MSMoE31
Numerical Computation with Functions and Chebfun  Part II of III
For Part I, see MSMoD31
For Part III, see MSTuD31
Date: August 10
Time: 16:0018:00
Room: 405
(Note: Click title to show the abstract.)
Organizer:
Trefethen, Lloyd N. (Univ. of Oxford)
Guettel, Stefan (The Univ. of Manchester)
Abstract: A recent theme in algorithms and software is efficient numerical computation with functions in a manner that "feels symbolic" since the accuracy is high and underlying discretizations (Chebyshev, Fourier,...) are hidden from the user. Projects of this kind include Chebfun, pychebfun, ApproxFun, and PaCAL. A pervasive theme in this work is the use of continuous analogues of familiar discrete mathematical objects and algorithms. This minisymposium will
present new developments in the areas of (1) differential and integral equations, (2) working with functions, and (3) rootfinding and linear algebra.
MSMoE311
16:0016:30
A fast and wellconditioned spectral method for solving singular integral equations
Slevinsky, Richard Mikael (Univ. of Oxford)
Abstract: From fracture mechanics and fluid dynamics to acoustic and electromagnetic scattering, boundary integral equations reduce the dimensionality of the underlying partial differential equations by one. The tradeoff for this reduction in complexity is the introduction of singular integral kernels. In this work, we use several remarkable properties of Chebyshev polynomials including their spectral convergence, their Hilbert and Cauchy transforms, and low rank bivariate approximations to construct a fast and wellconditioned spectral method.
MSMoE312
16:3017:00
Rectangular differentiation matrices
Xu, Kuan (Univ. of Oxford)
Abstract: The emergence of rectangular spectral collocation methods offers a novel but more flexible and robust way to implement boundary conditions. Moreover, it has also changed the way we view and interpret differential operators ¨C differential operators are, in fact, rectangular, not square. This talk will introduce the explicit constructions of rectangular differentiation matrices, followed by comparison with other construction methods. Properties and applications of rectangular differentiation matrices will also be discussed.
MSMoE313
17:0017:30
High accuracy Chebyshev coefficients via contour integrals
Austin, Anthony (Universtiy of Oxford)
Trefethen, Lloyd N. (Univ. of Oxford)
Abstract: Following Bornemann's work on computing Taylor coefficients to high precision by contour integrals over circles of large radius, Wang and Huybrechs have recently published a paper about computing Chebyshev coefficients to high precision by contour integrals over Bernstein ellipses of large parameter. Under certain circumstances, these methods make it possible to compute coefficients in ordinary floatingpoint arithmetic down at the level of $10^{100}$ or below. We investigate the use of such methods for generalpurpose computation with functions as in Chebfun, an in particular, the design of a simple if not optimal Chebfun "turbo" option.
MSMoE314
17:3018:00
Linearizations for computing roots of rational functions
Nakatsukasa, Yuji (Univ. of Tokyo)
Vanni, Noferini (Univ. of Manchester)
Townsend, Alex (MIT)
Abstract: The roots of a rational function in quotient form are simply those of the numerator polynomial, which can be computed via linearization. The situation is less straightforward when the rational function is given in other forms, such as partial or continued fractions. This work presents linearizations applicable to such rational functions. The linearizations have elements obtained directly from those of the rational function, resulting in significantly improved numerical stability compared with a polynomialization approach.
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Footnote: Code: TypeDateTimeRoom No.
Type : IL=Invited Lecture, SL=Special Lectures, MS=Minisymposia, IM=Industrial Minisymposia, CP=Contributed Papers, PP=Posters
Date: Mo=Monday, Tu=Tuesday, We=Wednesday, Th=Thursday, Fr=Friday
Time : A=8:309:30, B=10:0011:00, C=11:1012:10, BC=10:0012:10, D=13:3015:30, E=16:0018:00, F=19:0020:00, G=12:1013:30, H=15:3016:00
Room No.: TBA
