MIT Mathematics

Welcome!

18.085 Spring 2010

Class stuff:

MWF 12-1 4-370

Lecturer: Alan Edelman
Office: 2-343
e-mail: edelman@mit.edu

TA: Dorian Croitoru
Office: 2-251
Office Hours: Thursday 1--2:30
email: dorian@math.mit.edu
phone: 3-7566

(Half) TA: Nan Li
Office: 2-333
email: nan@math.mit.edu
phone: 3-7826

Tomorrow (Wednesday's) exam will cover Fourier series, a discrete convolution computation, circulant matrices, and a Fourier integral.

2010: We will follow the 2008 outline or indeed the video lectures fairly closely perhaps with a slight more emphasis on computing. For 2010 exams will be in class, open book and open notes, not evenings, no computers. Homeworks will be due on Fridays in spring 2010.
The exam dates will take place on the classes numbered 13, 26, and 39. The dates are Wednesday March 3, Friday April 9, and Wednesday May 12.
Strang's Videos (terrific)

Quizzes and Solutions for Spring 2010

Quiz 1 (pdf) and Solutions (pdf) .
Quiz 2 (pdf) and Solutions (pdf)
Quiz 3 (pdf) and Solutions (pdf)

HOMEWORK for Spring 2010

#11 (Final HW) Due Friday April 30 (Solutions to HW 11)
4.3: 6,7,9,20
4.4: 1,2,4,5,6,7
4.5: 3,4,5,7,11

#10 Due Friday April 23 (Solutions to HW 10)
Note: we may not cover this material in class, but it's very useful and similar in many ways to material we saw in class.
4.2: 1,2,3,7,8,10,20,21
Hint for problem 8: Write down T4(x) and combine until you get x^4.

By popular request (matlab to watch Fourier Series for square wave):

x=pi*[-1:.01:1];
for n=3:5:51, k=(1:2:n)'; plot(x, sum(diag(1./k)*sin(k*x)));title(['n=' num2str(n)]); pause(0.1); end

#9 (Solutions to HW 9)
Section 4.1: 2,3,4,5,7,8,9
Hints: 4: assume f(x) is general, not any one particular function
Hints: 9: Note the taylor series for 2*log(1+z) is 2(z-z^2/2+...). It's not that imaginary part of the taylor series is 2 log(1+z). Guess this could have been written clearer, but if you knew the series, you'd be ok. The real problem is a bit of a trig identity.

#8 (Solutions to HW 8) Due April 12 1pm
Section 3.4: 1,2,3,12,13
Section 3.5: 8,9,10

#7 (Solutions to #7) Due April 2 1pm
Section 3.2: 6. 8, 9. 10
Section 3.3: 1,2,3,4,5,6, 9
Hint on 3.2,6: I think we must remember that w=u" to think clearly here. Perhaps also it is good to realize that we will need all the functions to cancel out to get higher derivatives 0.
Hint on 3.3,9c: I think s(x,y)=arctan(y/x) might be a good stream function

#6 (Solutions to #6) Due March 19 1pm 3.1:3,4,7,10,12,13,15

#5 (Solutions to #5) Due March 12 (Friday) 1pm
2.7:4,5,8,9,10,13 (Note 12 is a copy of 8)

#4 (Solutions to #4) Due March 8 (monday), 1pm 12. (My survey of students was that extra time would be nice given the exam, but the homeworks are useful for mastering the material).
2.2: 7
2.3: 3,4,10
Problem 2.2:7 requires understanding that lambda=exp(i*theta). If you expand lambda as a power series of h, it looks like the power series of exp(i*h) for a few terms. The problem asks you to recognize where they differ.
Problem 2.3:3 is long but I hope very instructive. The U matrix in the Householder code has the information to multiply the Householder's in reverse order.
I got the exact answer using matlab's symbolic toolbox and then rounded only at the end. I got

    1.000000000705105e+000
   -2.964328419073236e-007
   -7.683186917434977e+000
   -2.206297290004058e-004
    9.840487549806722e+000
   -9.324611932899178e-003
   -5.011975877477831e+000
   -4.685413770053830e-002
    1.417778585705657e+000
    2.287193842956169e-002
   -3.135724049839490e-001
    7.196408559132141e-002

