MIT Mathematics: 18.085

Office: E17-420

e-mail: gs@math.mit.edu

The first help session will take place on Wednesday, September 10, from 4-5PM in 66-168. Please bring your questions! In addition, Florent will offer an office hour from 3-4PM every Monday, from 3-4PM.
The course Piazza is now up! Please register by following this link: http://piazza.com/mit/fall2014/18085
First homework has been posted! This will be due next Thursday, Sept. 11 at the start of class. Please write your NAME neatly on the assignment. Include any necessary compute printouts. Collaboration is encouraged, but copying is not allowed.

A = [1 3 2] is a 1 by 3 matrix and A' = 3 by 1 transpose
- Find K = A' * A and explain why this K is singular (many answers, all OK)
- Find all solutions to Ku=0
- Compute K^4 = A'AA'AA'AA'A by hand in a smart order
Page 16 of the text multiplies our 2nd difference matrix K (make it infinite to avoid boundary effects) by vectors ones = 1,1,1,.. linear 0,1,2,3,.. squares 0,1,4,9,...
- Do the same for cubes 0,1,8,27,.. and quartic 0,1,16,81,..
- Which ones give the wrong derivative and what is the error
- Multiply these vectors by the FORWARD 1st difference matrix D+ and the CENTERED first difference matrix D_0 (infinite) and compare to the derivatives of x^2, x^3, x^4.
Solve -u'' = delta(x-a) with u(0)=0 and u'(1)=0 (free). Graph u(x) and explain what is happening at x=a between 0,1
Problem 1.2.9 on page 24 is about the 4th difference with coefficients 1,-4,6,-4,1. Where did those numbers come from and should you divide by a power of h=delta x ?
Problem 1.2.19 on page 25 is about -u'' + u' = 1. Choose a finite difference approximation. Solve for U.
Invert our T matrix with T_11=1 and diagonals -1,2,-1 for n=3 and n=4. From the pattern guess a formula for any n. Can you show that your guess is the correct inverse (any n)?

PLEASE PRINT YOUR NAME CLEARLY
Good to underline your last name.
The Class List with class numbers will be created from these homeworks.
Homeworks are for learning. Discussion is OK. Write clear answers for the graders: please.
Problems will often come from the 18.085 textbook on CSE. (Printouts / Graphs for MATLAB questions.)
Section 1.1 is available on math.mit.edu/cse
A printing of 1.1.27 has a typo (my name for a mistake)
These are the simplest corrections: Verify that
- K comes from A0 A0 ' (prime = transpose)
- T comes from A1 A1 '
- B comes from A2 A2 '
- (Those have 2 then 1 then 0 boundary conditions. They change from positive definite to semidefinite.)
- problem set 1.2 from CSE textbook

Quotes collected by my 18.085 class
Goals for the Course: See applications of calculus, ODE, linear algebra, and discrete methods without going into too much proof.
Textbook: Computational Science and Engineering (Wellesley-Cambridge, 2007).
Grades: Homework 25%, 3 quizzes 75%, no final exam. Please email Prof. Strang about conflicts with quiz dates.
Use of MATLAB for tedious calculations is encouraged, however you need to know how to do the basic algorithms taught in the course by hand (at least for small matrices) for the quizzes.

Welcome!