Abstract: Three Numerov-type methods with phase-lag of order eight and ten are developed for the numerical integration of the one-dimensional Schrodinger equation. One has a large interval of periodicity and the other two areP-stable. Extensive numerical testing on the resonance problem indicates that these new methods are generally more accurate than other previously developed finite difference methods for this problem.

Abstract: A subsetS⊂X of feasible solutions of a multicriteria optimization problem is called e-optimal w.r.t. a vector-valued functionf:X→Y $$ \subseteq $$ ℝ K if for allx∈X there is a solutionz x∈S so thatf k(z x)≤(1+e)f k (x) for allk=1,...,K. For a given accuracy e>0, a pseudopolynomial approximation algorithm for bicriteria linear programming using the lower and upper approximation of the optimal value function is given. Numerical results for the bicriteria minimum cost flow problem on NETGEN-generated examples are presented.

Abstract: We describe a preconditioned conjugate gradient solution strategy for a multiprocessor system with message passing architecture. The preconditioner combines two techniques, a Schurcomplement preconditioning over “coupling boundaries” between the subdomains and an arbitrary choice of classic preconditioning for the inner degrees of freedom on each subdomain. All computational work on the single subdomains is carried out in parallel by distributing the subdomain data over the processor network before starting the finite element solution process (including generating the element matrices and assemblying the local subdomain stiffness matrix). The resulting spectral condition number of the entire preconditioner is estimated. For the important example of choosing MIC(0)-*-preconditioning on the subdomains, the condition number obtained is essentially the product of the two condition numbers involved.

Abstract: Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat).

Abstract: New interpolants of the explicit Runge-Kutta method for the initial value problem are proposed. These interpolants are based on values of the solution and its derivative from two successive integration steps. In this paper, three interpolants withO(h6) local error (l.e.), for the fifth order solution, of the methods Fehlberg 4(5) (RKF 4(5)), Dormand and Prince 5(4) (RKDP 5(4)) and Verner 5(6) (RKV 5(6)) without extra cost are derived. An interpolant withO(h7) (l.e.) for the sixth order solution of the Verner's method with only one extra function evaluation per integration step is also constructed. The above advantages are obtained without any cost in the magnitude of the error.

Abstract: The finite element analysis in engineering applications comprises three phases: domain discretization, equation solving and error analysis. The domain discretization or mesh generation is the pre-processing phase which plays an important role in the achievement of accurate solutions. In this paper, the improvement of one particularly promising technique for generating two-dimensional meshes is presented. Our technique shows advantages and efficiency over some currently available mesh generators.

Abstract: We study convergence and order conditions of Rosenbrock type methods applied to differential-algebraic systems of the formB(y)y′=a(y), with singular matrixB. An embedded pair of methods of order 3(2) is constructed.

Abstract: The problem of computing the electrostatic potential in a one dimensional multilayer semiconductor device with quantized electrons density is analysed using results of monotone operator theory and perturbation calculus. An error estimate is proved for the discretization with Lagrange finite elements of degree one. A practical implementation of the method, using a quasi-Newton algorithm is presented together with some numerical results.

Abstract: Given a simple polygonP ofn vertices, we present an algorithm that finds the pair of points on the boundary ofP that maximizes theexternal shortest path between them. This path is defined as theexternal geodesic diameter ofP. The algorithm takes0(n2) time and requires0(n) space. Surprisingly, this problem is quite different from that of computing theinternal geodesic diameter ofP. While the internal diameter is determined by a pair of vertices ofP, this is not the case for the external diameter. Finally, we show how this algorithm can be extended to solve theall external geodesic furthest neighbours problem.

Abstract: Complex numerical methods often contain subproblems that are easy to state in mathematical form, but difficult to translate into software. Several algorithmic isues of this nature arise in implementing a Newton iteration scheme as part of a finite-difference method for two-point boundary value problems. We describe the practical as well as theoretical considerations behind the decisions included in the final code, with special emphasis on two “watchdog” strategies designed to improve reliability and allow early termination of the Newton iterates.

Abstract: We approximate the Cauchy problem by a problem in a bounded domain Ω R =(−R,R) withR>0 sufficiently large, and the boundary conditions on ∂Ω R are imposed in terms of the far field behavior of solutions to the Cauchy problem. Then we solve this approximate problem by the finite element method for the spatial variable and the difference method for the time variable. Moreover a coupled numerical scheme for the Cauchy problem is presented. The error estimates are established.

Abstract: In this paper the area true approximation of histograms by rational quadraticC1-splines is considered under constraints like convexity or monotonicity. For the existence of convex or monotone histosplines sufficient and necessary conditions are derived, which always can be satisfied by choosing the rationality parameters appropriately. Since the mentioned problems are in general not uniquely solvable histo-splines with minimal mean curvature areconstructed.

Abstract: Most convergence concepts for discretizations of nonlinear stiff initial value problems are based on one-sided Lipschitz continuity. Therefore only those stiff problems that admit moderately sized one-sided Lipschitz constants are covered in a satisfactory way by the respective theory. In the present note we show that the assumption of moderately sized one-sided Lipschitz constants is violated for many stiff problems. We recall some convergence results that are not based on one-sided Lipschitz constants; the concept of singular perturbations is one of the key issues. Numerical experience with stiff problems that are not covered by available convergence results is reported.

Abstract: Several time-optimal and spacetime-optimal systolic arrays are presented for computing a process dependence graph corresponding to the Aitken algorithm. It is shown that these arrays also can be used to compute the generalized divided differences, i.e., the coefficients of the Hermite interpolating polynomial. Multivariate generalized divided differences are shown to be efficiently computed on a 2-dimensional systolic array. The techniques also are applied to the Neville algorithm, producing similar results.

