Mathematical Research Papers

Hestenes and Eduard Stiefel, "Methods of conjugate gradients for solving linear systems," Journal of Research of the National Bureau of Standards 49 (1952), 409-436. Our aim each week will be to have a lively discussion and a good time. (15) The Historian will distribute to the class a handout containing information he/she has obtained about the historical context of the paper. Each student should read all the readings each week and be prepared to participate in discussions.

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Reinsch (1967) smoothing with splines Cox (1972) calculation with B-splines de Boor (1972) calculation with B-splines OTHER Aitken (1932) Aitken extrapolation Cooley & Tukey (1965) the fast Fourier transform Greengard & Rokhlin (1987) fast multipole methods ODEs Curtiss & Hirschfelder (1952) stiffness and BD formulas Dahlquist (1956) stability and convergence Dahlquist (1963) A-stability Butcher (1965) Runge-Kutta methods Gear (1969) stiff ODEs Wanner, Hairer & Norsett (1978) order stars and stability theorems ELLIPTIC PDEs Peaceman & Rachford (1955) ADI Douglas (1955) ADI Strang (1971 or 1973) finite elements and approx.

Schoenberg (1946) splines Powell (1967) near-optimality of Chebyshev interp.

These authors are for the most part extraordinary people, including some about whom most numerical analysts know little (such as Hirschfelder, one of the leading American chemists of this century). Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik," Mathematische Annalen 100 (1928), 32-74. \ctr \ul \hangindent 20pt de Boor (1972): ``On calculating with $B$-splines'' \bs \ul \hangindent 20pt Cox (1972): ``The numerical evaluation of $B$-splines'' \hangindent 20pt M.

We were struck by how young many of the authors were when they wrote these papers (average age: 34), and by how short an influential paper can be (Householder: 3.3 pages, Cooley & Tukey: 4.4). For example, Curtiss and Hirschfelder inexplicably define stiffness in terms of exponentially diverging trajectories, not converging ones; nevertheless they invent the right cure for the problem in the shape of backward differentiation formulas. Translated as: "On the partial difference equations of mathematical physics," IBM Journal of Resarch and Development 11 (1967), 215-234.

Powell, "A rapidly convergent descent method for minimization," Computer Journal 6 (1963), 163-168. The grade will be A for those students who remain involved with the course throughout the semester and contribute to its success.

theory Buzbee, Golub & Nielsen (1970) fast Poisson via cyclic reduction Hockney (1965) fast Poisson via FFT Fedorenko (1961) multigrid methods Brandt (1977) multigrid methods PARABOLIC AND HYPERBOLIC PDEs Courant, Friedrichs & Lewy (1928) the CFL condition Crank & Nicolson (1947) finite differences for parabolic PDE O'Brien, Hyman & Kaplan (1951) Von Neumann stability analysis Lax & Richtmyer (1956) general stability theory Lax & Wendroff (1960,1962,1964) methods for solving conservation laws Kreiss (1962) more general stability theory Orszag (1971) spectral methods Kreiss and Oliger (1972) spectral methods Gustafsson, Kreiss & Sundstrom (1972) stability of boundary conditions Chorin (1973) vortex methods for CFD Engquist & Majda (1977) absorbing boundary conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [4] copy of handout to class describing course organization [contains some idiosyncratic Trefethen Te X macros; sorry] \input mac \ms\vfill Instructor: Nick Trefethen, Upson 4148, 255-4222, [email protected]: one two-hour meeting each week at a time and place to be determined Prerequisites: (1) ideally, at least two of CS 621, 622, 624 or their equivalents; and\hb \hbox All students, even those on reduced tuition, must register to take the course for a grade (i.e., no auditors).

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