Timo Eirola

On Smooth Decompositions of Matrices

In this paper we consider smooth orthonormal decompositions of smooth time varying matrices. Among others, we consider QR-, Schur-, and singular value decompositions, and their block-analogues. Sufficient conditions for existence of such decompositions are given and differential equations for the factors are derived. Also generic smoothness of these factors is discussed.

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.

Accelerating with rank-one updates

Abstract Consider the iteration xk + 1 = xk + H(b> − Axk) for solving Ax = b (A is n x n nonsingular). We discuss rank-one updates to improve H as an approximation to A−1 during the iteration. The update kills and reduces singular values of I − AH and thus speeds up the convergence. The algorithm terminates after at most n sweeps, and if all n sweeps are needed, then A−1 has been computed.

Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods

SummaryWe consider the question of whether multistep methods inherit in some sense quadratic first integrals possessed by the differential system being integrated. We also investigate whether, in the integration of Hamiltonian systems, multistep methods conserve the symplectic structure of the phase space.

Invariant curves of one-step methods

This paper considers the invariant sets of numerical one-step integration methods in a neighbourhood of a hyperbolic periodic solution of a nonlinear ODE. Using results from the dynamical systems theory it was possible to show that for the usual one-step methods the invariant sets areCk-circles (closed curves) for small enough stepsizeh. Here we give a direct proof for that and also show that they areO(hp)Ck-close to the true periodic trajectory, wherep is the order of the method.

Two concepts for numerical periodic solutions of ODE's

In computing a numerical approximation for a hyperbolic periodic orbit of an ordinary differential equation one can look either for periodic points of the mapping corresponding the numerical method or for an invariant set of that mapping. Here the one-step methods are considered. It is shown that both of these concepts for numerical approximation of periodic solutions are meaningful in the sense that the corresponding sets exist and are C^k-circles, provided the method is accurate enough. Also both of these are accurate to the order of the method. Further some relations of these concepts are studied.

Solution Methods for

We consider methods, both direct and iterative, for solving an

\mathbb R

{\mathbb R}

-Linear Problems in

-linear system

\mathbb C^n

Mz+ M_{\#}\overline{z}= b