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NARC - Suggested Classical Numerical Analysis Papers for Presenting

This list is based on a posting by Nick Trefethen [1] and a course offered at Stanford that reviewed these papers including the latter expanded list.

Primary list (via Trefethen)

  1. Cooley & Tukey (1965) the Fast Fourier Transform
  2. Courant, Friedrichs & Lewy (1928) finite difference methods for PDE
  3. Householder (1958) QR factorization of matrices
  4. Curtiss & Hirschfelder (1952) stiffness of ODEs; BD formulas
  5. de Boor (1972) calculations with B-splines
  6. Courant (1943) finite element methods for PDE
  7. Golub & Kahan (1965) the singular value decomposition
  8. Brandt (1977) multigrid algorithms
  9. Hestenes & Stiefel (1952) the conjugate gradient iteration
  10. Fletcher & Powell (1963) optimization via quasi-Newton updates
  11. Wanner, Hairer & Norsett (1978) order stars and applications to ODE
  12. Karmarkar (1984) interior pt. methods for linear prog.
  13. Greengard & Rokhlin (1987) multipole methods for particles

Primary Citation List

  1. James W. Cooley and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Mathematics of Computation 19 (1965), 297-301.

  2. 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.

  3. A. S. Householder, "Unitary triangularization of a nonsymmetric matrix," Journal of the Association of Computing Machinery 5 (1958), 339-342.

  4. C. F. Curtiss and J. O. Hirschfelder, "Integration of stiff equations," Proceedings of the National Academy of Sciences 38 (1952), 235-243.

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

  6. R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations," Bulletin of the American Mathematical Society 49 (1943), 1-23.

  7. G. Golub and W. Kahan, "Calculating the singular values and pseudo-inverse of a matrix," SIAM Journal on Numerical Analysis 2 (1965), 205-224.

  8. A. Brandt, "Multi-level adaptive solutions to boundary-value problems," Mathematics of Computation 31 (1977), 333-390.

  9. 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.

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

  11. G. Wanner, E. Hairer and S. P. Norsett, "Order stars and stability theorems," BIT 18 (1974), 475-489.

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

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

Full List of Sources

  • Linear Algebra - systems of equations and least-squares

    • Frankel (1950) optimal omega for SOR iteration
    • Hestenes & Stiefel (1952) the conjugate gradient iteration
    • Young (1954) theory of classical iterative methods
    • Householder (1958) QR decomposition
    • Wilkinson (1961) error analysis for systems of eqs.
    • 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.

  • Integration

    • Golub & Welsch (1969) Gauss quadrature rules
    • de Boor (1971) adaptive quadrature algorithms

  • Approximation

    • Remes (1934) Remes algorithm for Chebyshev approx.
    • Schoenberg (1946) splines
    • Powell (1967) near-optimality of Chebyshev interp.
    • 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. 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