A general preconditioning is presented in this paper for iterative solution of a problem approximated by spectral collocation methods. In this preconditioning, a parameter is introduced in the mass matrix of the finite element preconditioning so that the finite difference preconditioning and the finite element preconditioning are the special cases of this preconditioning corresponding to the specific parameters respectively. The eigenvalues of this preconditioned matrix are calculated and the condition number is investigated. The results reveal that there exsits an optimal parameter corresponding to a minimun condition number which is still less than that of the finite element precondition. The efficiency of this optimal preconditioning is demonstrated by the Richardson iteration. Around 73% - 82% iterations of the finite element preconditioning still can be reduced by this optimal parameter preconditioning.

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We present a numerical method based on a homotopy algorithm for a certain class of two-parameter eigenvalue problems. Numerical algorithm uses the predictor-corrector method for tracing homotopy curves. We show that the predictor-corrector iteration process shows quadratic convergence. Numerical examples are also given.

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Numerical experiments of the initial boundary value problem for a nonlinear degenerate parabolic equation of nondivergent form are presented in the case of dimension N=2, under zero Dirichlet condition. The emphasis is on a detailed behavior of solutions near the blow-up time. Precise growth rate of solutions is obtained. The numerical results indicate that the blow-up set and the blow-up rate are just those which the theory conjectures.

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The authors have previously developed a hybrid method combining the moire-interferometry measurement and the finite element method. This hybrid moire-interferometry and finite-element method made it possible to visualize the stress distribution in a plastic regime. However, the moire-interferometry data often contain the errors associated with the experimental measurement and subsequent fringe analysis. In the present study, first, a variational principle is derived to minimize the errors associated with the moire-interferometry measurement. This variational principle assures also the satisfaction of the equilibrium equation that is one of the governing equations in solid mechanics. On the basis of this variational principle, a hybrid moire-interferometry and finite-element method is developed for the case of elastic deformation field. Concepts of restoration energy and restoration force are introduced. The present hybrid method demonstrates automatic elimination of experimental measurement errors and smooth visualization of stress distribution. Furthermore, the present hybrid method automatically achieves the path independence of the Jintegral, restoring the path-dependence caused by the measurement errors.

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In this paper, we present a fraction-free algorithm for computing Rational Normal Forms of square matrices over the polynomial ring. The original algorithm is based on Danilevskii's method. Its principal transformations are similar to Gaussian elimination. When they are carried out exactly by computer algebra system, the difficulty lies in that the elements of intermediate matrices extremely swell and it requires much CPU-time to compute g.c.d.s and l.c.m.s for reducing rational expressions. In order to avoid such difficulty, we give a fraction-free algorithm analogous to Bareiss'single-step fraction-free elimination. We implement it on the computer algebra system REDUCE3.5. The experimental result shows the efficiency of our algorithm.

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In this paper, we investigate a synthesis problem of a type of symmetrical double-barrier structures. We derive analytical expressions for determining well width and barrier width assuming all other parameters, including desired resonant level and material properties, are specified for each case. Conditions for existence and number of solutions are discussed and resolved definitely. We also clarify possibility and the way to adjust the quantum number of the specified resonant level. The results and underlying ideas are expected to contribute to establishing design methodology of quantum size effect devices.

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