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Real and Integer Extended Rank Reduction Formulas and Matrix Decompositions: A Review

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Numerical Analysis and Optimization

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 134))

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Abstract

We have recently developed an extended rank reducing process for rank reduction of a matrix leading to various matrix decompositions containing the Abaffy-Broyden-Spedicato (ABS) and Wedderburn processes. Notably, the extended process contains both the Wedderburn biconjugation process and the scaled extended ABS class of algorithms. The process provides a general finite iterative approach for constructing factorizations of a matrix and its transpose under a common framework of a general decomposition having various useful structures such as triangular, orthogonal, diagonal, banded and Hessenberg and many others. One main new result is the derivation of an extended rank reducing process for an integer matrix leading to the so-called Smith normal form. For this process, to solve the arising quadratic Diophantine equations, we have proposed two algorithms. Here, we report some numerical results on randomly generated test problems showing a better performance of one algorithm, based on a recent ABS algorithm, in controlling the size of the solution. We also report results obtained by our algorithm on the Smith normal form having a more balanced distribution of the intermediate values as compared to the ones obtained by Maple.

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Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Nezam Mahdavi-Amiri

  2. Department of Mathematics, University of Qom, Qom, Iran

    Effat Golpar-Raboky

Authors
  1. Nezam Mahdavi-Amiri
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  2. Effat Golpar-Raboky
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Corresponding author

Correspondence to Nezam Mahdavi-Amiri .

Editor information

Editors and Affiliations

  1. Department of Mathematics College of Science, Sultan Qaboos University, Muscat, Oman

    Mehiddin Al-Baali

  2. Faculty of Engineering, University of Calabria, Arcavacada di Rende, Italy

    Lucio Grandinetti

  3. Department Mathematics and Statistics College of Science, Sultan Qaboos University, Muscat, Oman

    Anton Purnama

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Mahdavi-Amiri, N., Golpar-Raboky, E. (2015). Real and Integer Extended Rank Reduction Formulas and Matrix Decompositions: A Review. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. Springer Proceedings in Mathematics & Statistics, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-17689-5_10

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