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تسجيل مستخدم جديدTime and space efficient generators for quasiseparable matrices
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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 PDEs for particle interaction with the Fast Multi-pole Method (FMM), 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 2015. 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.
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The class of quasiseparable matrices is defined by a pair of bounds, called the quasiseparable orders, on the ranks of the maximal sub-matrices entirely located in their strictly lower and upper triangular parts. These arise naturally in applications
, as e.g. the inverse of band matrices, and are widely used for they admit structured representations allowing to compute with them in time linear in the dimension and quadratic with the quasiseparable order. We show, in this paper, the connection between the notion of quasisepa-rability and the rank profile matrix invariant, presented in [Dumas & al. ISSAC15]. This allows us to propose an algorithm computing the quasiseparable orders (rL, rU) in time O(n^2 s^($omega$--2)) where s = max(rL, rU) and $omega$ the exponent of matrix multiplication. We then present two new structured representations, a binary tree of PLUQ decompositions, and the Bruhat generator, using respectively O(ns log n/s) and O(ns) field elements instead of O(ns^2) for the previously known generators. We present algorithms computing these representations in time O(n^2 s^($omega$--2)). These representations allow a matrix-vector product in time linear in the size of their representation. Lastly we show how to multiply two such structured matrices in time O(n^2 s^($omega$--2)).
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When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.
Let $f_1,...,f_s in mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $I$, where $mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of matrices of traces for the factor algebra $A :=
CC[x_1, ..., x_m]/ I$, i.e. matrices with entries which are trace functions of the roots of $I$. Such matrices of traces in turn allow us to compute a system of multiplication matrices ${M_{x_i}|i=1,...,m}$ of the radical $sqrt{I}$. We first propose a method using Macaulay type resultant matrices of $f_1,...,f_s$ and a polynomial $J$ to compute moment matrices, and in particular matrices of traces for $A$. Here $J$ is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when $I$ has finitely many projective roots in $mathbb{P}^m_CC$. We also extend previous results which work only for the case where $A$ is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of $A$. Here we need the assumption that $s=m$ and $f_1,...,f_m$ define an affine complete intersection. This second method also works if we have higher dimensional components at infinity. A new explicit description of the generators of $sqrt{I}$ are given in terms of Bezoutians.
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