Angelika Bunse-Gerstner

A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

A method is presented to solve the real algebraic Riccati equation - XNX + XA + A^{T}X + K = 0 , where K = K^{T} and N = N^{T} . The solution for the corresponding eigenvalue problem Mx = lambda x , where M is a Hamiltonian matrix, is computed by an algorithm similar to the QR algorithm. Special symplectic matrices are used for the transformation of M such that the Hamiltonian form is preserved during the computations.

An analysis of the HR algorithm for computing the eigenvalues of a matrix

Abstract The HR algorithm, a method of computing the eigenvalues of a matrix, is presented. It is based on the fact that almost every complex square matrix A can be decomposed into a product A = HR of a so-called pseudo-Hermitian matrix H and an upper triangular matrix R . This algorithm is easily seen to be a generalization of the well-known QR algorithm. It is shown how it is related to the power method and inverse iteration, and for special matrices the connection between the LR and HR algorithms is indicated.

Matrix factorizations for symplectic QR-like methods

Abstract Symplectic QR -like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the QR process. A fundamental problem for the development of such methods is the choice of a suitable type of decomposition A = SR corresponding to the QR decomposition, where S is symplectic or unitary symplectic. Decompositions of this type are studied with regard to their application in a QR -like process.

Schur parameter pencils for the solution of the unitary eigenproblem

Abstract Let U − λV be an n × n pencil with unitary matrices U and V . An algorithm is presented which reduces U and V simultaneously to unitary block diagonal matrices G o = Q H UP and G e = Q H VP with block size at most two. It is an O ( n 3 ) process using Householder eliminations, and it is backward stable. In the special case V = I the block diagonal matrices G o , G H e can be normalized so that their entries are just the Schur parameters of the Hessenberg condensed form of U . We call G o − λG e a Schur parameter pencil. It can also be derived from U,V by a Lanczos-like process. For the solution of the eigenvalue problem for G o − λG e a QR -type algorithm can be developed based on this unitary reduction of a pencil U − λV to a Schur parameter pencil. The condensed form is preserved throughout the process. Each iteration step needs only O ( n ) operations. This method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the QR method for the Hermitian eigenvalue problem.

A quaternion QR algorithm

SummaryThis paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.

An algorithm for the symmetric generalized eigenvalue problem

Abstract A new method is presented for the solution of the matrix eigenvalue problem Ax=λBx, where A and B are real symmetric square matrices and B is positive semidefinite. It reduces A and B to diagonal form by congruence transformations that preserve the symmetry of the problem. This method is closely related to the QR algorithm for real symmetric matrices.

Using Model Reduction Methods within Incremental Four-Dimensional Variational Data Assimilation

Abstract Incremental four-dimensional variational data assimilation is the method of choice in many operational atmosphere and ocean data assimilation systems. It allows the four-dimensional variational data assimilation (4DVAR) to be implemented in a computationally efficient way by replacing the minimization of the full nonlinear 4DVAR cost function with the minimization of a series of simplified cost functions. In practice, these simplified functions are usually derived from a spatial or spectral truncation of the full system being approximated. In this paper, a new method is proposed for deriving the simplified problems in incremental 4DVAR, based on model reduction techniques developed in the field of control theory. It is shown how these techniques can be combined with incremental 4DVAR to give an assimilation method that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard t...

On the similarity transformation to tridiagonal form

For every real square matrix A there exists a nonsingular real matrix X, for which X −1 AX is a tridiagonal matrix, and under certain conditions X is uniquely determined. Simple proofs are presented for the existence and uniqueness of this transformation.

Regularization of descriptor systems

Algebraic conditions which ensure that a descriptor system can be regularized by derivative-plus-proportional output feedback are presented. The results are established using a canonical form which can be constructed by a numerically stable algorithm. The feedback can be selected so that the closed-loop system is strongly controllable and observable, has index at most one, and is optimally conditioned. >

BERECHNUNG DER EIGENWERTE EINER MATRIX MIT DEM HR-VERFAHREN

The HR-process, a method of computing the eigenvalues of a real square matrix, is presented. It leaves the pseudosymmetric form of a matrix invariant and may be considered as a generalization of the well-known QR-process. Computations are much faster with the HR-process than with the QR-process for the case of large non-symmetric matrices of tridiagonal form and for matrices which can be reduced to such a form in a well-behaved manner.