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Skew-symmetric matrix polynomials and their Smith forms
Get PDFWe characterize the Smith form of skew-symmetric matrix polynomials over an arbitrary field \F,
showing that all elementary divisors occur with even multiplicity.
Restricting the class of equivalence transformations to unimodular congruences,
a Smith-like skew-symmetric canonical form
for skew-symmetric matrix polynomials is also obtained.
These results are used to analyze the eigenvalue and elementary divisor structure of matrices
expressible as products of two skew-symmetric matrices,
as well as the existence of structured linearizations
for skew-symmetric matrix polynomials.
By contrast with other classes of structured matrix polynomials (e.g., alternating or palindromic polynomials),
every regular skew-symmetric matrix polynomial
is shown to have a structured strong linearization.
While there are singular skew-symmetric polynomials of even degree for which a structured linearization is impossible,
for each odd degree we develop a skew-symmetric companion form that uniformly provides a structured linearization
for every regular and singular skew-symmetric polynomial
of that degree.
Finally, the results are applied to the construction of minimal symmetric factorizations of skew-symmetric rational matrices
Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm
Get PDFA popular method for computing the matrix logarithm is the
inverse scaling and squaring method,
which essentially carries out the steps of the scaling and squaring
method for the matrix exponential in reverse order.
Here we make several improvements to the method,
putting its development on a par with our recent version
[\emph{SIAM J. Matrix Anal.\ Appl.}, 31 (2009), pp.\ 970--989]
of the scaling and squaring method for the exponential.
In particular,
we introduce backward error analysis to replace the previous forward
error analysis;
obtain backward error bounds in terms of the quantities
, for several small integer , instead of ;
and use special techniques to compute the argument of the
Pad\'e approximant more accurately.
We derive one algorithm
that employs a Schur decomposition,
and thereby works with triangular matrices,
and another that requires only matrix multiplications and the solution
of multiple right-hand side linear systems.
Numerical experiments show the new algorithms to be generally faster and more
accurate than their existing counterparts
and suggest that the Schur-based method is the method of choice for computing
the matrix logarithm
Implementation of QR Updating Algorithms on the GPU
Get PDFThe least squares problem is an extremely useful device to represent an approximate solution to overdetermined systems, and a QR factorisation is a common method for solving least squares problems. It is often the case that multiple least squares solutions have to be computed with only minor changes in the underlying data. In this case, knowledge of the difference between the old data set and the new one can be used to update an existing QR factorisation at a reduced computational cost. However, fairly recent developments have introduced the widespread use of massively parallel computational devices known as GPUs. GPUs have allowed QR factorisations, and subsequently, least squares solutions to be calculated in a greatly reduced time. The purpose of this project is to investigate the viability of the implementation of QR updating algorithms on the GPU and attempt gain speedup with a GPU based updating algorithm over both existing sequential QR updating algorithms, and full GPU QR factorisations. The conclusion of the investigation is that GPU based updating algorithms gain speedups over their sequential analogues for almost all problem sizes, whereas the proposed algorithms only gain speedups over the full GPU QR factorisation under certain conditions
Minimal indices and minimal bases via filtrations
Get PDFIn this note we develop a new way of formulating
the notions of minimal basis and minimal indices,
based on the concept of a filtration of a vector space.
The goal is to provide useful new tools
for working with these important concepts,
as well as to gain deeper insight
into their fundamental nature.
This approach also readily reveals a strong minimality property of minimal indices, from which follows
a characterization of the vector polynomial bases
in rational vector spaces.
The effectiveness of this new formulation is further illustrated by proving two fundamental properties:
the invariance of the minimal indices
of a matrix polynomial under field extension,
and the direct sum property of minimal indices
Implementing QR Factorization Updating Algorithms on GPUs
Get PDFLinear least squares problems are commonly solved by QR factorization. When multiple solutions have to be computed with only minor changes in the underlying data, knowledge of the difference between the old data set and the new one can be used to update an existing factorization at reduced computational cost. This paper investigates the viability of implementing QR updating algorithms on GPUs. We demonstrate that GPU-based updating for removing columns achieves speed-ups of up to 13.5x compared with full GPU QR factorization. Other updates achieve speed-ups under certain conditions, and we characterize what these conditions are
The hyperbolic Schur decomposition (extended)
Get PDFWe propose a hyperbolic counterpart of the Schur decomposition, with the emphasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices
IFISS: A computational laboratory for investigating incompressible flow problems
Get PDFThe IFISS Incompressible Flow & Iterative Solver Software package contains software which can be run with MATLAB or Octave to create a computational laboratory for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions, together with state-of-the-art preconditioned iterative solvers for the resulting discrete linear equation systems.
In this paper we give a flavour of the code's main features and illustrate its applicability using several case studies. We aim to show that IFISS can be a valuable tool in both teaching and research
Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane
Get PDFThis paper provides a review of important results concerning the Geometrical Theory of Diffraction and Geometrical Optics. It also reviews the properties of the existing solution for the problem of diffraction of a time harmonic plane wave by a half-plane. New mathematical expressions are derived for the wave fields involved in the problem of diffraction of a time harmonic plane wave by a quarter-plane, including the secondary radiated waves. This leads to a precise representation of the diffraction coefficient describing the diffraction occurring at the corner of the quarter-plane. Our results for the secondary radiated waves are an important step towards finding a formula giving the corner diffraction coefficient everywhere
Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach
Get PDFWe revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, {SIAM J. Matrix Anal. Appl.}, 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials.
We also show that every pencil in the double ansatz space is intrinsically connected to a B\'{e}zout matrix, which we use to prove the eigenvalue exclusion theorem.
In addition our exposition allows for any polynomial basis and for any field.
The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpration as a space of B\'{e}zout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearization in the important practical case of a Chebyshev basis