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Conversions between barycentric, RKFUN, and Newton representations of rational interpolants
Get PDFWe derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa, O. Sète, L. N. Trefethen, arXiv preprint 1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems
Chamber Graphs of some Geometries that are Almost Buildings
Get PDFThe global structure of the chamber graph of certain rank 3 geometries that are almost buildings is determined. Computer files containing extensive details of these graphs accompany this paper
Conversions between barycentric, RKFUN, and Newton representations of rational interpolants
Get PDFWe derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa, O. Sète, L. N. Trefethen, arXiv preprint 1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems
Root polynomials and their role in the theory of matrix polynomials
No full textWe give a coherent theory of root polynomials, an algebraic tool useful for the analysis of matrix polynomials. In particular, we first survey results previously appeared in the literature, giving a formal proof for those that lacked one. We next extend some of these results providing some new concepts and related theorems, thus simplifying and expanding the theory. Then, we give some applications of root polynomials, such as the recovery of Jordan chains from linearizations of matrix polynomials, or the behaviour of Jordan chains under rational reparametrization. We also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials
Harnessing GPU Tensor Cores for Fast FP16 Arithmetic to Speed up Mixed-Precision Iterative Refinement Solvers
No full textLow-precision floating-point arithmetic is a powerful tool for accelerating scientific computing applications, especially those in artificial intelligence. Here, we present an investigation showing that other high-performance computing (HPC) applications can also harness this power. Specifically, we use the general HPC problem, , where is a large dense matrix, and a double precision (FP64) solution is needed for accuracy. Our approach is based on mixed-precision (FP16 -> FP64) iterative refinement, and we generalize and extend prior advances into a framework, for which we develop architecture-specific algorithms and highly tuned implementations. These new methods show how using half-precision Tensor Cores (FP16-TC) for the arithmetic can provide up to 4 times speedup. This is due to the performance boost that the FP16-TC provide as well as to the improved accuracy over the classical FP16 arithmetic that is obtained because the GEMM accumulation occurs in FP32 arithmetic
Information geometry for control of some stochastic processes
No full textA basic requirement in control systems is a metric that measures discrepancies between actual and desired states.
For statistically influenced systems information geometric methods provide natural
Riemannian metrics on smooth spaces of states; such manifolds arise in minimum-phase
linear systems and multi-input systems with known stochastic noise.
Commonly recurring practical situations
are `nearly' Poisson or `nearly' Uniform with
a complementarity in the geometry of these two; another involves multivariate Gaussians
and their mixtures.
Similarly we
encounter `nearly' independent Poisson, and `nearly' independent Gaussian processes. For such cases we have information geometric results and examples.
Some of these methods are applicable to control systems for statistically influenced processes, such as monitoring essential features in continuous
production of threads, films, foils and
fibre networks, and batch processing of stochastic textures
Shaking and whirling: dynamics of spiders and their webs
Get PDFAs one of a set of defence strategies, orb-web spiders shake their webs. Other spiders whirl. Mathematical models are introduced which describe these phenomena and throw light on expected oscillation frequencies and behaviour that could be compared with experiments. The models also suggest dynamical interpretations for the design of webs. In particular a new interpretation of the function of stabilimenta is given
James Hardy Wilkinson 27 September 1919 - 5 October 1986
Get PDFA personal perspective on James Hardy Wilkinson; the person, his career and his contributions