Details:
Title  Rankprofile revealing Gaussian elimination and the CUP matrix decomposition  Author(s)  ClaudePierre Jeannerod, Clément Pernet, Arne Storjohann  Type  Article in Journal  Abstract  Abstract Transforming a matrix over a field to echelon form, or decomposing the matrix as a product of structured matrices that reveal the rank profile, is a fundamental building block of computational exact linear algebra. This paper surveys the wellknown variations of such decompositions and transformations that have been proposed in the literature. We present an algorithm to compute the CUP decomposition of a matrix, adapted from the LSP algorithm of Ibarra, Moran and Hui (1982), and show reductions from the other most common Gaussian elimination based matrix transformations and decompositions to the CUP decomposition. We discuss the advantages of the CUP algorithm over other existing algorithms by studying time and space complexities: the asymptotic time complexity is rank sensitive, and comparing the constants of the leading terms, the algorithms for computing matrix invariants based on the CUP decomposition are always at least as good except in one case. We also show that the CUP algorithm, as well as the computation of other invariants such as transformation to reduced column echelon form using the CUP algorithm, all work in place, allowing for example to compute the inverse of a matrix on the same storage as the input matrix.  Keywords  Gaussian elimination, LU matrix decomposition, Echelon form, Reduced echelon form, Rank, Rank profile, Fast linear algebra, Inplace computations  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717113000631 
Language  English  Journal  Journal of Symbolic Computation  Volume  56  Number  0  Pages  46  68  Year  2013  Edition  0  Translation 
No  Refereed 
No 
