44 research outputs found
Perturbation theory for the eigenvalues of factorised symmetric matrices
AbstractWe obtain eigenvalue perturbation results for a factorised Hermitian matrix H=GJG∗ where J2=I and G has full row rank and is perturbed into G+δG, where δG is small with respect to G. This complements the earlier results on the easier case of G with full column rank. Applied to square factors G our results help to identify the so-called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices
On some properties of the Lyapunov equation for damped systems
We consider a damped linear vibrational system whose dampers
depend linearly on the viscosity parameter v. We show that the
trace of the corresponding Lyapunov solution can be represented as
a rational function of v whose poles are the eigenvalues of a
certain skew symmetric matrix. This makes it possible to derive an
asymptotic expansion of the solution in the neighborhood of zero
(small damping)
Block diagonalization of nearly diagonal matrices
In this paper we study the effect of block diagonalization of a nearly diagonal matrix by iterating the related Riccati equations. We show that the iteration is fast, if a matrix is diagonally dominant or scaled diagonally dominant and the block partition follows an appropriately defined spectral gap. We also show that both kinds of diagonal dominance are not destroyed after the block diagonalization
Spectral gap of segments of periodic waveguides
We consider a periodic strip in the plane and the associated quantum
waveguide with Dirichlet boundary conditions. We analyse finite segments of the
waveguide consisting of periodicity cells, equipped with periodic boundary
conditions at the ``new'' boundaries. Our main result is that the distance
between the first and second eigenvalue of such a finite segment behaves like
.Comment: 3 page
Novel Modifications of Parallel Jacobi Algorithms
We describe two main classes of one-sided trigonometric and hyperbolic
Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian
matrices. These types of algorithms exhibit significant advantages over many
other eigenvalue algorithms. If the matrices permit, both types of algorithms
compute the eigenvalues and eigenvectors with high relative accuracy.
We present novel parallelization techniques for both trigonometric and
hyperbolic classes of algorithms, as well as some new ideas on how pivoting in
each cycle of the algorithm can improve the speed of the parallel one-sided
algorithms. These parallelization approaches are applicable to both
distributed-memory and shared-memory machines.
The numerical testing performed indicates that the hyperbolic algorithms may
be superior to the trigonometric ones, although, in theory, the latter seem
more natural.Comment: Accepted for publication in Numerical Algorithm
A GPU-based hyperbolic SVD algorithm
A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm,
using a massively parallel graphics processing unit (GPU), is developed. The
algorithm also serves as the final stage of solving a symmetric indefinite
eigenvalue problem. Numerical testing demonstrates the gains in speed and
accuracy over sequential and MPI-parallelized variants of similar Jacobi-type
HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are
discussed.Comment: Accepted for publication in BIT Numerical Mathematic
Block-Diagonalization of Operators with Gaps, with Applications to Dirac Operators
We present new results on the block-diagonalization of Dirac operators on
three-dimensional Euclidean space with unbounded potentials. Classes of
admissible potentials include electromagnetic potentials with strong Coulomb
singularities and more general matrix-valued potentials, even non-self-adjoint
ones. For the Coulomb potential, we achieve an exact diagonalization up to
nuclear charge Z=124 and prove the convergence of the Douglas-Kroll-He\ss\
approximation up to Z=62, thus improving the upper bounds Z=93 and Z=51,
respectively, by H.\ Siedentop and E.\ Stockmeyer considerably. These results
follow from abstract theorems on perturbations of spectral subspaces of
operators with gaps, which are based on a method of H.\ Langer and C.\ Tretter
and are also of independent interest
Conditional Wegner Estimate for the Standard Random Breather Potential
We prove a conditional Wegner estimate for Schr\"odinger operators with
random potentials of breather type. More precisely, we reduce the proof of the
Wegner estimate to a scale free unique continuation principle. The relevance of
such unique continuation principles has been emphasized in previous papers, in
particular in recent years.
We consider the standard breather model, meaning that the single site
potential is the characteristic function of a ball or a cube. While our methods
work for a substantially larger class of random breather potentials, we discuss
in this particular paper only the standard model in order to make the arguments
and ideas easily accessible
Scattering theory for Klein-Gordon equations with non-positive energy
We study the scattering theory for charged Klein-Gordon equations:
\{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x,
D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)=
f_{1}, {array}. where: \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq
n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x),
describing a Klein-Gordon field minimally coupled to an external
electromagnetic field described by the electric potential and magnetic
potential . The flow of the Klein-Gordon equation preserves the
energy: h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+
\bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x)
\d x. We consider the situation when the energy is not positive. In this
case the flow cannot be written as a unitary group on a Hilbert space, and the
Klein-Gordon equation may have complex eigenfrequencies. Using the theory of
definitizable operators on Krein spaces and time-dependent methods, we prove
the existence and completeness of wave operators, both in the short- and
long-range cases. The range of the wave operators are characterized in terms of
the spectral theory of the generator, as in the usual Hilbert space case