Resources - Math 579 - Spring 2004

General Numerical Analysis Information

• Numerical Analysis, by Richard L. Burden and J. Douglas Faires, PWS Publishing, various editions. A good general upper-level undergraduate text for numerical analysis.
• Analysis of Numerical Methods, by Eugene Isaacson and Herbert B. Keller, Dover Publishing. A classic reference to all the basics of numerical analysis; the often cited source of proofs for many results.

Roundoff Error and Floating Point Arithmetic

• Numerical Computing with IEEE Floating Point Arithmetic, by Michael L. Overton, SIAM. Extended version of class handout.
• What Every Computer Scientist Should Know About Floating Point Arithmetic, by David Goldberg, Computing Surveys, March 1991. 70+ pages about the consequences of the IEEE standard for floating point arithmetic; probably has more than you probably ever need to know.
• Floating Point Disasters from Prof. Doug Arnold's website. Discussion of two real disasters caused by floating point problems.

Fast Fourier Transform

• James W. Cooley and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Mathematics of Computation 19 (1965), 297-301.
• Heideman, Johnson, and Burrus (1984) "Gauss and the history of the fast Fourier transform"
• Cooley (1987) "How the FFT gained acceptance"
• Burrus (1990) "Notes on the FFT"

Finite Difference Methods for PDEs

• R. Courant, K. O. Friedrichs and H. Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik," Mathematische Annalen 100 (1928), 32-74. Translated as: "On the partial difference equations of mathematical physics," IBM Journal of Resarch and Development 11 (1967), 215-234.
• Lax (1967): "Hyperbolic difference equations: A review of the Courant-Friedrichs-Lewy paper in the light of recent developments"

Multipole Methods

• L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," Journal of Computational Physics 72 (1987), 325-348.

Conjugate Gradient Method

• Magnus R. 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.

Optimization via Quasi-Newton Updates

• R. Fletcher and M. J. D. Powell, "A rapidly convergent descent method for minimization," Computer Journal 6 (1963), 163-168.

Multigrid

• A. Brandt, "Multi-level adaptive solutions to boundary-value problems," Mathematics of Computation 31 (1977), 333-390.
• William L. Briggs, {\it A Multigrid Tutorial}, SIAM, 1987.

Backward Error Analysis

• J.H. Wilkinson, "Modern Error Analysis," SIAM Review 13 (1971), 548-568. A historical look at error analyses.
• J. vonNeuman and H.H. Goldstine, "Numerical inverting of matrices of high order," Bull. Amer. Math. Soc 53 (1947), 1021-1099. Defining paper covering errors based on working with computer numbers. Uses some backward error analysis but is primarily forward-based.
• J.W. Givens, "Numerical computation of the characteristic values of a real symmetric matrix," Report ORNL-1574 (1954). Considered by many the first paper to focus primarily on backward error analysis.

Convergence Acceleration

• A.C. Aitken, "On Bernoulli's numerical solution of algebraic equations," Proc. Royal Society of Edinburgh 46 (1926), 289-305.
• L.F. Richardson, "The approximate arithmetical solution by finite difference of physical problems involving differential equations, with an application to the stress in a masonry dam," Philos. Trans. Roy. Soc. London, Series A 210 (1910), 307-357.
• C. Brezinski, "Covergence acceleration during the 20th century," Journal of Computationa and Applied Mathematics 122 (2000), 1-21.

Stability and Convergence for ODE methods

• G. Dahlquist, "Convergence and stability in the numerical integration of ordinary differential equations," Mathematica Scandinavica, Vol 4 (1956), 33-53.

Interior Point Methods for Linear Programing

• N. Karmarkar, "A new polynomial-time algorithm for linear programming," Combinatorica 4 (1984), 373-395.

Finite Element Method

• R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations," Bulletin of the American Mathematical Society 49 (1943), 1-23.
• Philippe G. Ciarlet, {\it The Finite Element Methods for Elliptic Problems}, North-Holland, 1980 (other editions available).
• Claes Johnson, {\it Numerical Solution of Partial Differential Equations by the Finite Element Method}, Cambridge, 1987.

Gaussian Quadrature Rules

• Golub and Welsch, "Gauss....," Mathematics of Computation 23 (1969), 221-230.

B-Splines

• C. de Boor, "On calculating with B-splines," Journal of Approximation Theory 6 (1972), 50-62.

QR Decomposition

• A. S. Householder, "Unitary triangularization of a nonsymmetric matrix," Journal of the Association of Computing Machinery 5 (1958), 339-342.
• Golub and Van Loan, {\it Matrix Computations}, Johns Hopkins University Press, 3rd Edition.