Teaching / Course archive
Math 578: Fall 2009
Numerical analysis 1
Course Record
Overview
Numerical analysis course covering linear systems, iterative methods, eigenvalue problems, nonlinear equations, least squares, approximation, interpolation, and quadrature.
Matlab was used as the official mathematical software and programming tool.
Schedule and Topics
| Date | Topic |
|---|---|
| Sep 2 | Introduction to numerical analysis. |
| Sep 4 | Floating point arithmetic, well-posedness, conditioning, stability, and backward stability. |
| Sep 9 | Gaussian elimination and LU decomposition. |
| Sep 16 | QR decomposition, Gram-Schmidt process, Householder reflection, and least-squares problems. |
| Sep 23 | Fixed point iterations and the Banach fixed point theorem. |
| Sep 30 | Relaxation schemes, Richardson iteration, and steepest descent. |
| Oct 7 | Krylov subspaces, Arnoldi, GMRES, and preconditioning. |
| Oct 19 | Nonlinear equations in one variable: Newton-Raphson and bisection. |
| Oct 28 | Roots of polynomials, Horner's scheme, deflation, Sturm sequences, and Euclid's algorithm. |
| Nov 4 | Matrix eigenvalue problems, Hessenberg matrices, and power iterations. |
| Nov 18 | Polynomial interpolation, Lagrange coefficients, Newton polynomials, and divided differences. |
| Nov 20 | Numerical quadrature, Gauss quadrature, orthogonal polynomials, and least squares approximation. |
| Nov 27 | Trigonometric polynomials and fast Fourier transform. |
| Dec 2 | Review. |
| Dec 10 | Final exam. |
Topic List
- Direct methods for linear systems
- Iterative methods, Krylov subspaces, GMRES, conjugate gradients, and preconditioning
- Nonlinear equations and Newton-type methods
- Matrix eigenvalue problems
- Polynomial interpolation, quadrature, least squares, and uniform approximation
- Trigonometric polynomials and FFT
Books and Related Pages
Books
- Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri, Numerical mathematics. Springer.
- Robert Plato, Concise numerical mathematics. AMS.
- Lloyd N. Trefethen and David Bau III, Numerical linear algebra. SIAM.
- Germund Dahlquist and Ake Bjorck, Numerical methods. Dover.
- Kendall Atkinson and Weimin Han, Theoretical numerical analysis. Springer.
Assignments and Projects
Assignments and Exams
- Five homework assignments, with analytical and computational components.
- Midterm exam covering linear systems, conditioning, fixed point methods, stationary and nonstationary iterations, conjugate gradients, and GMRES.
- Cumulative final exam.
Archive Note
Matlab code names are recorded in the recovered page, but the code files are not present in the local public archive.