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The purpose of this paper is to present an algorithm for matrix multiplication based on a formula discovered by Pan [7]. For matrices of order up to 10 000, the nearly optimum tuning of the algorithm results in a rather clear non-recursive one- or two-level structure with the operation count comparable to that of the Strassen algorithm [9]. The algorithm takes less workspace and has better numerical stability as compared to the Strassen algorithm, especially in Winograd's modification [2]. Moreover, its clearer and more flexible structure is potentially more suitable for efficient implementation on modern supercomputers. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The method of elastic multibody systems is frequently used to describe the dynamical behavior of the mechanical subsystems in multi-physics simulations. One important issue for the simulation of elastic multibody systems is the error-controlled reduction of the flexible body's degrees of freedom. By the use of second order frequency-weighted Gramian matrix based reduction techniques the distribution of the loads is taken into account a-priori and very accurate models can be obtained within a predefined frequency range and even a-priori error bounds are available. However, the calculation of the frequency-weighted Gramian matrices requires high computational effort. Hence, appropriate approximation schemes have to be used to find the dominant eigenspace of these matrices. In the current contribution, the matrix integral needed for calculating the Gramian matrices is approximated by quadratures using integral kernel snapshots. The number and location of these snapshots have a strong influence on the reduction results. Sophisticated snapshot selection methods based on Greedy algorithms from the reduced basis methods are used to construct the optimal location of snapshot frequencies. The method can be viewed as an automatic determination of optimal frequency weighting and as an adaptive learning of quadrature rules. One ingredient of Greedy algorithms is the need of error measures. To gain computational advantage two different error estimators are derived and used in the Greedy algorithm instead of the absolute or relative error. [ABSTRACT FROM AUTHOR]

In this work, we apply the ideas of domain decomposition and multi-grid methods to PDE-based eigenvalue problems represented in two equivalent variational formulations. To find the lowest eigenpair, we use a “subspace correction” framework for deriving the multiplicative algorithm for minimizing the Rayleigh quotient of the current iteration. By considering an equivalent minimization formulation proposed by Mathew and Reddy, we can use the theory of multiplicative Schwarz algorithms for non-linear optimization developed by Tai and Espedal to analyse the convergence properties of the proposed algorithm. We discuss the application of the multiplicative algorithm to the problem of simultaneous computation of several eigenfunctions also formulated in a variational form. Numerical results are presented. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]