Leaf Anatomy and Systematics of Polygaleae (Polygalaceae)

Polygalaceae are comprised of approximately 1000 species found worldwide in a variety of habitats. Over half of the species in the family were traditionally grouped into the polyphyletic genus Polygala and need resolution. Leaves from 21 species in 13 genera within tribe Polygaleae were field collected or taken from specimens at the Missouri Botanical Garden Herbarium (MO) to find phylogenetically useful characteristics of the foliar anatomy. The leaves were rehydrated using concentrated ammonium hydroxide, embedded in paraffin, and sectioned at 10 μm thickness using a rotary microtome. To observe epidermal features, rehydrated leaves were also cleared using 3% potassium hydroxide. Of the characteristics observed, 13 showed consistent variation and were mapped onto a previously published molecular phylogeny to find diagnosable features for the various clades of Polygaleae. While foliar anatomy showed a high degree of homoplasy, there were a few informative characteristics. The most distinct clade was that of Phlebotaenia and Rhinotropis, which were united based on having a plane margin, fibrous vascular bundles, an absence of hairs on both epidermises, and a thick adaxial cuticle

Computing with functions in spherical and polar geometries I. The sphere

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.Comment: 23 page

Numerical Computing with Functions on the Sphere and Disk

A new low rank approximation method for computing with functions in polar and spherical geometries is developed. By synthesizing a classic procedure known as the double Fourier sphere (DFS) method with a structure-preserving variant of Gaussian elimination, approximants to functions on the sphere and disk can be constructed that (1) preserve the bi-periodicity of the sphere, (2) are smooth over the poles of the sphere (and origin of the disk), (3) allow for the use of FFT-based algorithms, and (4) are near-optimal in their underlying discretizations. This method is used to develop a suite of fast, scalable algorithms that exploit the low rank form of approximants to reduce many operations to essentially 1D procedures. This includes algorithms for differentiation, integration, and vector calculus. Combining these ideas with Fourier and ultraspherical spectral methods results in an optimal complexity solver for Poisson\u27s equation, which can be used to solve problems with 108 degrees of freedom in just under a minute on a laptop computer. All of these algorithms have been implemented and are publicly available in the open-source computing system called Chebfun [21]

Computing numerically with rational functions

180 pagesNew numerical methods using rational functions are presented for applications in linear algebra and signal processing. Classical results from Zolotarev are applied to develop a collection of low rank methods and theoretical results for computing with matrices that have special displacement structures using the alternating direction implicit (ADI) method. This includes a new low rank method for solving Sylvester and Lyapunov matrix equations with right hand sides that have decaying singular values, spectrally accurate low rank solvers for certain elliptic partial differential equations with smooth right-hand sides, and explicit bounds on the singular values of special families of structured matrices. Methods from conformal mapping and adaptive rational approximation are applied to build approximate solutions to Zolotarev's problem on sets where solutions are not known. This leads to new bounds on the numerical ranks of matrices, and it generalizes the regime in which ADI-based methods can be applied. The approximate solutions supply quasi-optimal ADI shift parameters for solving Sylvester matrix equations. A superfast rank-structured solver for Toeplitz linear systems is designed with ADI-based compression methods, and theoretical arguments are supplied that justify the effectiveness of rank-structured solvers for Toeplitz and related linear systems. The solvers are competitive with the state of the art, and rational approximation arguments are used to derive explicit error bounds on the numerical ranks of important submatrices for various weakly admissible hierarchical formats. A data-driven rational approximation framework is developed for reconstructing signals from samples with poorly separated spectral content. This approach combines a variant of Prony's method with a modified version of the AAA algorithm to construct representations of signals in both frequency and time space. The approximation methods are automatic and adaptive, requiring no tuning or manual parameter selection, and they are robust to various forms of corruption, including additive Gaussian noise, perturbed sampling grids, and missing data. A collection of algorithms and an accompanying software package for adaptively computing with these representations is introduced that includes procedures for differentiation/integration, rootfinding/polefinding, convolution, filtering, extrapolation, and more