If you have the tollbox you can type

AAA=sym(0:49).'/49;
AAA=(AAA*ones(1,12) ).^ (ones(50,1)*(0:11));
bbb= 0:sym(8/100):4-sym(8/100);
bbb=cos(bbb).';
digits(100);
xxx=double(vpa(AAA.'*AAA)\vpa(AAA.'*bbb));

2.4: 2,5,6,7,9

#3 (Solutions to #3) for Friday February 26
1.6: 15, 16, 17, 21, 23
2.1: 4,5,6
2.2: 5
Recommended but do not hand in: Do the MATLAB homework in Problem set 3 by scrolling all the way down to "First MATLAB Homework in 18.085" from the previous semester.

#2 (Solutions to #2) for Friday February 19
1.3: 5,6,7,13
1.4: 2,6,12,15
1.5: 2,10,11,12
1.6: 12

#1 (Solutions to #1) for Friday February 12 (Please submit in class, or by 1pm near 2-343) (Please write your name extra extra neatly)
1.1: 1,7,10,12
Hint on 10: The idea is to check that UU' truly is H and then invert using the rule inv(A*B)=inv(B)*inv(A).
1.2: 3,4,7,18

Last semester's stuff:
[announcements] [homework assignments] [quizzes and solutions] [links] [resources and old exams]

Course Topics

Applied Linear Algebra

Applied Differential Equations

Fourier Methods

Algorithms

Course outline 2008

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 40%, 3 evening quizzes 60%, no final.

Homework: Due Wednesdays. Please use MATLAB notation to describe algorithms. 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.

Announcements

Exam 3 and 18.085 grades are all done and the Exam 3 average was 80.

The graded exams are available from Debbie Bower in 2-108 (desk at right)

I hope you enjoyed the course and will remember those A'CA ideas forever.

It was a pleasure for me to work with you. Very best wishes!
Gilbert Strang

Exam 3 - Solutions

Professor Strang's favorite -1,2,-1 matrix in cupcakes

Bigger version

Exam 2 - Solutions

[top]

HOMEWORK

Please include your 18.085 number (if it was circled on hwk 1)
at the top of your first homework page to help us order the hwks alphabetically
THANK YOU

#10 for Fri Dec 4 (Quiz TUES 8th IS IN WALKER !!!!) - Solutions

Section 4.4: 1, 2, 3, 7, 8, 9, 10

Section 4.5: 1, 2, 8, 10, 11, 12, 14, 22, 24, 25

#9 for Mon Nov 23 - Solutions

Section 4.1: 1, 6, 10, 13, 14, 18

Section 4.2: 4

Section 4.3: 2, 6, 8, 10, 15, 22

Problem 4.3.22 was added in the latest printing of the book. Here are the two important parts of the problem:
a) Write out F2D = kron(F,F) for N=2. It will be 4 by 4
d) Why does F2D times a vector u2D need only O(N^2 log N) operations?

MATLAB Homework

The homework is about the important "Gibbs phenomenon".
Create a figure like Fig. 4.3 showing the partial sums of the
Fourier series for the SQUARE WAVE in Exercise 4.1.2.
It jumps from -1 to +1 at x = 0.

Can you determine the amount of the overshoot? In what way does
this same overshoot number appear in Fig. 4.2 for the delta function?
What is the size of the overshoot if the step fcn jumps from 0 to A ?

Find a relation between the WIDTH of the overshoot (the first lobe
above +1) and the NUMBER of terms in the partial sum for the square wave.

New question (to me): Beyond the overshoot in Fig. 4.3
it looks to me that there is an UNDERSHOOT below +1. Is this true
in your graph of partial sums? What depth of undershoot?
I hope you can blow up the figure near the jump.

Extra: Also graph and blow up the Fourier series for the ramp
function f(x) = max(0,x) near the corner (not a jump!) at x = 0.

