377 research outputs found
Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication
AbstractRepresentations of real Toeplitz and Toeplitz-plus-Hankel matrices are presented that involve real trigonometric transformations (DCT, DST, DHT) and diagonal matrices. These representations can be used for fast matrix-vector multiplication. In particular, it is shown that the multiplication of an n Ă— n Toeplitz-plus-Hankel matrix by a vector requires only 4 transformations of length n plus O(n) operations
Matrix models for classical groups and ToeplitzHankel minors with applications to Chern-Simons theory and fermionic models
We study matrix integration over the classical Lie groups
and , using symmetric function theory and the equivalent formulation
in terms of determinants and minors of ToeplitzHankel matrices. We
establish a number of factorizations and expansions for such integrals, also
with insertions of irreducible characters. As a specific example, we compute
both at finite and large the partition functions, Wilson loops and Hopf
links of Chern-Simons theory on with the aforementioned symmetry
groups. The identities found for the general models translate in this context
to relations between observables of the theory. Finally, we use character
expansions to evaluate averages in random matrix ensembles of Chern-Simons
type, describing the spectra of solvable fermionic models with matrix degrees
of freedom.Comment: 32 pages, v2: Several improvements, including a Conclusions and
Outlook section, added. 36 page
Quantum algorithm and circuit design solving the Poisson equation
The Poisson equation occurs in many areas of science and engineering. Here we
focus on its numerical solution for an equation in d dimensions. In particular
we present a quantum algorithm and a scalable quantum circuit design which
approximates the solution of the Poisson equation on a grid with error
\varepsilon. We assume we are given a supersposition of function evaluations of
the right hand side of the Poisson equation. The algorithm produces a quantum
state encoding the solution. The number of quantum operations and the number of
qubits used by the circuit is almost linear in d and polylog in
\varepsilon^{-1}. We present quantum circuit modules together with performance
guarantees which can be also used for other problems.Comment: 30 pages, 9 figures. This is the revised version for publication in
New Journal of Physic
Matrix algebras and displacement decompositions
A class xi of algebras of symmetric nxn matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of xi are then exploited in a general displacement decomposition of an arbitrary nxn matrix A. Any algebra of xi is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of L1 space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp.1-25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp.49-99]) finally leads to the identification in xi of three new (non-Hessenberg) matrix algebras close to H, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas
On the Spectral Properties of Matrices Associated with Trend Filters
This paper is concerned with the spectral properties of matrices associated
with linear filters for the estimation of the underlying trend of a time
series. The interest lies in the fact that the eigenvectors can be interpreted
as the latent components of any time series that the filter smooths through the
corresponding eigenvalues. A difficulty arises because matrices associated with
trend filters are finite approximations of Toeplitz operators and therefore
very little is known about their eigenstructure, which also depends on the
boundary conditions or, equivalently, on the filters for trend estimation at
the end of the sample. Assuming reflecting boundary conditions, we derive a
time series decomposition in terms of periodic latent components and
corresponding smoothing eigenvalues. This decomposition depends on the local
polynomial regression estimator chosen for the interior. Otherwise, the
eigenvalue distribution is derived with an approximation measured by the size
of the perturbation that different boundary conditions apport to the
eigenvalues of matrices belonging to algebras with known spectral properties,
such as the Circulant or the Cosine. The analytical form of the eigenvectors is
then derived with an approximation that involves the extremes only. A further
topic investigated in the paper concerns a strategy for a filter design in the
time domain. Based on cut-off eigenvalues, new estimators are derived, that are
less variable and almost equally biased as the original estimator, based on all
the eigenvalues. Empirical examples illustrate the effectiveness of the method
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