Schwarz methods as preconditioners is given in this paper. Both cases of with and without coarse mesh are considered. It is pointed out that an advantage of matrix analysis is to obtain more exact upper bound. Our numerical tests access the estimations.

Recently, M. Hanke and M. Neumann$^{[4]}$ have derived a necessary and sufficient condition on a splitting of $A=U-V$, which leads to a fixed point system, such that the iterative sequence converges to the least squares solution of minimum 2-norm of the system $Ax=b$. In this paper, we give a necessary and sufficient condition on the splitting such that the iterative sequence converges to the weighted Moore-Penrose solution of the system $Ax=b$ for every $x_0\in C^n$ and every $b\in C^m$. We also provide a necessary and sufficient condition such that the iterative sequence is convergent for every $x_0\in C^n$.

The aim of this paper is to lay a algebraic geometry foundation for constructing smoothing interpolants on curved side element. Some interpolation theorems in polynomial space are given. The main results effectively for CAGD are presented.

In this paper, we have proved that the lower bound of the number of real multiplications for computing a length $2^{t}$ real GFT(a,b) $(a=\pm 1/2,b=0\ or\ b=\pm 1/2,a=0)$ is $2^{t+1}-2t-2$ and that for computing a length $2^{t}$ real GFT(a,b)$(a=\pm 1/2, b=\pm 1/2)$ is $2^{t+1}-2$. Practical algorithms which meet the lower bounds of multiplications are given.

In this paper, a new step-size skill for a projection and contraction method$^{[10]}$ for linear programming is generalized to an iterative method$^{[22]}$ for solving nonlinear projection equation. For linear programming, our scheme is the same as that of$^{[10]}$. For complementarity problem and related problems, we give an improved algorithm by considering the new step-size skill and ALGORITHM B discussed in [22]. Numerical results are provided.

In the sense of the nonlinear multisplitting and based on the principle of sufficiently using the delayed information, we propose models of asynchronous parallel accelerated overrelaxation iteration methods for solving large scale system of nonlinear equations. Under proper conditions, we set up the local convergence theories of these new method models.