#8 for Fri Nov 6 - Solutions

Section 3.3: 7, 8, 11, 16, 24 (harder), 27

Section 3.4: 4, 17, 18

Section 3.5: 1, 2 (the book explains the kron command)

#7 for Fri October 30 - Solutions

Section 3.1: 1, 2, 5, 9, 11, 14, 17

Section 3.2: 5, 7, 17, 19

#6 for Wed October 21 - Solutions

Section 2.7: 1, 2, 3, 6, 7, 11

MATLAB1 : Create the matrix A for Problem 5 (Truss E, with 30-30-120 triangle at the top) Check det (A'*A) Find 2 solutions to Au = 0.

MATLAB2: Bar 2 of a 4-node truss connects joints 1 and 3 at a 45 degree angle. That row of A is row2 = [-1 -1 0 0 1 1 0 0]/sqrt(2)] With stiffness c2 = 6, print the stiffness matrix E2 = c2*row2' *row2. The true element matrix e2 is 4 by 4, without all the zeros in E2. Print e2 and execute a Matlab command to place e2 into E2.

#5 for Wed October 14 - Solutions

Section 2.4: 7, 8, 9, 11, 14, 15, 17, 18

#4 due the evening of Thursday October 8 (when you hand in the exam)

Section 2.2: 5 + this question:

Solve u'' = -Tu for T = [1 -1; -1 2]. Looking for the solution with 4 constants A B as on page 116 (which was fixed-fixed)

Section 2.3: 1, 7, 8, 9, 12

Section 2.4: 1, 3

Please keep this homework for reference during the exam / plan to collect later

#3 for Wed September 30 - Solutions

Section 1.6: 20, 22, 24, 27

Section 2.1: 1, 3, 7, 8, plus this question

First MATLAB Homework in 18.085

Find the displacements x(1),...,x(100) of 100 masses connected by
springs all with c = 1. You may take each force f(i) = .01
and consider two boundary conditions at the bottom:
(a) Spring 101 connects the last mass to a support as in Figure 1.7
(b) Mass 100 hangs free at the end of the line of springs.

Submit GRAPHS of the displacements in these two cases.

Here are a few MATLAB hints, mostly correct: d = ones(100,1) is a
column vector of 100 ones and diag(d) is a diagonal matrix (in fact I)
with d on the diagonal. diag(d,1) puts d on an off-diagonal, maybe
this matrix has order 101. After computing the vector x try

plot(x,'+')
xlabel('mass number')
ylabel('displacement')
print

Suggested problems not to turn in
Section 1.5: 9
Section 1.6: 15, 16

NOTE: The mysterious numbers on the graded Hwk 1 are in alphabetical
order -- if you write your number at the top of future homeworks, that
helps to put them quickly in the right order. (Some don't yet have a
number, still to do)

#2 for Wed September 23 - Solutions

Section 1.3: 1, 5, 9, 11

Section 1.4: 1, 4, 7, 15

Section 1.5: 1, 7, 9, 13

Section 1.6: 4, 6

#1 for WED September 16:

Section 1.1: 2, 5, 17, 20

Section 1.2: 1, 4, 7, 16

from the CSE textbook

NOTES
Apologies that question 1.2.1 involves delta functions !!
The slope of u(x) jumps from A to B.
So the next derivative is a delta function at x = 0 of magnitude ... (see page 38)
No boundary conditions in this question, it runs from minus infinity to infinity.
And the last line of 1.2.14 needs a multiplication by -1. One more error is hidden in the book.

PLEASE PRINT YOUR NAME CLEARLY
The class list will be created from the homeworks -- and Hwk 1 will not be graded in full detail, it is a start for this course.

[top]

Quizzes and Solutions

[top]

Previous Course Content

This section contains information from the previous courses, videos, practice exams, and extra homework and solutions. (Please report broken links so I can fix them).

18.085 from Fall 2008 is now completely online at OCW-18.085

Videos: The special event for Fall 2008 is that the lectures will be recorded for OpenCourseWare You will already find a partial earlier set on the website but the course has since evolved. The videos of 18.06 Linear Algebra have been successful on OCW (please use them for help!!). The Lord Foundation gave a new grant for 18.085.