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: Cleve Moler:  Numerical Computing with Matlab Press et. al,   Numerical Recipes (2nd edition online) You'll also need the FileOpen Plugin for adobe reader Mathews       Numerical Analysis, Numerical Methods project     Mathews:     Internet Matlab Resources Numerical Methods Michael Heath: Scientific Computing: An Introductory Survey, 2nd ed. with Java Demos Faires and Burden: Numerical Methods Chapra and Canales: Numerical Methods for Engineers Chapra: Applied Numerical Methods with MATLAB Trefethen and Bau: Numerical Linear Algebra Numerical Analysis: Burden and Faires:  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.

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

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

Homework  #4

Due Wednesday 5 March
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
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
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
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