20,475 research outputs found
Ten Digit Algorithms
This paper was presented as the A R Mitchell Lecture at the 2005 Dundee Biennial Conference on Numerical Analysis, 27 June 2005
Predictions for Scientific Computing Fifty Years from Now
This essay is adapted from a talk given June 17, 1998 at the conference "Numerical Analysis and Computers - 50 Years of Progress" held at the University of Manchester, England in commemoration of the 50th anniversary of the Mark 1 computer
Householder triangularization of a quasimatrix
A standard algorithm for computing the QR factorization of a matrix A is Householder triangularization. Here this idea is generalized to the situation in which A is a quasimatrix, that is, a âmatrixâ whose âcolumnsâ are functions defined on an interval [a,b]. Applications are mentioned to quasimatrix leastsquares fitting, singular value decomposition, and determination of ranks, norms, and condition numbers, and numerical illustrations are presented using the chebfun system
Ten Digit Problems
Most quantitative mathematical problems cannot be solved exactly, but there are powerful algorithms for solving many of them numerically to a specified degree of precision like ten digits or ten thousand. In this article three difficult problems of this kind are presented, and the story is told of the SIAM 100-Dollar, 100-Digit Challenge. The twists and turns along the way illustrate some of the flavor of algorithmic continuous mathematics
Is Gauss quadrature better than Clenshaw-Curtis?
We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at . Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at is only half as high, but which is nevertheless equally accurate near
Numerical Analysis
Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud
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In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud
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With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre SĂŒli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright
Drugs research: an overview of evidence and questions for policy
In 2001 the Joseph Rowntree Foundation embarked upon a programme of research that explored the problem of illicit drugs in the UK. The research addressed many questions that were often too sensitive for the government to tackle. In many cases, these studies represented the first research on these issues.
This study gives an overview of the projects in the programme. The topics covered include:
* The policing of drug possession.
* The domestic cultivation, purchasing and heavy use of cannabis.
* Non-problematic heroin use, heroin prescription and Drug Consumption Rooms.
* The impact of drugs on the family.
* Drug testing in schools and in the workplac
Computing the Gamma function using contour integrals and rational approximations
Some of the best methods for computing the gamma function are based on numerical evaluation of Hankel's contour integral. For example, Temme evaluates this integral based on steepest-decent contours by the trapezoid rule. Here we investigate a different approach to the integral: the application of the trapezoid rule on Talbot-type contours using optimal parameters recently derived by Weideman for computing inverse Laplace transforms. Relatedly, we also investigate quadrature formulas derived from best approximations to exp(z) on the negative real axis, following Cody, Meinardus and Varga. The two methods are closely related and both converge geometrically. We find that the new methods are competitive with existing ones, even though they are based on generic tools rather than on specific analysis of the gamma function
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