Abstract: In this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Holder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠C k [a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.

Abstract: The question discussed in this paper is “When should a code switch between stiff and non-stiff options, when should it switch between one order and another and how should it adjust its stepsize from one step to the next.” Criteria are proposed for switching between the options that become available as the integration progresses and these are presented in algorithmic form.

Abstract: Solving an initial value problem by a Rosenbrock method produces, in general, the numerical solution at (in advance unknown) gridpoints. Applications requiring frequent output (as graphics, delay differential equations, problems with driving equations) normally restrict the stepsize control of these codes and increase the computational overhead considerably. In this paper we introduce a class of continuous extensions for Rosenbrock-type methods. These extensions furnish a continuous numerical solution without affecting the efficiency of the codes. — Author's Abstract

Abstract: In this note we extend the validity of the mesh-independence principle for nonlinear operator equations and their discretizations to include operators whose derivatives are only Holder continuous.

Abstract: We derive a method for solvingN+m nonlinear algebraic equations inN+m unknownsy≠R m andz≠R N of the formA(y)z+b(y)=0, where the(N+m) × N matrixA(y) and vectorb(y) are continuously differentiable functions ofy alone. By exploiting properties of an orthonormal basis for null(A T (y)) the problem is reduced to solvingm nonlinear equations iny only. These equations are solved by Newton's method inm variables. Details of computational implementation and results are provided.

Abstract: Interval Analysis methods have been applied for obtaining the global optimum of the multimodal multivariable functions. We discuss here the multicriterion optimization problem, where several objective functions must be optimized in conflicting situations.

Abstract: For (scalar) nonlinear two-point boundary value problems of the second order, we present a programmed algorithm for proving the existence of a solution within calculated bounds. This algorithm is based on the theory of (functional-) differential inequalities applied to certain transformed problems.

Abstract: This paper shows that the nonlinear ABS algorithm [1] can be viewed as inexact Newton method [3]. From this point of view, the truncated nonlinear ABS algorithm is established and its convergence property is discussed.

Abstract: In this paper we are considering iterative methods for bounding the inverse of a matrix, which make use of interval arithmetic. We present a class of methods as a combination of ordinary Schulz's methods for only approximating the inverse matrix (see [3]) and of well-known interval Schulz's methods (see [1]). Two convergence theorems are proved. Our methods are shown to be asymptotically of the same order of convergence as the ordinary Schulz's methods being part of them. Therefore we are getting considerably more efficient interval methods by our approach than by the classical interval Schulz's methods in [1] or [5]. A numerical example is given.

Abstract: A classical problem of geometry is the following: given a convex polygon in the plane, find an inscribed polygon of shortest circumference. In this paper we generalize this problem to arbitrary polygonal paths in space and consider two cases: in the “open” case the wanted path of shortest length can have different start and end point, whereas in the “closed” case these two points must coincide. We show that finding such shortest paths can be reduced to finding a shortest path in a planar “channel”. The latter problem can be solved by an algorithm of linear-time complexity in the open as well in the closed case. Finally, we deal with constrained problems where the wanted path has to fulfill additional properties; in particular, if it has to pass straight through a further point, we show that the length of such a constrained polygonal path is a strictly convex function of some angle α, and we derive an algorithm for determining such constrained polygonal paths efficiently.

Abstract: In this paper, which carries on the considerations in [1], the structure of the global error is studied for some time discretization schemes, applied to a class of stiff initial value problems as they typically arise from the semi-discretization of parabolic initial/boundary value problems (method of lines). The implicit Euler and trapezoidal schemes and a locally one-dimensional splitting method are considered, and ‘perturbed’ asymptotic error expansions are derived which are valid independent of the stiffness (independent of the meshwidth in space). The key point within the analysis is a careful, quantitative description of the remainder term in such an expansion. The results are applicable in the method of lines setting and enable the prediction of the behavior of extrapolation algorithms for the class of problems under consideration. These theoretical considerations are illustrated by numerical examples.

Abstract: In a series of foregoing papers we have studied the structure of the global discretization error for the implicit Euler scheme and the implicit midpoint and trapezoidal rules applied to a general class of nonlinear stiff initial value problems. Full asymptotic error expansions (in the conventional sense) exist only in special situations; for the general case, asymptotic expansions in a weaker sense have been derived. In the present paper we demonstrate how these results can be used for an analysis of acceleration techniques applied to stiff problems. In particular, extrapolation and defect correction algorithms are considered. Various numerical results are presented and discussed.

Abstract: A time-discrete pseudospectral algorithm is suggested for the numerical solution of a nonlinear third order equation arising in fluidization. The nonlinear stability and convergence of the new scheme are analyzed. Numerical comparisons with available finite-difference methods are also reported which clearly indicate the superiority of the new scheme.

Abstract: We propose a modification of an algorithm introduced by Martinez (1987) for solving nonlinear least-squares problems. Like in the previous algorithm, after the calculation of an approximated Gauss-Newton directiond, we obtain the next iterate on a two-dimensional subspace which includesd. However, we simplify the process of searching the new point, and we define the plane using a scaled gradient direction, instead of the original gradient. We prove that the new algorithm has global convergence properties. We present some numerical experiments.

Abstract: We propose a method for simultaneous computation of verified bounds for the matrix functions exp (A) and ∫ 0 1 exp (As) ds where the inclusion of the integral is obtained during the computation of verified bounds for exp (A) at very little additional cost. Highly accurate results of our method are achieved by the use of advanced computer arithmetic and an implementation of dynamic precision by means of staggered correction representation.