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Algebra and Geometry Combined Explains How the Mind Does Math

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Parallel Processing and Applied Mathematics (PPAM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8384))

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Abstract

This paper updates my talk on Cache Blocking for Dense Linear Algorithms since 1985 given at PPAM 11; see [11]. We again apply Dimension Theory to matrices in the Fortran and C programming languages. New Data Structures (NDS) for matrices are given. We use the GCD algorithm to transpose a \(n\) by \(m\) matrix \(A\) in CMO order, standard layout, in-place. Algebra and Geometry are used to make this idea concrete and practical; it is the reason for title of our paper: make a picture of any matrix by the GCD algorithm to convert it into direct sum of square submatrices. The picture is Geometry and the GCD algorithm is Algebra. Also, the in-place transposition of the GKK and TT algorithms will be compared. Finally, the importance of using negative integers will be used to give new results about subtraction and finding primitive roots which also make a priori in-place transpose more efficient.

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Notes

  1. 1.

    nb is the order of a square submatrix \(A_{ij}\) of \(A\) that enters a core.

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Authors and Affiliations

  1. IBM T.J. Watson Research Center, Ossining, USA

    Fred G. Gustavson

  2. Umeå University, Umeå, Sweden

    Fred G. Gustavson

Authors
  1. Fred G. Gustavson
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Correspondence to Fred G. Gustavson .

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Editors and Affiliations

  1. Institute of Computer and Information Science, Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Department of Computer Science, Knoxville, Tennessee, USA

    Jack Dongarra

  3. Institute of Computer and Information Science, Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

  4. Technical University of Denmark Informatics and Mathematical Modelling, Kongens Lyngby, Denmark

    Jerzy Waśniewski

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Gustavson, F.G. (2014). Algebra and Geometry Combined Explains How the Mind Does Math. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2013. Lecture Notes in Computer Science(), vol 8384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55224-3_1

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  • DOI: https://doi.org/10.1007/978-3-642-55224-3_1

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  • Print ISBN: 978-3-642-55223-6

  • Online ISBN: 978-3-642-55224-3

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