6 research outputs found
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
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 in lower staircase form using flops and 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
Fast Hessenberg reduction of some rank structured matrices
We develop two fast algorithms for Hessenberg reduction of a structured
matrix where is a real or unitary diagonal
matrix and . The proposed algorithm for the
real case exploits a two--stage approach by first reducing the matrix to a
generalized Hessenberg form and then completing the reduction by annihilation
of the unwanted sub-diagonals. It is shown that the novel method requires
arithmetic operations and it is significantly faster than other
reduction algorithms for rank structured matrices. The method is then extended
to the unitary plus low rank case by using a block analogue of the CMV form of
unitary matrices. It is shown that a block Lanczos-type procedure for the block
tridiagonalization of induces a structured reduction on in a block
staircase CMV--type shape. Then, we present a numerically stable method for
performing this reduction using unitary transformations and we show how to
generalize the sub-diagonal elimination to this shape, while still being able
to provide a condensed representation for the reduced matrix. In this way the
complexity still remains linear in and, moreover, the resulting algorithm
can be adapted to deal efficiently with block companion matrices.Comment: 25 page
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