153 research outputs found
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
From approximating to interpolatory non-stationary subdivision schemes with the same generation properties
In this paper we describe a general, computationally feasible strategy to
deduce a family of interpolatory non-stationary subdivision schemes from a
symmetric non-stationary, non-interpolatory one satisfying quite mild
assumptions. To achieve this result we extend our previous work [C.Conti,
L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to
full generality by removing additional assumptions on the input symbols. For
the so obtained interpolatory schemes we prove that they are capable of
reproducing the same exponential polynomial space as the one generated by the
original approximating scheme. Moreover, we specialize the computational
methods for the case of symbols obtained by shifted non-stationary affine
combinations of exponential B-splines, that are at the basis of most
non-stationary subdivision schemes. In this case we find that the associated
family of interpolatory symbols can be determined to satisfy a suitable set of
generalized interpolating conditions at the set of the zeros (with reversed
signs) of the input symbol. Finally, we discuss some computational examples by
showing that the proposed approach can yield novel smooth non-stationary
interpolatory subdivision schemes possessing very interesting reproduction
properties
Computing a Hurwitz factorization of a polynomial
AbstractA polynomial is called a Hurwitz polynomial (sometimes, when the coefficients are real, a stable polynomial) if all its roots have real part strictly less than zero. In this paper we present a numerical method for computing the coefficients of the Hurwitz factor f(z) of a polynomial p(z). It is based on a polynomial description of the classical LR algorithm for solving the matrix eigenvalue problem. Similarly with the matrix iteration, it turns out that the proposed scheme has a global linear convergence and, moreover, the convergence rate can be improved by considering the technique of shifting. Our numerical experiments, performed with several test polynomials, indicate that the algorithm has good stability properties since the computed approximation errors are generally in accordance with the estimated condition numbers of the desired factors
Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials
Subdivision schemes are iterative methods for the design of smooth curves and
surfaces. Any linear subdivision scheme can be identified by a sequence of
Laurent polynomials, also called subdivision symbols, which describe the linear
rules determining successive refinements of coarse initial meshes. One
important property of subdivision schemes is their capability of exactly
reproducing in the limit specific types of functions from which the data is
sampled. Indeed, this property is linked to the approximation order of the
scheme and to its regularity. When the capability of reproducing polynomials is
required, it is possible to define a family of subdivision schemes that allows
to meet various demands for balancing approximation order, regularity and
support size. The members of this family are known in the literature with the
name of pseudo-splines. In case reproduction of exponential polynomials instead
of polynomials is requested, the resulting family turns out to be the
non-stationary counterpart of the one of pseudo-splines, that we here call the
family of exponential pseudo-splines. The goal of this work is to derive the
explicit expressions of the subdivision symbols of exponential pseudo-splines
and to study their symmetry properties as well as their convergence and
regularity.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
Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems
In this paper we consider the application of polynomial root-finding methods to the
solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed results at a modest increase in computational cost. Numerical experiments illustrate that under some restriction on the conditioning the novel iterations can approximate and/or refine the eigenvalues of a tridiagonal matrix with high relative accuracy
Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices
It is well known that if a matrix solves the
matrix equation , where is a linear bivariate polynomial,
then is normal; and can be simultaneously reduced in a finite
number of operations to tridiagonal form by a unitary congruence and, moreover,
the spectrum of is located on a straight line in the complex plane. In this
paper we present some generalizations of these properties for almost normal
matrices which satisfy certain quadratic matrix equations arising in the study
of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure
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