Ralph Byers

Solving the algebraic Riccati equation with the matrix sign function

Abstract This paper presents some improvements to the matrix-sign-function algorithm for the algebraic Riccati equation. A simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions. Scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2-by-2 algebraic Riccati equations. Numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations. Refinement also produces an error estimate. The resulting procedure is numerically stable. It compares favorably with current Schur vector-based algorithms.

Numerical methods for simultaneous diagonalization

A Jacobi-like algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations is presented. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the off-diagonal entries to zero. Its asymptotic convergence rate is shown to be quadratic and numerically stable. It preserves the special structure of real matrices, quaternion matrices, and real symmetric matrices.

Approaches to robust pole assignment

Abstract Robust pole assignment is a non-linear optimization problem in many variables. We describe numerical methods for determining robust or well-conditioned so-lutions to the problem of pole assignment by state feedback. The solutions are chosen to minimize various objective functions based on the condition number of the eigenvector matrix. Careful choice of parametrization and objective function avoids singularities and artificial variable constraints; explicit formulae for the gradient and hessian permit rigorous stopping criteria and rapid local convergence. Several computational examples are included.

Numerical computation of an analytic singular value decomposition of a matrix valued function

SummaryThis paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t)T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVDs and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

An exact line search method for solving generalized continuous-time algebraic Riccati equations

We present a Newton-like method for solving algebraic Riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of Newtons method. It avoids the problem of a disastrously large first step and accelerates convergence when Newton steps are too small or too long. The additional work to perform the line search is small relative to the work needed to calculate the Newton step.

A chart of numerical methods for structured eigenvalue problems

We consider eigenvalue problems for real and complex matrices with two of the following algebraic properties: symmetric, Hermitian, skew symmetric, skew Hermitian, symplectic, conjugate symplectic, J-symmetric, J-Hermitian, J-skew symmetric, J-skew Hermitian. In the complex case we found numerically stable algorithms that preserve and exploit both structures in 24 out of the 44 nontrivial cases with such a twofold structure. Of the remaining 20, we found algorithms that preserve part of the structure of 9 pairs. In the real case we found algorithms for all pairs studied. The algorithms are constructed from a small set of numerical tools, including orthogonal reduction to Hessenberg form, simultaneous diagonalization of commuting normal matrices, Francis’ QR algorithm, the Quaternion QR-algorithm and structure revealing, symplectic, unitary similarity transformations.

Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems,

Feedback design for regularizing descriptor systems

H_{\infty}

The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance

optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/or skew-Hamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skew-Hamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skew-Hamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure-preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure.

The Multishift QR Algorithm. Part II: Aggressive Early Deflation

This paper surveys numerical techniques for the regularization of descriptor (generalized state-space) systems by proportional and derivative feedback. We review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the Weierstras canonical form. Three condensed forms display the controllability and observability properties of a descriptor system. The condensed forms are obtained through orthogonal equivalence transformations and rank decisions, so they may be computed by numerically stable algorithms. In addition, the condensed forms display whether a descriptor system is regularizable, i.e., when the system pencil can be made to be regular by derivative and/or proportional output feedback, and, if so, what index can be achieved. Also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback.