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Bounds for the Distance to the Nearest Correlation Matrix
Get PDFIn a wide range of practical problems correlation matrices are formed in such a way that, while symmetry and a unit diagonal are assured, they may lack semidefiniteness. We derive a variety of new upper bounds for the distance from an arbitrary symmetric matrix to the nearest correlation matrix. The bounds are of two main classes: those based on the eigensystem and those based on a modified Cholesky factorization. Bounds from both classes have a computational cost of flops for a matrix of order but are much less expensive to evaluate than the nearest correlation matrix itself. For unit diagonal with for all the eigensystem bounds are shown to overestimate the distance by a factor at most . We show that for a collection of matrices from the literature and from practical applications the eigensystem-based bounds are often good order of magnitude estimates of the actual distance; indeed the best upper bound is never more than a factor larger than a related lower bound. The modified Cholesky bounds are less sharp but also less expensive, and they provide an efficient way to test for definiteness of the putative correlation matrix. Both classes of bounds enable a user to identify an invalid correlation matrix relatively cheaply and to decide whether to revisit its construction or to compute a replacement, such as the nearest correlation matrix
Hamiltonian relative equilibria with continuous isotropy
Get PDFIn symmetric Hamiltonian systems, relative equilibria usually arise in continuous families. The geometry of these families in the setting of free actions of the symmetry group is well-understood. Here we consider the question for non-free actions. Some results are already known in this direction, and we use the so-called bundle equations to provide a systematic treatment of this question which both consolidates the known results, extending the scope of the results to deal with non-compact symmetry groups, as well as producing new results. Specifically we address questions about the stability, persistence and bifurcations of these relative equilibria
Coadjoint orbits of the special Euclidean group
Get PDFIn this report we classify the coadjoint orbits for compact semisimple Lie groups by establishing a correspondence between orbits and subsets of Dynkin diagrams. Particular
attention is given to the special unitary and orthogonal groups for which the orbits are complex flags and real Hermitian flags respectively. The orbits for non-compact and nonsemisimple affine groups are discussed and in the example of the special Euclidean group a geometric bijection between adjoint and coadjoint orbits is found
Matrix Depot: An Extensible Test Matrix Collection for Julia
Get PDFMatrix Depot is a Julia software package that provides easy access to a large and diverse collection of test matrices. Its novelty is threefold. First, it is extensible by the user, and so can be adapted to include the user's own test problems. In doing so it facilitates experimentation and makes it easier to carry out reproducible research. Second, it amalgamates in a single framework three different types of matrix collections, comprising parametrized test matrices, regularization test problems, and real-life sparse matrix data. Third, it fully exploits the Julia language. It uses multiple dispatch to help provide a simple interface and, in particular, to allow matrices to be generated in any of the numeric data types supported by the language
Polynomial Zigzag Matrices, Dual Minimal Bases, and the Realization of Completely Singular Polynomials
Get PDFMinimal bases of rational vector spaces are a well known
and important tool in systems theory.
If minimal bases for two subspaces of rational -space
are displayed as the rows of polynomial matrices
Z_1(\la)_{k \times n} and Z_2(\la)_{m \times n}, respectively,
then and are said to be
dual minimal bases
if the subspaces have complementary dimension,
i.e., ,
and Z_1^{}(\la) Z_2^T(\la) = 0.
In other words, each Z_j(\la) provides a minimal basis
for the nullspace of the other.
It has long been known that for any dual minimal bases Z_1(\la) and Z_2(\la),
the row degree sums of and are the same.
In this paper we show that this is the only constraint on the row degrees,
thus characterizing the possible row degrees of dual minimal bases.
The proof is constructive, making extensive use
of a new class of sparse, structured polynomial matrices
that we have baptized zigzag matrices.
Another application of these polynomial zigzag matrices
is the constructive solution of the following inverse problem for minimal indices --
given a list of left and right minimal indices and a desired degree ,
does there exist a completely singular matrix polynomial
(i.e., a matrix polynomial with no elementary divisors whatsoever)
of degree
having exactly the prescribed minimal indices?
