E4300: Computational Math 1

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.

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	Student version from Columbia Bookstore Matlab Tutorials (Mathworks site)

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	Lecture01 Template pdf Lecture01.pdf Moler Ch1 Some Modeling Notes	Introduction and Motivation: Modeling, Methods and Matlab --the fundamental tools and problems in scientific computation
24-29 Jan	Lecture02 Template pdf Lecture02.pdf Lecture03 Template pdf Lecture03.pdf Calculus Review (Mathews) Taylor Series Taylor Truncation Error	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	Polynomial Evaluation ExpChallenge TestCode
31 Jan- 6 Feb	Lecture04 Template pdf Lecture04.pdf Lecture05 Template pdf Lecture05.pdf Lecture06 Template pdf Lecture06.pdf Moler Ch4 NR Chapter 9.0-9.4 Mathews --root finding modules	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	Homework #2 ProblemSet02.tar.gz Due 15 Feb	My Rootfinding Demos My old Rootfinding Demos Heath's Demos
7-14 Feb	Lecture07 Template pdf Lecture07.pdf Lecture08 Template pdf Lecture08.pdf Moler Ch3 NR chapter 3 Mathews Chebyshev Polynomials, (html ) My Interpolation Notes	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	Homework #3 ProblemSet03.tar.gz Due 26 Feb	Lagrange Interpolation demos Heath's Interpolation demos Moler's NCM Interpolation demos
19-26 Feb	Lecture09.pdf (Ethan Coon) correction to Lecture09 Lecture10.pdf (Ethan Coon) Lecture11 Template pdf Lecture11.pdf Moler Ch6 NR Ch4 Mathews -Quadrature Modules Mathews-NewtonCotes Formulas Mathews - Gauss-Legendre Quadrature rules	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	Lecture12 Template pdf Lecture12.pdf Lecture13 Template pdf Lecture13.pdf Lecture14 Template pdf Lecture14.pdf Moler Ch7 NR chapter 16 MMM Notes Ch4	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	Lecture15 Template pdf Lecture15.pdf	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	Lecture16 Template pdf Lecture16.pdf Lecture17 Template pdf Lecture17.pdf Lecture18 Template pdf Lecture18.pdf Lecture19 Template pdf Lecture19.pdf Linear Algebra Review Moler Ch2 Trefethen and Bau My Linear Algebra WebSite	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	Moler's LU routines matlab LU demos matlab QR demo
8-10 April	Lecture20 Template pdf Lecture20.pdf Lecture21 Template pdf Lecture21.pdf Trefethen and Bau	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)		matlab Eigen demos
15-29 April	Lecture22 Template pdf Lecture22.pdf Lecture23 Template pdf Lecture23.pdf Lecture24 Template pdf Lecture24.pdf Lecture25 Template pdf Lecture25.pdf Lecture26 Template pdf Lecture26.pdf Numerical Recipes: Ch 17 Trefethen: Spectral methods Ch1 &6 Mathews: Shooting FiniteDifference Galerkin	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	BVP Demos Shooting Finite Difference Non-Linear FD Collocation Comparison of Chebyshev vs FD matrices Galerkin FEM Trefethen's Spectral Methods site
6 May	Lecture27.pdf	Review Session Study Guide v1.2
13 May	Final Exam Tuesday 13 May, 1:10-4:00 PM, 535 Mudd

Introduction and Motivation:

Sources of Error and Life in Floating point land:

Root finding and optimization for f(x)

Numerical Quadrature and Differentiation

Solution of ODE's #1: Initial Value Problems

Solving Non-linear systems of Equations

Numerical Linear Algebra #1

Numerical Linear Algebra #2

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