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In this paper we take a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of very small dimension to a known vector which is, in turn, computed accurately by exploiting high-order rational Chebyshev and Padé approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Further parallelism is introduced by expanding the rational approximations into partial fractions. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.
We propose a method for the solution of linear systems AX B where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order n s with s of moderate size. The method uses a single Krylov subspace per step as a generator of approximations, a projection process, and a Richardson acceleration technique. It thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right-hand sides. Numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right-hand sides.
Summary.A wide range of computational kernels in data mining and information retrieval from text collections involve techniques from linear algebra. These kernels typically operate on data that is presented in the form of large sparse term-document matrices (tdm). We present TMG, a research and teaching toolbox for the generation of sparse tdm's from text collections and for the incremental modification of these tdm's by means of additions or deletions. The toolbox is written entirely in MATLAB, a popular problem solving environment that is powerful in computational linear algebra, in order to streamline document preprocessing and prototyping of algorithms for information retrieval. Several design issues that concern the use of MATLAB sparse infrastructure and data structures are addressed. We illustrate the use of the tool in numerical explorations of the effect of stemming and different term-weighting policies on the performance of querying and clustering tasks.
Motivated by a recent method of Freund , who introduced a quasi-minimal residual (QMR) version of the CGS algorithm, we propose a QMR variant of the Bi-CGSTAB algorithm of van der Vorst, which we call QMRCGSTAB for solving nonsymmetric linear systems. The motivation for both QMR variants is to obtain smoother convergence behavior of the underlying method. We illustrate this by numerical experiments, which also show that for problems on which Bi-CGSTAB performs better than CGS, the same advantage carries over to QMRCGSTAB.
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