We show that such a matrix polynomial exists
if and only if divides the sum of the minimal indices.
The constructed realization is simple,
and explicitly displays the desired minimal indices
in a fashion analogous
to the classical Kronecker canonical form
of singular pencils
Tropical roots as approximations to eigenvalues of matrix polynomials
Get PDFThe tropical roots of
are points at which the maximum is attained at least twice.
These roots, which can be computed in only operations, can be good approximations to the moduli of
the eigenvalues of the matrix polynomial , in particular when the norms of the matrices vary widely.
Our aim is to investigate this observation and its applications.
We start by providing annuli defined in terms of the tropical roots of that contain the eigenvalues of .
Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of .
Our tropical localization of eigenvalues are less tight than
eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret.
Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the Ehrlich-Aberth method
and our results further justify this choice.
Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for
on (a) the conditioning of linearizations of tropically scaled and
(b) the backward stability of eigensolvers based on linearizations of tropically scaled .
We anticipate that the tropical roots of , on which
the tropical scalings are based, will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems
such as that implemented in the MATLAB function \texttt{polyeig}
Adjoint and coadjoint orbits of the Euclidean group
Get PDFWe give a geometric description of the adjoint and coadjoint orbits of the special Euclidean group. We implement the method of little subgroups as introduced by Rawnsley in 1975 and the method of types by Burgoyne and Cushman in 1977 to classify these orbits completely. The orbits are diffeomorphic to affine flag manifolds, whose definition and geometry we also explore. Since coadjoint orbits are naturally symplectic, such manifolds provide us with interesting examples of symplectic homogeneous spaces.
As discovered by Cushman and van der Kallen in 2006, we identify a bijection between the coadjoint and adjoint orbits of the Euclidean group. Furthermore, we show that orbits corresponding under this bijection are homotopy equivalent. Whether the bijection for other groups, and especially the Poincare group, preserves homotopy type of orbits remains an open question
Efficient adaptive stochastic Galerkin methods for parametric operator equations
Get PDFThis paper is concerned with the design and implementation of efficient solution algorithms for elliptic PDE problems with correlated random data. The energy orthogonality that is built into stochastic Galerkin approximations is cleverly exploited to give an innovative energy error estimation strategy that utilizes the tensor product structure of the approximation space. An associated error estimator is constructed and shown theoretically and numerically to be an effective mechanism for driving an adaptive refinement process. The codes used in the numerical studies are available online
Are resultant methods numerically unstable for multidimensional rootfinding?
Get PDFHidden-variable resultant methods are a class of algorithms for solving
multidimensional polynomial rootfinding problems. In two dimensions, when
significant care is taken, they are competitive practical rootfinders. However, in higher
dimensions they are known to miss zeros, calculate roots to low precision,
and introduce spurious solutions. We show that
the hidden-variable resultant method based on
the Cayley (Dixon or Bézout) resultant is inherently
and spectacularly numerically unstable by a factor that grows exponentially with
the dimension. We also show that the Sylvester resultant for solving
bivariate polynomial systems can square the condition number of the
problem.
In other words, two popular hidden-variable resultant methods are numerically
unstable, and this mathematically explains the difficulties that
are frequently reported by practitioners. Along the way, we prove that the
Cayley resultant is a generalization of Cramer's rule for solving linear systems
and generalize Clenshaw's algorithm to an evaluation scheme for polynomials
expressed in a degree-graded polynomial basis
Existence of symmetric central configurations
Get PDFCentral configurations have been of great interest over many years, with the earliest examples due to Euler and Lagrange. There are numerous results in the literature demonstrating the existence of central configurations with specific symmetry properties, using slightly different techniques in each. The aim here is to describe a uniform approach by adapting to the symmetric case the well-known variational argument showing the existence of central configurations. The principal conclusion is that there is a central configuration for every possible symmetry type, and for any symmetric choice of masses. Finally the same argument is applied to the class of balanced configurations introduced by Albouy and Chenciner