This course is based on one originated by Nick Trefethen; you can find
information on that
course here.
Over the semester, we will read a series of papers on Numerical
Analysis; a superset of the papers that we may read is below.
Misc.
An Algorithm for the Machine
Calculation of Complex Fourier Series, by
J.W. Cooley and J.W. Tukey, Math. Comput. 19, 297-301
(1965).
To be added: Aitken
Numerical Integration
Calculation of Gauss Quadrature
Rules, by G.H. Golub and J.H. Welsch, Math. Comput. 23,
221-230 (1969).
ODEs
A Special Stability Problem for
Linear Multistep Methods, by G.G. Dahlquist, BIT 3, 27-43
(1963).
On the Attainable Order of Runge-Kutta
Methods, by J.C. Butcher, Math. Comput. 19, 408-417 (1965).
PDEs
On the Partial
Difference Equations of Mathematical Physics, by R. Courant,
K. Friedrichs, and H. Lewy, Mathematische Annalen 100, 32-74
(1928).
Variational Methods for the Solution
of Problems of Equilibrium and Vibrations, by R. Courant,
Bulletin of the American Mathematical Society 49, 1-23 (1943).
Survey of the Stability of Linear
Finite Difference Equations, by P.D. Lax and R.D. Richtmyer,
Communications on Pure and Applied Mathematics 9, 267-293 (1956).
Systems of Conservation Laws,
by P. Lax and B. Wendroff, Communications on Pure and Applied
Mathematics 13, 217-237 (1960).
Optimization
Maximization of a Linear Function of
Variables Subject to Linear Inequalities, by G.B. Dantzig,
Activity Analysis of Production and Allocation,
pp. 339-347. Cowles Commission Monograph No. 13. John Wiley &
Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951.
Application of the Simplex Method to
a Transportation Problem, by G.B. Dantzig,
Activity Analysis of Production and Allocation,
pp. 359-373. Cowles Commission Monograph No. 13. John Wiley &
Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951.
A New Polynomial-Time Algorithm for
Linear Programming, by N. Karmarkar, Combinatorica 4, 373-395
(1984).
The Convergence of a Class of
Double-Rank Minimization Algorithms 1. General Considerations,
by C.G. Broyden, J. Inst. Maths Applics 6, 76-90 (1970).
A New Approach to Variable Metric
Algorithms, by R. Fletcher, The Computer Journal 13, 317-322
(1970).
A Family of Variable-Metric Methods
Derived by Variational Means, by D. Goldfarb, Math. Comp. 24,
23-26 (1970).
Conditioning of Quasi-Newton Methods
for Function Minimization, by D.F. Shanno, Math. Comp. 24,
647-656 (1970).
Direct Methods in Linear Algebra
Error Analysis of Direct Methods of
Matrix Inversion, by J.H. Wilkinson,
J. Assoc. Comput. Mach. 8, 281-330 (1961).
Computation of Plane Unitary Rotations
Transforming a General Matrix to Triangular Form, by
W. Givens, J. Soc. Indust. Appl. Math. 6, 26 - 50 (1958).
Unitary Triangularization of a
Nonsymmetric Matrix, by A.S. Householder,
J. Assoc. Comput. Mach. 5, 339-342 (1958).
Gaussian Elimination is not
Optimal, by V. Strassen, Numer. Math. 13, 354-356 (1969).
Nested Dissection of a Regular Finite
Element Mesh, by A. George, SIAM J. Numer. Anal. 10, 345-363
(1973).
Iterative Methods in Linear Algebra
Iterative Methods for Solving Partial
Difference Equations of Elliptic Type, D. Young,
Trans. Amer. Math. Soc. 76, 92-111 (1954).
Methods of Conjugate Gradients
for Solving Linear Systems, M.R. Hestenes and E. Stiefel,
Journal of Research of the National Bureau of Standards 49,
409-436 (1952).
A Generalized Conjugate
Gradient Method for the Numerical Solution of Elliptic Partial
Differential Equations, by P. Concus, G.H. Golub, and
D.P. O'Leary, in Sparse Matrix Computations, J.R. Bunch and
D.J. Rose, eds., Academic Press, 1976, pp. 309-332.
An Iterative Solution
Method for Linear Systems of Which the Coefficient Matrix is a
Symmetric M-Matrix, by J.A. Meijerink and H.A. van der Vorst,
Math. Comp. 31, 148-162 (1977).
GMRES: A Generalized Minimal
Residual Algorithm for Solving Nonsymmetric Linear Systems, by
Y. Saad and M.H. Schultz, SIAM J. Sci. Stat. Comput. 7, 856-869
(1986).
Multigrid Methods
A Relaxation Method for Solving
Elliptic Difference Equations, by R.P. Fedorenko,
Z. Vycisl. Mat. i Mat. Fiz. 1, 922-927, 1961.
The Speed of Convergence of One
Iterative Process, by R.P. Fedorenko,
Z. Vycisl. Mat. i Mat. Fiz. 4, 559-564, 1964.
Multi-Level Adaptive Solutions to
Boundary-Value Problems, by A. Brandt, Math. Comp. 31, 333-390
(1977).
Algebraic Multigrid (AMG)
for Automatic Multigrid Solution with Application to Geodetic
Computations, by A. Brandt, S. McCormick, and J. Ruge,
Tech. Report Inst. Comp. Studies., Colo. State Univ., 1982.
Algebraic multigrid (AMG)
for sparse matrix equations, by A. Brandt, S. McCormick, and
J. Ruge, in Sparsity and its applications (Loughborough, 1983),
257-284, Cambridge Univ. Press, Cambridge, 1985.
Eigenvalues and SVD:
On the Speed of Convergence of Cyclic
and Quasicyclic Jacobi Methods for Computing Eigenvalues of
Hermitian Matrices, by P. Henrici,
J. Soc. Indust. Appl. Math. 6, 144-162 (1958).
The QR Transformation: A Unitary
Analogue to the LR Transformation - Part 1, by J.G.F. Francis,
Comput. J. 4, 265-271 (1961).
The QR Transformation - Part 2, by
J.G.F. Francis, Comput. J. 4, 332-345 (1962).
Calculating the Singular Values and
Pseudo-Inverse of a Matrix, by G. Golub and W. Kahan, J. SIAM
Numer. Anal. Ser. B 2, 205-224 (1965).
Tensors:
Tensor Decompositions and
Applications, by T.G. Kolda and B.W. Bader, SIAM Review 51,
455-500 (2009).
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