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Mosaic-Skeleton approximations

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Abstract

If a matrix has a small rank then it can be multiplied by a vector with many savings in memory and arithmetic. As was recently shown by the author, the same applies to the matrices which might be of full classical rank but have a smallmosaic rank. The mosaic-skeleton approximations seem to have imposing applications to the solution of large dense unstructured linear systems. In this paper, we propose a suitable modification of brandt's definition of an asymptotically smooth functionf(x,y). Then we considern×n matricesA n =[f(x (n) i ,y (n) j )] for quasiuniform meshes {x (n) i } and {y (n) j } in some bounded domain in them-dimensional space. For such matrices, we prove that the approximate mosaic ranks grow logarithmically inn. From practical point of view, the results obtained lead immediately toO(n logn) matrix-vector multiplication algorithms.

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Authors and Affiliations

  1. Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina 8, 117333, Moscow, Russia

    Eugene Tyrtyshnikov

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  1. Eugene Tyrtyshnikov
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The work was supported in part by the Russian Fund of Basic Research and also by the Volkswagen-Stiftung.

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Tyrtyshnikov, E. Mosaic-Skeleton approximations. Calcolo 33, 47–57 (1996). https://doi.org/10.1007/BF02575706

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  • DOI: https://doi.org/10.1007/BF02575706

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