Dai-Yuan (DY) conjugate gradient method is an effective method for solving large-scale unconstrained optimization problems. In this paper, a new DY method, possessing a spectral conjugate parameter βk, is presented. An attractive property of the proposed method is that the search direction generated at each iteration is descent, which is independent of the line search. Global convergence of the proposed method is also established when strong Wolfe conditions are employed. Finally, comparison experiments on impulse noise removal are reported to demonstrate the effectiveness of the proposed method.

The Conjugate Residual method, one of the iterative methods for solving linear systems, is applied to the problems with a dense coefficient matrix on distributed memory parallel computers. Based on an assumption on the computation and communication times of the proposed algorithm for parallel computers, it is shown that the optimal number of processing elements is proportional to the problem size N. The validity of the prediction is confirmed through numerical experiments on Hitachi SR2201.

We propose a set of new algorithms for linear programming. These algorithms are derived by accelerating the method of averaged convex projections for linear inequalities. We provide strict proofs for the convergence of our algorithms. The algorithms are so simple that they can be calculated by super-parallel processing. To this effect, we propose networks for implementing the algorithms. Furthermore, we provide illustrative examples to demonstrate the capability of our algorithms.

A parallel overlapping preconditioner is applied to ICCG method and the effect of the parallel preconditioning on the convergence of the method is investigated by solving large scale block tridiagonal linear systems arising from the discretization of Poisson's equation. Compared with the original ICCG method, the parallel preconditioned ICCG method can solve the problems in high parallelism with slight increasing the number of iterations. Furthermore, the speedup and the efficiency are evaluated for the parallel preconditioned ICCG method by substituting the experimental results into formulae of complexity. For example, when a domain of simulation is discretized on a 250250 rectangular grid and the preconditioner is divided into 249 smaller ones, its speedup is 146.3 with the efficiency 0.59.