computing stable eigendecompositions of matrices abstract if a matrix t is known only to within a tolerance because of measurement or roundoff errors then it may be difficult to compute an eigendecomposition of t since its invariant subspaces are discontinuous functions of its entries in this paper we show how to compute a stable decomposition of an uncertain matrix t which varies continuously and boundedly as t varies in a ball of radius

some new algorithms for the spectral dichotomy methods abstract given a regular matrix pencil b a having no eigenvalues on the circle cr of center 0and radius r we describe a new algorithm to compute iteratively the deflating subspaces of b a corresponding to the eigenvalues inside and outside cr along with a spectral condition number that indicates the numerical quality of the computed deflating subspaces we then generalize the proposed algorithm to the case where cr is replaced by a straight line or an ellipse

a natural gradient descent algorithm for the solution of discrete algebraic lyapunov equations based on the geodesic distance a new framework to calculate the numerical solution of the discrete algebraic lyapunov equation is proposed by using the geometric structures on the riemannian manifold specifically two algorithms based on the manifold of positive definite symmetric matrices are provided one is a gradient descent algorithm with an objective function of the classical euclidean distance the other is a natural gradient descent algorithm with an objective function of the geodesic distance on the curved riemannian manifold furthermore these two algorithms are compared with a traditional iteration method simulation examples show that the convergence speed of the natural gradient descent algorithm is the fastest one among three algorithms

quantum stochastic processes for maps on hilbert c modules we discuss pairs of maps where is a map between c algebras and is a module map between hilbert c modules which are generalization of representations of hilbert c modules a covariant version of stinespring s theorem for such a pair is established and quantum stochastic processes constructed from pairs t t of families of such maps are studied we prove that the quantum stochastic process j jt constructed from a quantum dynamical semigroup t is a j map for the quantum stochastic process j jt constructed from the given quantum dynamical semigroup t and that j is covariant if the quantum dynamical semigroup is covariant

type i product systems of hilbert modules abstract christensen and evans showed that in the language of hilbert modules a bounded derivation on a von neumann algebra with values in a two sided von neumann module i e a sufficiently closed two sided hilbert module are inner then they use this result to show that the generator of a normal uniformly continuous completely positive cp semigroup on a von neumann algebra decomposes into a suitably normalized cp part and a derivation like part the backwards implication is left open in these notes we show that both statements are equivalent among themselves and equivalent to a third one namely that any type i tensor product system of von neumann modules has a unital central unit from existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered fock modules we thus find the analogue of arveson s result that type i product systems of hilbert spaces are symmetric fock spaces on the way to our results we have to develop a number of tools interesting on their own right inspired by a very similar notion due to accardi and kozyrev we introduce the notion of semigroups of completely positive definite kernels cpd semigroups being a generalization of both cp semigroups and schur semigroups of positive definite c valued kernels the structure of a type i system is determined completely by its associated cpd semigroup and the generator of the cpd semigroup replaces arveson s covariance function as another tool we give a complete characterization of morphisms among product systems of time ordered fock modules in particular the concrete form of the projection endomorphisms allows us to show that subsystems of time ordered systems are again time ordered systems and to find a necessary and sufficient criterion when a given set of units generates the whole system as a byproduct we find a couple of characterizations of other subclasses of morphisms we show that the set of contractive positive endomorphisms are order isomorphic to the set of cpd semigroups dominated by the cpd semigroup associated with type i system

circular trichotomy of the spectrum of regular matrix pencils we introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside on and outside the unit circle the method extends the iteration used in the context of spectral dichotomy where the assumption on the absence of eigenvalues on the unit circle is removed it constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1 equal to 1 and larger than or equal to 1 an orthogonalization process is then used to extract the desired delating subspaces the resulting algorithm is an inverse free easy to implement and sufficiently fast the derived convergence estimates reveal the key parameters which determine the rate of convergence the method is tested on several numerical examples

ab algorithm and its modifications for the spectral problems of linear pencils of matrices this paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular

zero temperature properties of randomly self interacting polymers we study the zero temperature behavior of several simple models for randomly self interacting polymers in one and 1 1 dimensions results are based on exact enumeration and closed form expressions

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

energy level structure of two dimensional self interacting random chains the authors study the energy level structure of the lowest lying states for a two dimensional version of the random chain model proposed by iori marinari and parisi 1991 the authors find that the multifold degeneracy of the ground state is lifted as the random noise interaction is switched on and a solitary global minimum emerges in a glassy phase as the strength of the noise term is increased this indicates that the model mimics the characteristics expected of a native protein they also show that the transition from the globule state with no noise term to the glassy phase takes place smoothly

computation of pseudospectra via spectral projectors abstract in this note we discuss new techniques for analyzing the pseudospectra of matrices and propose a numerical method for computing the spectral projector associated with a group of eigenvalues enclosed by a polygonal curve numerical tests are reported

elliptic dichotomy of a matrix spectrum abstract given a regular linear matrix pencil a b and an ellipse in the complex plane we describe an algorithm for computing the right invariant subspaces of a b associated to the eigenvalues inside and outside the ellipse the algorithm builds a particular matrix pencil of order twice the order of the pencil a b to which circular dichotomy techniques can be applied efficiently the algorithm allows also the computation of the canonical form of the pencil a b

efficient computation of the spectral projections of regular matrix pairs we consider the problem of computing the spectral projections of a regular matrix pair associated to eigenvalues that are inside the unit circle we discuss two algorithmic variants that both rely on the generalized schur decomposition with re ordering from there they both obtain a block diagonalization but in different ways one does so by solving a generalized sylvester equation while the other uses a reverse ordering of the eigenvalues in the schur decomposition the block diagonalization allows accurate computation of the needed left and right spectral projections numerical comparisons show that this technique is a strong competitor to the spectral dichotomy approach

cp h extendable maps between hilbert modules and cph semigroups abstract one may ask which maps between hilbert modules allow for a completely positive extension to a map acting block wise between the associated extended linking algebras in these notes we investigate in particular those cp extendable maps where the 22 corner of the extension can be chosen to be a homomorphism the cp h extendable maps we show that they coincide with the maps considered by asadi 4 by bhat ramesh and sumesh 9 and by skeide 28 we also give an intrinsic characterization that generalizes the characterization by abbaspour tabadkan and skeide 1 of homomorphically extendable maps as those which are ternary homomorphisms for general strictly cp extendable maps we give a factorization theorem that generalizes those of asadi of bhat ramesh and sumesh and of skeide for cp h extendable maps this theorem may be viewed as a unification of the representation theory of the algebra of adjointable operators and the ksgns construction then we examine semigroups of cp h extendable maps so called cph semigroups as an application we illustrate their relation with a new sort of generalized dilation of cp semigroups cph dilations

conformational properties of random heteropolymers in the folded phase we investigated the behaviour at zero temperature of a random self interacting chain using a steepest descent algorithm we mainly studied the energy minima landscape showing that there are few relevant energy minima at this temperature we estimate the radius of their basin of attraction

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

parallel algorithm for solving some spectral problems of linear algebra abstract an algorithm is described for computing the deflating subspaces of a regular linear matrix pencil b a more precisely the algorithm is intended to compute the projection matrices p and i p onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle this algorithm can be considered as an orthogonal version of the matrix sign function method it possesses a quadratic convergence rate and simultaneously with the projection operators operators computes a condition number of the problem we point out the extent to which this algorithm can be parallelized applications are made to solving the riccati equation

extended hamiltonian algorithm for the solution of discrete algebraic lyapunov equations in this paper we use a second order learning algorithm for solving the numerical solution of the discrete algebraic lyapunov equation specifically extended hamiltonian algorithm based on the manifold of positive definite symmetric matrices is provided furthermore this algorithm is compared with the euclidean gradient algorithm the riemannian gradient algorithm and the two traditional iteration methods simulation examples show that the convergence speed of the extended hamiltonian algorithm is the fastest one among these algorithms