Implicit QR for Companion-like Pencils

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems . The modified QZ algorithm computes the generalized eigenvalues of certain NxN rank structured matrix pencils using O(N^2) ops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method

A CMV--based eigensolver for companion matrices

In this paper we present a novel matrix method for polynomial rootfinding. By exploiting the properties of the QR eigenvalue algorithm applied to a suitable CMV-like form of a companion matrix we design a fast and computationally simple structured QR iteration.Comment: 14 pages, 4 figure

Zerofinding of analytic functions by structured matrix methods

We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV--like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples

A CMV--based eigensolver for companion matrices

In this paper we present a novel matrix method for polynomial rootfinding. The roots are approximated by computing the eigenvalues of a permuted version of the companion matrix associated with the polynomial in block upper Hessenberg form with possibly nonsquare subdiagonal blocks. It is shown that this form, referred to as a lower staircase form of the companion matrix in reference to its characteristic appearance, is well suited for the application of the QR eigenvalue algorithm. In particular, each matrix generated under this iteration is block upper Hessenberg and, moreover, all its submatrices located in a specified upper triangular portion are of rank two at most with entries represented by means of four given vectors. By exploiting these properties we design a fast and computationally simple structured QR iteration which computes the eigenvalues of a companion matrix of size $n$ in lower staircase form using $O(n^2)$ flops and $O(n)$ memory storage. This iteration is theoretically faster than other fast variants of the QR iteration for companion matrices in customary Hessenberg form. Numerical experiments show the efficiency and the accuracy of the proposed approach

Computing with quasiseparable matrices

International audienceThe 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. ISSAC'15]. This allows us to propose an algorithm computing the quasiseparable orders (rL, rU) in time O(n^2 s^(ω−2)) where s = max(rL, rU) and ω 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^(ω−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^(ω−2))

Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

Abstract(#br)In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair $$(A, \, B)$$ ( A , B ) with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair $$(A,\, B)$$ ( A , B ) , where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector..

Implicit QR for companion-like pencils

International audienceA fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zero-finding problems. The modified QZ algorithm computes the generalized eigenvalues of certain $N\times N$ rank structured matrix pencils using $O(N^2)$ flops and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method. - See more at: http://www.ams.org/journals/mcom/2016-85-300/S0025-5718-2015-03020-8/#sthash.JxbTFYBc.dpu