Mathematical Research Papers

Mathematical Research Papers-23
Contents of this file: [1] Trefethen NA-net posting of [2] bibliographic citations for 13 "classic papers" [3] longer list of papers we considered reading [4] copy of handout to class describing course organization [5] weekly assignments These items are separated by dashed lines. Wanner, Hairer & Norsett (1978) order stars and applications to ODE 12. Next, the Mathematician gave a presentation of some of the central ideas of the paper. Hirschfelder, "Integration of stiff equations," Proceedings of the National Academy of Sciences 38 (1952), 235-243. In other cases it is hoped the Experimentalists will explore nontrivial applications or unexplained phenomena. (15) Finally, the Professor will add whatever comments he deems appropriate.----------------------------------------------------------------- | Prof. Third and fourth, two Experimentalists reported the results of Matlab, C, or Fortran experiments conducted to illustrate some of the properties of the algorithm under discussion. To me and at least some of the students, this course provided a satisfying vision of the broad scope of numerical analysis and a sense of excitement at what a diversity of beautiful and powerful ideas have been invented in this field. \vfill Some of the roles above may sometimes be played by pairs of students rather than individuals.

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This journal will publish the highest quality papers in all of the traditional areas of applied and theoretical areas of mathematics and computer science, and it will actively seek to publish seminal papers in the most emerging and interdisciplinary areas in all of the mathematical sciences.

Research in the Mathematical Sciences wishes to lead the way by promoting the highest quality research of this type.

Norsett, "Order stars and stability theorems," BIT 18 (1974), 475-489. INTEGRATION Golub & Welsch (1969) Gauss quadrature rules de Boor (1971) adaptive quadrature algorithms APPROXIMATION Remes (1934) Remes algorithm for Chebyshev approx.

Karmarkar, "A new polynomial-time algorithm for linear programming," Combinatorica 4 (1984), 373-395. Golub (1965) least-squares problems Strassen (1969) Gaussian elimination is not optimal George (1973) nested dissection Gill, Golub, Murray & Saunders (1974) updating matrix factorizations Concus, Golub & O'Leary (1976) preconditioned conjugate gradients Meijerink & van der Vorst (1977) incomplete LU preconditioning Skeel (1980) iterative refinement and stability Saad & Schultz (1986) GMRES for nonsymmetric systems LINEAR ALGEBRA - EIGENVALUES AND SVD Jacobi (1846) Jacobi's method for matrix eigenvalues Henrici (1958) convergence of the Jacobi method Rutishauser (1958) the LR algorithm Kublanovskaya (1961) the QR algorithm Francis (1961) the QR algorithm Golub & Kahan (1965) computation of the SVD Moler & Stewart (1973) QZ algorithm for gen'd eigenvalues Cuppen (1981) divide and conquer for eigenvalues OPTIMIZATION Dantzig (1951) simplex method for linear programming Davidon (1959) variable metric methods Fletcher & Powell (1963) DFP quasi-Newton update formula Broyden/Fletcher/Goldfarb/Shanno (`70) BFGS quasi-Newton update formula Karmarkar (1984) interior pt methods for linear prog.

Evidently four of the most exciting modern developments in numerical analysis -- multigrid iterations, conjugate gradient iterations, interior point methods, and multipole methods -- have in common that they depend on the approximate computation of quantities that might in principle be computed exactly. Tukey, "An algorithm for the machine calculation of complex Fourier series," Mathematics of Computation 19 (1965), 297-301.

Most readers of this note will have thought of other classic authors and papers that should have been on the list. We are saving up ideas for the next run of CS 722 in a couple of years. of Computer Science Cornell University - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [2] bibliographic citations for 13 "classic papers" Fuller bibliographic citations: 1. 722 met once a week for two hours, and in the course of the semester we read thirteen papers: 1. His/her assignment is to speak with the class about the technical aspects of the paper.The thirteen papers were selected partly for their variety; they touch upon nearly all the main problems of numerical computation. In particular, it may be more fun for an Experimentalist to be a pair rather than a solo.The Mission of the Journal will be to publish high-quality original articles that make a significant contribution to the research areas of both theoretical and applied mathematics and theoretical computer science.This journal is an efficient enterprise where the editors play a central role in soliciting the best research papers, and where editorial decisions are reached in a timely fashion. There will be thirteen weekly meetings, each organized around a classic paper and related readings. Non-students may also participate provided they agree to act like students. Kahan, "Calculating the singular values and pseudo-inverse of a matrix," SIAM Journal on Numerical Analysis 2 (1965), 205-224. Ideally, he/she will understand all the details of the paper, though it is recognized that this will not always be possible. Brandt, "Multi-level adaptive solutions to boundary-value problems," Mathematics of Computation 31 (1977), 333-390. Each week's meeting will be organized about the following principal players: \par \ss \vfill A rough agenda will be as follows: \def\item #1. This handout should include a plot of citations as a function of time (e.g.\ from the )~~Examples of other interesting information might be the original review in or the biographical entries from obituaries from the historical remarks found in numerical analysis textbooks, results of a conversation with a relevant Cornell faculty member, a survey of impact on software libraries, etc. (30) The Mathematician is responsible for reading the main paper with exceptional care. 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.

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