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Time and space efficient generators for quasiseparable matrices. (English) Zbl 1381.65035
Summary: The class of quasiseparable matrices is defined by the property that any submatrix entirely below or above the main diagonal has small rank, namely below a bound called the order of quasiseparability. These matrices arise naturally in solving partial differential equations for particle interaction with the Fast Multi-pole Method, or computing generalized eigenvalues. From these application fields, structured representations and algorithms have been designed in numerical linear algebra to compute with these matrices in time linear in the matrix dimension and either quadratic or cubic in the quasiseparability order. Motivated by the design of the general purpose exact linear algebra library LinBox, and by algorithmic applications in algebraic computing, we adapt existing techniques introduce novel ones to use quasiseparable matrices in exact linear algebra, where sub-cubic matrix arithmetic is available. In particular, we will show, the connection between the notion of quasiseparability and the rank profile matrix invariant, that we have introduced in [J.-G. Dumas et al., in: Proceedings of the 40th international symposium on symbolic and algebraic computation, ISSAC 2015. New York, NY: Association for Computing Machinery (ACM). 149–156 (2015; Zbl 1345.65019)]. It results in two new structured representations, one being a simpler variation on the hierarchically semiseparable storage, and the second one exploiting the generalized Bruhat decomposition. As a consequence, most basic operations, such as computing the quasiseparability orders, applying a vector, a block vector, multiplying two quasiseparable matrices together, inverting a quasiseparable matrix, can be at least as fast and often faster than previous existing algorithms.

MSC:
65F30 Other matrix algorithms (MSC2010)
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Software:
LinBox
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