Wednesday, September 2Introduction to numerical analysis.
Friday, September 4Floating point arithmetics, well-posedness and well conditioning, stability, backward stability.
Wednesday, September 9Gaussian elimination. LU decomposition.
Friday, September 11Gaussian elimination with partial pivoting. Cholesky decomposition.
Wednesday, September 16QR decomposition. Gram-Schmidt process. Householder reflection. Least-squares problems.
Friday, September 18Conditioning 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 25Stationary iterative methods for linear systems. Splitting methods. Jacobi and Gauss-Seidel.
Wednesday, September 30Relaxation schemes. Richardson. Nonstationary iterative methods. Steepest descent.
Friday, October 2Conjugate directions. Conjugate gradients.
Wednesday, October 7Krylov subspaces. Arnoldi. GMRES. Preconditioning.
Friday, October 9 - (Homework 2 due)Midterm.
Wednesday, October 14 - (No class)
Friday, October 16 - (No class)
Monday, October 19Nonlinear equations in R. The Newton-Raphson method. Bisection method.
Wednesday, October 21Secant method. Regula Falsi. Round-off error. Conditioning. Stopping criteria.
Friday, October 23Aitken's extrapolation. Nonlinear equations in Rn. Fixed point iterations. Newton's method.
Monday, October 26Inexact Newton methods. Broyden's method. Forcing global convergence. Continuation method. A glance at nonlinear unconstrained optimization.
Wednesday, October 28Roots of polynomials. Horner's scheme. Deflation. Sturm sequences. Euclid's algorithm.
Friday, October 30Bernoulli 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 6Inverse, and Rayleigh quotient iterations. Jacobi, QR, and LR iterations. Reduction to Hessenberg form.
Wednesday, November 11Polynomial methods for symmetric tridiagonal eigenvalue problems.
Friday, November 13Subspace 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 20Numerical quadrature. Interpolatory quadrature. Gauss quadrature. Orthogonal polynomials. Least squares approximation.
Wednesday, November 25Uniform approximation. Chebyshev polynomials. Applications to interpolation and the convergence theory of CG.
Friday, November 27Trigonometric polynomials. Fast Fourier transform.
Wednesday, December 2 - (Homework 5 due)Review.
Thursday, December 10Final exam.