A structure preserving flow for computing Hamiltonian matrix exponential

Canonical Forms for Hamiltonian and Symplectic Matrices and Pencils

This paper gives a constructive method for finding canonical forms for symplectic and Hamiltonian matrices. No restrictions are made on the eigen values or their multiplicity. Real canonical forms are treated in detail.

On the construction of real canonical forms of Hamiltonian matrices whose spectrum is an imaginary pair

Physical interpretation of the symplectic orthogonality of the eigensolutions of a hamiltonian or symplectic matrix

Max-plus representation for the fundamental solution of the time-varying differential Riccati equation

The effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity is investigated. Particular attention is paid tolinear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of ascalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary, complex orthogonal and symplectic matrices. Under reasonable assumptions on the scalar product, it is shown that there is little or no difference between structured and unstructured condition numbers and distance to singularity for matrices in Lie and Jordan algebras. Hence, for these classes of matrices, the usual unstructured perturbation analysis is sufficient. It is shown that this is not true in general for structures in automorphism groups. Bounds and computable expressions for the structured condition numbers for a linear system and matrix inversion are derived for these nonlinear structures.Structured backward errors for the approximate solution of linear systems are also considered. Conditions are given for the structured backward error to be finite. For Lie and Jordan algebras it is proved that, whenever the structured backward error is finite, it is within a small factor of or equal to the unstructured one. The same conclusion holds for orthogonal and unitary structures but cannot easily be extended to other matrix groups. This work extends and unifies earlier analyses.

An SVD-like matrix decomposition and its applications

This paper investigates the effect of structure‐preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo‐symmetric, persymmetric, skew‐symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.

Analyzing the positivity preservation of numerical methods for the Liouville-von Neumann equation

In recent years there has been a growing interest in the dynamics of matrix differential systems on a smooth manifold. Research effort extends to both theory and numerical methods, particularly on the manifolds of orthogonal and symplectic matrices. This paper concerns dynamical systems on the manifold OB (n) of square oblique rotation matrices, a constraint appearing in some minimization problems and in multivariate data analysis. Background and theoretical results on differential equations on OB (n) are provided. Moreover, numerical procedures preserving the structure of the solution are found among known quadratic invariant preserving methods. Numerical tests and simulations on the oblique Procrustes problem are also reported.

Direct application of the phase estimation algorithm to find the eigenvalues of the Hamiltonians

Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.

Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary wave or front solutions. In this paper, we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure -- with two-form denoted by Omega - associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts. We introduce the concept of the "symplectic Evans matrix", a matrix consisting of restricted "Omega-symplectic" forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter lambda to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large $|\lambda|$ behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case. The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation. By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained. The theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrodinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.

We consider three numerical methods – one based on power series, one on the Magnus series and matrix exponentials, and one a library initial value code – for solving a linear system arising in non‐selfadjoint ODE eigenproblems. We show that in general, none of these methods has a cost or an accuracy which is uniform in the eigenparameter, but that for certain special types of problem, the Magnus method does yield eigenparameter‐uniform accuracy. This property of the Magnus method is explained by a trajectory‐shadowing result which, unfortunately, does not generalize to higher order Magnus type methods such as those in [11,12].

We extend the regime-switching model to the rich class of time-changed Lévy processes and use the Fourier cosine expansion (COS) method to price several options under the resulting models. The extension of the COS method to price under the regime-switching model is not straightforward because it requires the evaluation of the characteristic function which is based on a matrix exponentiation which is not an easy task. For a two-state economy, we give an analytical expression for computing this matrix exponential, and for more than two states, we use the Carathéodory–Fejér approximation to find the option prices efficiently. In the new framework developed here, it is possible to allow switches not only in the model parameters as is commonly done in literature, but we can also completely switch among various popular financial models under different regimes without any additional computational cost. Calibration of the different regime-switching models with real market data shows that the best models are the regime-switching time-changed Lévy models. As expected by the error analysis, the COS method converges exponentially and thus outperforms all other numerical methods that have been proposed so far.