A new method for unconstrained optimization problems is presented. It belongs to the class of trust region method, in which the descent direction is sought by using the trust region steps within the restricted subspace. Because this subspace can be specified to include information about previous steps, the method is also related to a supermemory descent method without performing multiple dimensional searches. Trust region methods have attractive global convergence property. Supermemory information has good scale independence property. Since the method possesses the characteristics of both the trust region methods and the supermemory descent methods, it is endowed with rapid convergence. Numerical tests illustrate this point.

This paper deals with the stability analysis of $\theta -$methods for the numerical solution of delay differential equations (DDEs). We focus on the behaviour of such methods in the solution of the linear test equation $y^{\prime}(t)=a(t)y(t)+b(t)y(t-\tau )$, where $\tau >0$, $a(t)$ and $b(t)$ are functions from $R$ to $C$. It is proved that the linear $\theta -$method and the one-leg $\theta -$method are TGP-stable if and only if $\theta =1.$

In this paper, we study the point vortex method for 2-D Euler equation of incompressible flow on the half plane, and the explicit Euler's scheme is considered with the reflection method handling the boundary condition. Optimal error bounds for this fully discrete scheme are obtained.

We consider solving integral equations of the second kind defined on the half-line $[0,\infty)$ by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.

In this paper, the problem of computing zeros of a general degree bivariate Bernstein polynomial is considered. An efficient and robust algorithm is presented that takes into full account particular properties of the function considered. The algorithm works for rectangular as well as triangular domains. The outlined procedure can also be applied for the computation of the intersection of a Bézier patch and a plane as well as in the determination of an algebraic curve restricted to a compact domain. In particular, singular points of the algebraic curve are reliably detected.

In this paper, a modification of the bisection simplex method$^{[7]}$ is made for more general purpose use. Organized in an alternative simpler form, the modified version exploits information of the optimal value, as does the original bisection method, but no bracket on the optimal value is needed as part of input; instead, it only requires provision of an estimate $b_0$ of the optimal value and an estimate of the error bound of $b_0$ (it is not sensitive to these values though) . Moreover, a new, ratio-test-free pivoting rule is proposed, significantly reducing computational cost at each iteration. Our numerical experiments show that the method is very promising, at least for solving linear programming problems of such sizes as those tested.

Here a new kind of nonlinear wave, which is called $\delta-$wave, is described by some high resolution difference solutions for Riemann problems of one-dimensional (1-D) and two-dimensional (2-D) nonlinear hyperbolic systems in conservation laws. Some phenomena are numerically shown for the solutions of Riemann problems for 2-D gas dynamics systems.

A procedure of computing the position of the planar Stewart platform with coplanar ground points is presented avoiding the computation of Groebner basis by standard algorithm. The polynomial system resulted is triangularized. The number of arithmetic operations needed can be precisely counted.

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.
We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.

In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.