E4300: CM1 -- Numerical Methods

Time:

Tu-Th 1:10-2:25  Location: 535 Mudd
Instructor: Marc Spiegelman   (home page)
Office Hours: Tues/Thurs 4:00-5:00pm
 211 Mudd (APAM)
TAs: Daisuke Shiraki (principal TA), Yan Yan 
Wenjia Jing (1/2 time)
Office Hours:
Daisuke: Mondays 4-5pm
Wenjia: Wednesdays 3:30-4:30 pm
Yan Yan: Tuesdays 9:10-10:50am

All office hours in 287 Engineering Terrace
Useful (?)
Texts:
Online References/Resources:
Numerical Methods Numerical Analysis:
Grading: Homework 60%, Midterm 20% Final 20%

All Homeworks due by 5pm on the due date in the E4300 Homework box in 200 Mudd (APAM).
Homeworks can also be turned in during class.
Matlab Resources


Prerequisites:

Calculus, Vector Calculus, Linear Algebra and ODE's will be used extensively. Students must also have some programming experience to the level of COMS 1000x classes.  All programming exercises in this class will be in MATLAB and some experience with this language  will be useful. However,  I will teach most things that are necessary and try to provide sufficient examples. 




Syllabus
Dates Reading/Notes Subject Problem Sets Matlab Examples/Demos
22 Jan

Introduction and Motivation: 

Modeling, Methods and Matlab --the fundamental tools and problems in scientific computation


24-29 Jan

Sources of Error and Life in Floating point land: 

 model Error, Truncation Error, Roundoff Error
A short guide to IEEE floating point, The Nitty Gritty guide to floating point systems, An IEEE floating point calculator
Homework  #1
Due 5 Feb
31 Jan- 6 Feb

Root finding and optimization for f(x)

Fixed Point iteration
Brackets and existence
Basic Algorithms:  Bisection, Newton, Secant, inverse interpolation
Comparison and convergence rates
Combined Algorithms - Brent's method and fzero
Optimization of 1-D functions:
Basic algoriththms: golden section, Newton, parabolic interpolation
Due 15 Feb
7-14 Feb Interpolation and Approximation
Polynomial Interpolation, Lagrange and Monomial Basis
Pitfalls of large order: Chebyshev points
Heath's Interpolation demos
Piecewise Polynomial Interpolation: C0, C1, C2 (pchip and spline)
Matlab Interp routines
Data Approximation by Linear Least Squares

Due 26 Feb

19-26 Feb


Numerical Quadrature and Differentiation

Motivation: solution of IVPs and BVPs
Newton Cotes and error estimates: Mid--point, Trapezoidal, Simpsons
Arbitrary order and method of undetermined coefficients
Gauss Quadrature
Extended Newton Cotes
Adaptive Quadrature: quad routines

Homework  #4

Due Wednesday 5 March
Moler's NCM Quadrature Demos (quadtx, quadgui)
28 Feb-
  11 March

Solution of ODE's #1:   Initial Value Problems

Interlude:
Numerical Differentiation:  Finite Difference to Spectral methods

 ODE IVPs
Motivation
Linear systems and expm
Application of Quadrature:  Single step schemes: Euler, Midpoint, RK4 and errors
Error Control and Adaptive Stepping
Embedded RK schemes: ode45 (matlab ode suite)
Systems of ODE's
Stiff systems: Example and symptoms
Implicit methods: ODE23s
Homework  #5

Due Thursday 13 March
  • Matlab Demos for simple algorithms
  •  My Matlab Demos for the odexx routines
  • (README files)
11 March

Solving Non-linear systems of Equations

Motivation:
Existence and Uniqueness (Hah!)
n-Dimensional Taylor's theorem and Newton's method
Packages and Libraries (fsolve, PETSc)
n-D non-linear optimization
Non-linear least-squares: Gauss-Newton method



  • matlab Demos for nDimensional Newton's Method
17-21 March Spring Break!
27 March
Midterm 1:10-2:25pm
Study Guide
updated 12 March 2008
13 March-3 April

Numerical Linear Algebra #1

Motivation: Ax=b is everywhere
Hooray for backslash!
Existence, Uniqueness and Condition #: Vector and Matrix Norms
Direct Methods for NLA:
Gaussian Elimination and the LU:
Partial Pivoting and roundoff error
Special Matrices: Symmetric, Tridiagonal, Sparse matrices
Least Squares and QR
Orthogonalization by Householder Transformations (Givens?)
Homework  #6

due April 10
8-10 April

Numerical Linear Algebra #2

Introduction to Iterative methods for sparse matrices
Splitting Methods and the Iteration Matrix
(Jacobi, Gauss-Seidel)
Eigenvalues of the iteration matrix: spectral radius, power method, inverse power method with shifts
Other methods: Krylov Methods (CG, GMRES)

15-29 April

Solution of ODE's #2: Boundary Value Problems and intro to PDE's

Motivation: Numerical PDE's Discrete vs Continuous Approximations
2 point BVPs:
Shooting methods, Finite Difference, Collocation and Galerkin FEM
Adding time= PDE's
Method of lines, Explicit FD, Newton
Stability and beyond
Homework  #7

due May 2

Matlab Routine for Newton's method
newton.m

6 May
Review Session
 Study Guide v1.2


13 May
Final Exam Tuesday 13 May, 1:10-4:00 PM, 535 Mudd

marc spiegelman
Last modified: 18 Jan 2008