APPM 5610, Spring 2007
Numerical Analysis II
Instructor:
Gregory Beylkin
Office:
ECOT 323
Office
hours:
MWF: 10-11
Class hours/location:
MWF 9.00-9.50 @ECCR 133
Text for APPM 5600/5610
- K. Atkinson, Introduction to Numerical Analysis
Recommended Supplemental Texts:
- G. Golub and C. Van Loan, Matrix Computations,
Chapters 2-5,
7, 10.
- A. Iserles, A First
Course in the Numerical Analysis of Differential
Equations
- J. Stoer and R. Bulirsch, Introduction to Numerical
Analysis
(except Chapter 1 and Sections 4A, 7.7, 8.8-8.10)
- K. W. Morton and D. F. Mayers, Numerical Solution of
Partial
Differential Equations
Syllabus
The Matrix Eigenvalue Problem
- Theoretical preliminaries
- Diagonalizable, normal, self-adjoined (Hermitian) matrices
- Eigenvalue problem
- Schur's decomposion, Singular Value Decomposition (SVD)
- Eigenvalue location, error analysis, and stability
- The power and inverse power methods
- Householder reflections, Givens rotations, Hessenberg form of a
matrix
- QR Iteration
- Algorithms for a self-adjoined (Hermitian) tridiagonal matrix
Numerical Methods for Ordinary Differential Equations
- Existence, uniqueness, and stability theory
- Euler's method
- Linear multistep methods
- Predictor-corrector methods
- Convergence and stability theory for multistep methods
- Stiff ODEs
- Runge-Kutta methods
- Boundary value problems
Fourier Series, Fourier Integrals and the Fast Fourier Transform
(FFT) algorithm
Introduction to Linear PDEs
- Classification of linear PDE's
- Inital value and boundary value problems
- Finite Difference discretization of elliptic PDEs and associated
linear
algebra problems
- Algorithms for the Poisson equation
- Finite Difference discretization of hyperbolic PDEs
- Finite Difference discretization of parabolic PDEs: Crank-Nicolson and ADI methods
- Stability and convergence: CFL condition, von
Neumann stability analysis, Lax equivalence theorem
- Pseudospectral methods
- *** A brief introduction to multiresolution methods for the
Poisson
equation
Introduction to Linear Integral Equations
- ***
Integral equations of the potential theory
- ***
Discretization of integral equations and associated linear algebra
problems
- *** Fast methods for solving integral equations
*** Extra topics (to be covered if time permits).