APPM 5610, Numerical Analysis 2, Spring 2018
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Text for APPM 5600/5610
- K. Atkinson, Introduction to Numerical Analysis, Second Edition (only a few sections will be used for the course)
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:
- Ake Bjork, Numerical Methods in Matrix Computations, Springer, 2015 , Chapter 3.
- G. Golub and C. Van Loan, Matrix Computations, Chapters 2-5, 7, 10. (quite terse but has key algorithms)
- J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (except Chapter 1 and Sections 4A, 7.7, 8.8-8.10)
- J.C. Butcher, Ordinary Differential Equations, Wiley, 2008.
- K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations (finite difference methods)
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 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.
Lecture Times and Location
| Instructor | Room Number | Time |
|---|---|---|
| Gregory Beylkin | FLMG 103 | MW 3:00 to 4:15 |
Office Hours
| Instructor/TA | Room Number | Office Hours |
|---|---|---|
| Gregory Beylkin | ECOT 323 | MW 1:00 to 2:00 |
Homeworks
Homework 1 Due January 31
Homework 2 Due February 7
Homework 3 Due February 14
Homework 4 Due February 21
Homework 5 Due February 28
Homework 6 Due March 14
Homework 7 Due March 21
Homework 8 Due April 11
Homework 9 Due April 18
Homework 10 Due April 25