Heike Faßbender

Hamilton and Jacobi come full circle: Jacobi algorithms for structured Hamiltonian eigenproblems☆

We develop Jacobi algorithms for solving the complete eigenproblem for Hamiltonian and skew-Hamiltonian matrices that are also symmetric or skew-symmetric. Based on the direct solution of 4×4, and in one case, 8×8 subproblems, these structure preserving algorithms produce symplectic orthogonal bases for the invariant subspaces associated with a matrix in any one of the four classes under consideration. The key step in the construction of the algorithms is a quaternion characterization of the 4×4 symplectic orthogonal group, and the subspaces of 4×4 Hamiltonian, skew-Hamiltonian, symmetric and skew-symmetric matrices. In addition to preserving structure, these algorithms are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.

The Symplectic Eigenvalue Problem, the Butterfly Form, the SR Algorithm, and the Lanczos Method

Abstract We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2n × 2n symplectic matrix can be reduced to this condensed form which contains 8n − 4 nonzero entries and is determined by 4n − 1 parameters. The symplectic eigenvalue problem can be solved using the SR algorithm based on this condensed form. The SR algorithm preserves this form and can be modified to work only with the 4n − 1 parameters instead of the 4n2 matrix elements. The reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to Hessenberg form. A Lanczos-like algorithm for reducing a symplectic matrix to butterfly form is also presented.

On numerical methods for discrete least-squares approximation by trigonometric polynomials

Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in [0,2π) by trigonometric polynomials are presented. The algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only O(mn) arithmetic operations as compared to O(mn 2 ) operations needed for algorithms that ignore the structure of the problem. An algorithm which solves this problem with real-valued data and real-valued solution using only real arithmetic is given. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.

SR and SZ algorithms for the symplectic (butterfly) eigenproblem

Abstract SR and SZ algorithms for the symplectic (generalized) eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed. A 2 n × 2 n symplectic butterfly matrix has 8 n − 4 (generically) nonzero entries, which are determined by 4 n − 1 parameters. While the SR algorithm operates directly on the matrix entries, the SZ algorithm works with the 4 n − 1 parameters. The algorithms are made more compact and efficient by using Laurent polynomials, instead of standard polynomials, to drive the iterations.

Two connections between the SR and HR eigenvalue algorithms

Abstract The SR and HR algorithms are members of the family of GR algorithms for calculating eigenvalues and invariant subspaces of matrices. This paper makes two connections between the SR and HR algorithms: (1) An iteration of the SR algorithm on a 2 n × 2 n symplectic butterfly matrix using shifts μ i , μ −1 i , i = 1,…, k , is equivalent to an iteration of the HR algorithm on an n × n tridiagonal sign-symmetric matrix using shifts μ i + μ −1 i , i = 1,…, k . (2) An iteration of the SR algorithm on a 2 n × 2 n J -tridagonal Hamiltonian matrix using shifts μ i , − μ i , i = 1,…, k , is equivalent to an iteration of the HR algorithm on an n × n tridiagonal sign-symmetric matrix using shifts μ 2 i , i = 1,…, k .

On the solution of the rational matrix equation X = Q + LX -1 L T

We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation, where is symmetric positive definite and is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.

A fully adaptive rational global Arnoldi method for the model-order reduction of second-order MIMO systems with proportional damping

The model order reduction of second-order dynamical multi-input and multi-output (MIMO) systems with proportional damping arising in the numerical simulation of mechanical structures is discussed. Based on finite element modelling the systems describing the mechanical structures are large and sparse, either undamped or proportionally damped. This work concentrates on a new model reduction algorithm for such second order MIMO systems which automatically generates a reduced system approximating the transfer function in the lower range of frequencies. The method is based on the rational global Arnoldi method. It determines the expansion points iteratively. The reduced order and the number of moments matched per expansion point are determined adaptively using a heuristic based on some error estimation. Numerical examples comparing our results to modal reduction and reduction via the rational block Arnoldi method are presented.

Breaking Van Loan’s Curse: A Quest forStructure-Preserving Algorithms for Dense Structured Eigenvalue Problems

In 1981 Paige and Van Loan (Linear Algebra Appl 41:11–32, 1981) posed the open question to derive an \(\mathcal{O}(n^{3})\) numerically strongly backwards stable method to compute the real Hamiltonian Schur form of a Hamiltonian matrix. This problem is known as Van Loan’s curse. This chapter summarizes Volker Mehrmann’s work on dense structured eigenvalue problems, in particular, on Hamiltonian and symplectic eigenproblems. In the course of about 35 years working on and off on these problems the curse has been lifted by him and his co-workers. In particular, his work on SR methods and on URV-based methods for dense Hamiltonian and symplectic matrices and matrix pencils is reviewed. Moreover, his work on structure-preserving methods for other structured eigenproblems is discussed.

On the Solution of the Rational Matrix Equation

We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation, where is symmetric positive definite and is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.

Numerische Methoden zur passivitätserhaltenden Modellreduktion Numerical Methods for Passivity Preserving Model Reduction

Abstract Bei der Modellierung dynamischer Systeme entstehen heutzutage häufig Systeme hoher Ordnung (d.h. mit 10000 und mehr Gleichungen). Um eine numerische Simulation mit akzeptablem zeitlichem Umfang zu gewährleisten, reduziert man das gegebene dynamische System von Gleichungen zu einem System derselben Form, welches eine Lösung mit stark verkürzter Rechenzeit erlaubt. Häufig wird gefordert, dass das reduzierte System die selben Eigenschaften wie das unreduzierte Modell aufweist; wichtige Eigenschaften sind in diesem Zusammenhang insbesondere Stabilität und Passivität. Dieser Beitrag gibt einen Überblick über numerische Verfahren zur passivitätserhaltenden Modellreduktion.