Instructor
Gregory Beylkin
ECOT 323
beylkin@colorado.edu
Lectures
MW 3:00-4:15, ECCR 139
Office hours
MW 4:15-5:30, ECOT 323
Textbooks for 5600/5610.
- K. Atkinson, Introduction to Numerical Analysis,
Second Edition. Useful links for this book:
Chapters
1-5
Chapters
6-9
- A. Iserles, A First
Course in the Numerical Analysis of Differential
Equations
Recommended Supplemental Texts:
- G. Golub and C. Van Loan, Matrix Computations,
Chapters 2-5,
7, 10.
- 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
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 self-adjoined (Hermitian) tridiagonal
matrices
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
- *** A fast algorithm for (linear) two-point boundary value
problem
- *** An introduction to symplectic integrators
A review: Fourier Integrals, Fourier Series and 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 only if time permits.
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