Robert M. Corless - Compact Finite Differences and Cubic Splines
| Dates: | 22 November 2019 |
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| Times: | 14:00 - 15:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Speaker: | Robert M. Corless |
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Robert Corless from University of Western Ontario will be speaking at the seminar.
Abstract :
In this talk I uncover and explain—using contour integrals and residues—a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\’e scheme for generating fourth-order accurate compact finite differences gives exactly the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. I also introduce an apparently new spline-like interpolant that I call a compact cubic interpolant; this is similar to one introduced in 1972 by Swartz and Varga, but has higher order accuracy at the edges. I argue that for mildly nonuniform meshes the compact cubic approach offers some potential advantages, and even for uniform meshes offers a simple way to treat the edge conditions, relieving the user of the burden of deciding to use one of the three standard options: free (natural), complete (clamped), or “not-a-knot” conditions. Finally, I establish that the matrices defining the compact cubic splines (equivalently, the fourth-order compact finite difference formulas) are positive definite, and in fact totally nonnegative, if all mesh widths are the same sign.
Speaker
Robert M. Corless
Role: Professor in School of Mathematical and Statistical Sciences
Organisation: University of Western Ontario
Travel and Contact Information
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Frank Adams 1
Alan Turing Building
Manchester
