AM 261b

Introduction to Numerical Analysis


Final exam from 1999 (pdf format). This exam is not too dissimilar from the exam I am planning for this course. We did not do so much on quadrature this year, so probably that question will be different. (I don't know why this came down from the site---it should have stayed up).

Questions regarding the final exam:

  1. What resources can be used on the final exam? Answer: the text, and your notes (including any papers you downloaded to read), and any calculator manuals you need. You will need a calculator, preferably programmable; some calculators (such as the HP 48 series) have matrix manipulations that may save you a couple of minutes. You will not need math handbooks, but you can bring one if you want.
  2. Can I bring a laptop and use Matlab? Answer: No. The issue is one of fairness; if we could get everyone to have a copy of Matlab, then the exam could be different. But we tried that in the lab for the first assignment and it didn't work.
  3. Where do I pick up my assignments? Answer: at the applied math office, when it is ready [It's there now.].
  4. Will you have an office hour? Answer: Yes, Thursday 11--12 as usual.

Script from final day's lecture: lastday.m The main lesson: retrospective analysis of the output of ode45 or ode15s is possible in terms of the residual: you can assess the quality of the solution produced, even without knowing the correct answer, if the problem is not stiff.

m-file to compute residuals from output of ODE45 (from April 8, 2002): residual.m

Resulting graph from the commands: reswest.jpg

>> [t,y] = ode45('west',[0,1],-1/2,odeset('refine',64));
>> d = residual(t,y,'west',64);
>> plot(t,d)



Transcripts from April 1, 2002

Numerical solution of y' = x^2 + y^2, y(0) = 1 (a problem with a movable pole) from March 25 and 27:

Assignment 3 Due Thursday, March 28 (no late penalty till April 3) Text file for assignment (Typos pointed out by Ivan Saika-Voivod)

Assignment 2 (not in-class) Due Monday, March 13 8:00am (penalty 2^(-n) for n days late). All questions except the bonus question are from Recktenwald.

  1. Question 1, p. 445
  2. Questions 6 and 7, p. 446
  3. Question 28, p. 450
  4. Question 33, p. 452
  5. Questions 6 and 7, p. 512
  6. (Bonus) Write a Matlab m-file program that computes the Turing factoring PA = LDUR for a given input rectangular matrix A. The pivots giving the permutation matrix A should be chosen as the maximum in a given column, hence exchanging rows. The unit lower triangular matrix L and the unit upper triangular matrix U should be built up by vector operations. The matrix R should be the reduced row-echelon form of A, provided that det(D) is not zero (and since D is diagonal, this will be easy to check).

    Remarks added Feb 28, 2002

    A rectangular matrix A can be factored into a product of a nonsingular
    matrix F and the unique reduced row echelon form R, namely A = FR. Then the
    nonsingular (square) matrix F can be factored into PF = L (U1) where L is
    unit lower triangular, P is a permutation matrix, and U1 is upper triangular
    but with non-unit diagonal. U1 may be further factored to U1 = DU where D
    is diagonal and U is unit upper triangular. Putting all these together

    PA = LDUR

    where P is a permutation matrix
    L is unit lower triangular (ie lower triangular with 1's on the diagonal)
    D is diagonal
    U is unit upper triangular (ie uppert triangular with 1's on the diagonal)
    and R is the unique reduced row echelon form of A. IF A had happened to be
    square and nonsingular to start with, then R is just the identity matrix.

An even-tempered day (02.20.02):

First assignment solutions (m-files)

  1. Question 1 (fixed minor error at end of file 2.19.2)
  2. Question 2

Plan for labs January 31 and February 1: Introduction to the Matlab editor and programming in Matlab.

Plan for Wednesday January 30: more linear algebra, including special classes of matrices and their uses.

Transcript of class notes (Monday January 28)

Supplementary material for Monday January 28: A Maple worksheet on the SVD, similar to the Matlab session above but with more text. To run it, double click on its icon, and if you have Maple (at least release 6) then it will launch. Click on the plus signs to expose the text, and hit ENTER on each command line in turn. Maple is similar in many ways to Matlab, but there are important differences, so this worksheet is to be regarded as supplementary only.

Reading assignment: all of Recktenwald Chapter 7 (Review of Linear Algebra)

Transcripts of 2nd lab (Ivan Saika-Voivod), January 24--25.

  1. First section
  2. Second section

Transcript of first lab (Thursday, January 17, and Friday, January 18)

Transcript of class notes (Wednesday, January 16)

Many course notes can be found at

``Numerical Monsters'' by G.C. Essex, M. Davison, and C. Schultzky, ACM Sigsam Bulletin, volume 34, no. 4, issue 134, December 2000, pp. 16--32.

Responses to questions on 1st lecture (Thursday January 10)

David Goldberg, What Every Computer Scientist Should Know about Floating Point Arithmetic

First assignment (due Monday January 14, not for credit)

  1. Give your name and email address (I will create a course mailing list).
  2. Write a short paragraph describing your academic interests and your mathematical and computer science background (courses, and marks if you wish).
  3. Some pranksters cut into a one-mile section of railway track, and weld in an extra foot of track. This causes the track to bow up in a circular arch. How high is the top of the arch?

Meeting times (Winter 2002):

Classes: Monday and Wednesday 8-9am (KB 203), with an occasional Tuesday 8am as needed (will be announced).

NOTE: There is some confusion regarding the lecture schedule; there are two different versions of the timetable floating around, one with Thursday 9am and one with Wednesday 9am. The Wednesday 9am time is preferred: anyone who *cannot* make that time, please let me know ASAP.

No meeting Wednesday January 9.

Tutorials (Labs): ONE OF Th 9-10am (UC 2) and Friday 8-9am (UC 2)

No labs January 10 or 11.

Course Outline (PDF format: if you can't read it, get Adobe Acrobat acrobat button )