Note: This schedule is subject to revision during the term.
Wednesday, September 2
Introduction to numerical analysis.
Friday, September 4
Floating point arithmetics, well-posedness and well conditioning, stability, backward stability.
Wednesday, September 9
Gaussian elimination. LU decomposition.
Friday, September 11
Gaussian elimination with partial pivoting. Cholesky decomposition.
Wednesday, September 16
QR decomposition. Gram-Schmidt process. Householder reflection. Least-squares problems.
Friday, September 18
Conditioning and stability revisited. Stability of solving linear systems with QR decomposition. Stability of GEPP.
Wednesday, September 23 - (Homework 1 due)
Fixed point iterations. The Banach fixed point theorem.
Friday, September 25
Stationary iterative methods for linear systems. Splitting methods. Jacobi and Gauss-Seidel.
Wednesday, September 30
Nonstationary iterative methods.
Friday, October 2
Conjugate directions. Conjugate gradients.
Wednesday, October 7
Krylov subspaces. Arnoldi. GMRES. Preconditioning.
Friday, October 9 - (Homework 2 due)
Wednesday, October 14 - (No class)
Friday, October 16 - (No class)
Monday, October 19
Nonlinear equations in R. The Newton-Raphson method. Bisection method.
Wednesday, October 21
Secant method. Regula Falsi. Round-off error. Conditioning. Stopping criteria.
Friday, October 23
Aitken's extrapolation. Nonlinear equations in Rn. Fixed point iterations. Newton's method.
Monday, October 26
Inexact Newton methods. Broyden's method. Forcing global convergence. Continuation method. A glance at nonlinear unconstrained optimization.
Wednesday, October 28
Roots of polynomials. Horner's scheme. Deflation. Sturm sequences. Euclid's algorithm.
Friday, October 30
Bernoulli iteration. Muller's method. Bairstow's method.
Wednesday, November 4 - (Homework 3 due)
Matrix eigenvalue problems. Using the characteristic polynomial. Hessenberg matrix. Power iterations.
Friday, November 6
Inverse, and Rayleigh quotient iterations. Jacobi, QR, and LR iterations. Reduction to Hessenberg form.
Wednesday, November 11
Polynomial methods for symmetric tridiagonal eigenvalue problems.
Friday, November 13
Subspace iteration. Simultaneous iterations. Basic QR. Francis' Implicit QR method.
Wednesday, November 18 - (Homework 4 due)
Polynomial interpolation. Lagrange coefficients. Newton's polynomials. Divided differences.
Friday, November 20
Numerical quadrature. Interpolatory quadrature. Gauss quadrature. Orthogonal polynomials. Least squares approximation.
Wednesday, November 25
Uniform approximation. Chebyshev polynomials. Applications to interpolation and the convergence theory of CG.
Friday, November 27
Trigonometric polynomials. Fast Fourier transform.
Wednesday, December 2 - (Homework 5 due)
Thursday, December 10