Solving the algebraic Riccati equation with the matrix sign function

AbstractThis 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

A new scaling for Newton's iteration for the polar decomposition and its backward stability

This is the published version, also available here: http://dx.doi.org/10.1137/070699895.We propose a scaling scheme for Newton's iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can, for example, be the Frobenius norms of the matrix and its inverse. In exact arithmetic, for matrices with condition number no greater than $10^{16}$, with this scaling scheme no more than 9 iterations are needed for convergence to the unitary polar factor with a convergence tolerance roughly equal to $10^{-16}$. It is proved that if matrix inverses computed in finite precision arithmetic satisfy a backward-forward error model, then the numerical method is backward stable. It is also proved that Newton's method with Higham's scaling or with Frobenius norm scaling is backward stable

An Exact Line Search method for solving generalized continuous-time algebraic Riccati equations

©1998 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.We present a Newton-like method for solving algebraic Riccati equations that uses Exact Line Search to improve the sometimes erratic convergence behavior of Newton's 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

An Arithmetic for Matrix Pencils: Theory and New Algorithms

This paper introduces arithmetic-like operations on matrix pencils. The pencil-arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case. These operations give an unusual perspective on a variety of pencil related computations. We derive generalizations of monodromy matrices and the matrix exponential. A new algorithm for computing a pencil arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces

Structured condition numbers for invariant subspaces

Invariant subspaces of structured matrices are sometimes better conditioned with respect to structured perturbations than with respect to general perturbations. Sometimes they are not. This paper proposes an appropriate condition number c(S), for invariant subspaces subject to structured perturbations. Several examples compare c(S) with the unstructured condition number. The examples include block cyclic, Hamiltonian, and orthogonal matrices. This approach extends naturally to structured generalized eigenvalue problems such as palindromic matrix pencils

Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils

Working title was “Numerical Computation of Deflating Subspaces of Embedded Hamiltonian and Symplectic Pencils"We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear- quadratic optimal control problems, H infinity 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.All authors were partially supported by Deutsche Forschungsgemeinschaft, Research Grant Me 790/7-2 and Sonderforschungsbereich 393, “Numerische Simulation auf massiv parallelen Rechnern”

Dissecting antibodies induced by a chimeric yellow fever-dengue, live-attenuated, tetravalent dengue vaccine (CYD-TDV) in naive and dengue-exposed individuals

Sanofi Pasteur has developed a chimeric yellow fever–dengue, live-attenuated, tetravalent dengue vaccine (CYD-TDV) that is currently approved for use in several countries. In clinical trials, CYD-TDV was efficacious at reducing laboratory-confirmed cases of dengue disease. Efficacy varied by dengue virus (DENV) serotype and prevaccination dengue immune status. We compared the properties of antibodies in naive and DENV-exposed individuals who received CYD-TDV. We depleted specific populations of DENV-reactive antibodies from immune serum samples to estimate the contribution of serotype-cross-reactive and type-specific antibodies to neutralization. Subjects with no preexisting immunity to DENV developed neutralizing antibodies to all 4 serotypes of DENV. Further analysis demonstrated that DENV4 was mainly neutralized by type-specific antibodies whereas DENV1, DENV2, and DENV3 were mainly neutralized by serotype cross-reactive antibodies. When subjects with preexisting immunity to DENV were vaccinated, they developed higher levels of neutralizing antibodies than naive subjects who were vaccinated. In preimmune subjects, CYD-TDV boosted cross-reactive neutralizing antibodies while maintaining type-specific neutralizing antibodies acquired before vaccination. Our results demonstrate that the quality of neutralizing antibodies induced by CYD-TDV varies depending on DENV serotype and previous immune status. We discuss the implications of these results for understanding vaccine efficacy

Vertebral Tortuosity Is Associated With Increased Rate of Cardiovascular Events in Vascular Ehlers-Danlos Syndrome

Background Arterial tortuosity is associated with adverse events in Marfan and Loeys-Dietz syndromes but remains understudied in Vascular Ehlers-Danlos syndrome. Methods and Results Subjects with a pathogeni

BRAF Activation Initiates but Does Not Maintain Invasive Prostate Adenocarcinoma

Prostate cancer is the second leading cause of cancer-related deaths in men. Activation of MAP kinase signaling pathway has been implicated in advanced and androgen-independent prostate cancers, although formal genetic proof has been lacking. In the course of modeling malignant melanoma in a tyrosinase promoter transgenic system, we developed a genetically-engineered mouse (GEM) model of invasive prostate cancers, whereby an activating mutation of BRAFV600E–a mutation found in ∼10% of human prostate tumors–was targeted to the epithelial compartment of the prostate gland on the background of Ink4a/Arf deficiency. These GEM mice developed prostate gland hyperplasia with progression to rapidly growing invasive adenocarcinoma without evidence of AKT activation, providing genetic proof that activation of MAP kinase signaling is sufficient to drive prostate tumorigenesis. Importantly, genetic extinction of BRAFV600E in established prostate tumors did not lead to tumor regression, indicating that while sufficient to initiate development of invasive prostate adenocarcinoma, BRAFV600E is not required for its maintenance