Averaging operators for exponential splittings

In this paper, we are concerned with the derivation of a local error representation for exponential operator splitting methods when applied to evolutionary problems that involve critical parameters. Employing an abstract formulation of differential equations on function spaces, our framework includes Schrodinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients. We illustrate the general mechanism on the basis of the first-order Lie splitting and the second-order Strang splitting method. Further, we specify the local error representation for a fourth-order splitting scheme by Yoshida. From the given error estimate it is concluded that higher-order exponential operator splitting methods are favourable for the time-integration of linear Schrodinger equations in the semi-classical regime with critical parameter 0<e≪1, provided that the time stepsize h is sufficiently smaller than $\sqrt[p]{\varepsilon}$ , where p denotes the order of the splitting method.

We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more than two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schrödinger equation with an external magnetic field or with an unbounded potential.

Permutability of Toeplitz and Hankel matrices

Triangular factorization of general N × N Hermitian matrices requires O(N 3) operations. We present a procedure that factors certain structured matrices in O(N 2) operations. The family of structured matrices considered in this paper includes Toeplitz and Hankel matrices, as well as a variety of ‘Toeplitz-like’ and ‘Hankel-like’ matrices (e.g. inverses, sums and products of Toeplitz and Hankel matrices).

CONTENTS Introduction Chapter I. The algebraic structure of the spaces of Toeplitz and Hankel matrices § 1. Basic definitions § 2. Basic notation § 3. Statement of the theorems § 4. Auxiliary lemmas § 5. Proof of the theorems Chapter II. Fast computations in some problems of linear algebra connected with Toeplitz and Hankel matrices § 1. Basic definitions and notation § 2. Statement of the theorems § 3. Auxiliary lemmas § 4. Proof of the theorems Chapter III. The fast prediction of stochastic processes § 1. The fast prediction of stationary stochastic processes § 2. The fast prediction of piecewise constant random functions Chapter IV. The structure of Toeplitz and Hankel integral operators § 1. Definitions § 2. Notation § 3. Statement of the theorems References

Some identities for determinants of structured matrices

On algebras of Toeplitz plus Hankel matrices

Conditions for commuting a Toeplitz matrix and a Hankel matrix were obtained relatively recently (in 2015). The solution to the problem of describing all anti-commuting pairs (T, H), where T is a Toeplitz matrix and H is a Hankel matrix, is sketched below.

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

Kernel structure of block Hankel and Toeplitz matrices and partial realization

Prior work on high-order exponential operator splitting methods is extended to evolution equations defined by three linear operators. A posteriori local error estimators are constructed via a suitable integral representation of the local error involving the defect associated with the splitting solution and quadrature approximation via Hermite interpolation. In order to prove asymptotical correctness, a multiple integral representation involving iterated defects is deduced by repeated application of the variation-of-constant formula. The error analysis within the framework of abstract evolution equations provides the basis for concrete applications. Numerical examples for initial-boundary value problems of Schrödinger and of parabolic type confirm the asymptotical correctness of the proposed a posteriori error estimators.

We say that square matrices A and B of the same order quasi-commute if AB = σBA for a certain scalar σ. The classical definitions of commutation and anti-commutation are particular cases of the above definition. We give a complete description of pairs of quasi-commuting Toeplitz and Hankel matrices for σ ≠ ±1.

A Widom like formula for some Toeplitz plus Hankel determinants

We consider the Hecke algebra on homogeneous trees. We prove that it is a maximal abelian subalgebra of some operator algebras if the degree of the tree is greater than 2. There we show the influence of geometry of the tree on that fact. If the degree is 2 (for example, in the case of integers) then we show that operators which commute with the Hecke algebra can be uniquely represented as a sum of Hankel and Toeplitz matrices.

In the present work, we investigate the error behaviour of exponential operator splitting methods for nonlinear evolutionary problems. In particular, our concern is to deduce an exact local error representation that is suitable in the presence of critical parameters. Essential tools in the theoretical analysis including time-dependent nonlinear Schrodinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients are an abstract formulation of differential equations on function spaces and the formal calculus of Lie-derivatives. We expose the general mechanism on the basis of the least technical example method, the first-order Lie–Trotter splitting