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

This 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 SVD's 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.

Numerical Methods for Stiff Two-Point Boundary Value Problems

We consider the two-point boundary value problem for stiff systems of ordinary differential equations. For systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions, a priori estimates are obtained. Problems with turning points can be treated with this theory, and we discuss this in detail. We give robust difference approximations and present error estimates for these schemes. In particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions. Numerical examples are presented for both linear and nonlinear problems.

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

Combined phylogenetic analyses reveal interfamilial relationships and patterns of floral evolution in the eudicot order Fabales

Relationships between the four families placed in the angiosperm order Fabales (Leguminosae, Polygalaceae, Quillajaceae, Surianaceae) were hitherto poorly resolved. We combine published molecular data for the chloroplast regions matK and rbcL with 66 morphological characters surveyed for 73 ingroup and two outgroup species, and use Parsimony and Bayesian approaches to explore matrices with different missing data. All combined analyses using Parsimony recovered the topology Polygalaceae (Leguminosae (Quillajaceae + Surianaceae)). Bayesian analyses with matched morphological and molecular sampling recover the same topology, but analyses based on other data recover a different Bayesian topology: ((Polygalaceae + Leguminosae) (Quillajaceae + Surianaceae)). We explore the evolution of floral characters in the context of the more consistent topology: Polygalaceae (Leguminosae (Quillajaceae + Surianaceae)). This reveals synapomorphies for (Leguminosae (Quillajaceae + Surianaceae)) as the presence of free filaments and marginal/ventral placentation, for (Quillajaceae + Surianaceae) as pentamery and apocarpy, and for Leguminosae the presence of an abaxial median sepal and unicarpellate gynoecium. An octamerous androecium is synapomorphic for Polygalaceae. The development of papilionate flowers, and the evolutionary context in which these phenotypes appeared in Leguminosae and Polygalaceae, shows that the morphologies are convergent rather than synapomorphic within Fabales.

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.