AM9561, Graduate Introductory Numerical Methods Autumn 2010

Rob Corless

Middlesex College Room 272

Meeting Times

Midterm MC 204 Thursday 11.11.10 18:30--20:30.
(Lab) NCB 105, Mondays 9:30--10:20
(Lectures) Mondays 2:30--5:30 in MC204
Thanksgiving makeup lectures 12:30 Fridays MC204

TA: Nic Fillion

Course Textbook:

Introductory Graduate Survey of Numerical Methods, By Rob Corless and Nic Fillion, alpha version available Sept 20

Resources

Assignments

.pdf version

Assignment 1 (worth 5% of your mark). .pdf version , and the LaTeX source , which you will need for question 8.
Assignment 0 (not for credit) Write a short description of your scientific background and interests. Include contact email for inclusion in the course mailing list.

Labs

Rootfinding, Newton, and fractals. fractalboundary.m
Quadrature. Find the integral of f = @(v) ((1-v.*cot(v)).^2+v.^2)./(z+v.*csc(v).*exp(-v.*cot(v))) for z = 0.13, on the interval -pi < v < pi. Divide your answer by 2*pi. You should get W(z)/z, where W(z)*exp(W(z)) = z. Document how you get around the apparent bug in "quad" that won't let you integrate this from -pi to pi.
equival.m and chebval.m
Iterated square root demo, save as "Higham.m"
Maple worksheet exploring exact but slow computation; save as "slow.mw"

Topics so far

More on quadrature: Spectral Accuracy and the double exponential method . Added: class notes 15.11.10
Splines, PCHIP, and quadrature
Floats, polynomials . See also a technical note at the Mathworks explaining that Matlab doesn't quite comply with IEEE754.
15.10.2010 tinytitchy.m
AM 9561 4.10.2010 SVD.pdf

Old course material below this line

2009 Course Lectures Tuesday and Thursday 9:30--10:30 MC204

Computer Lab Tuesday 12:30--13:30 North Campus Building 105

In the scheduling meeting we had discussed a preference for the time 13:30--14:30, but balancing different needs I chose this time.

Office Hour: Thursday 10:30--11:30 (right after class)

TA Office Hour: Monday 11:30, MC 275B or Rotunda. (Roman Naryshkin)

Midterm

SSC 2110 has been booked for your AM 9561 midterm on Thursday, November 5 from 5:00 to 7:00pm.

Topics: Condition numbers in all contexts: arithmetic, matrix eigenproblems, solution of linear systems of equations, interpolation, quadrature. NEW : Results are in. Exams handed back 10.11.2009. Solutions

Extension to Assignment 2

Several people have requested more time for Assignment 2. I will at least extend the deadline till Thursday, and we will discuss tomorrow if it makes sense to extend it till next Tuesday. This will be announced also by email (I have finally created the class mailing list). Assignment 1 has now been marked (..except I am double-checking that all of them got marked) and so you should get them back soon.

Resources

Assignments

Assignment 4 (worth 5% of your mark). .pdf version . Helper files first.m and second.m and the file mypoisson.m .
Assignment 3 (worth 5% of your mark). .pdf version due Tuesday November 16, 2009. For interest, .tex source and I have updated the .bib file in the link below. For those of you who wish to use
Assignment 2 (worth 5% of your mark). .pdf version due Tuesday October 20, 2009. For interest, .tex source and I have updated the .bib file in the link below. For those of you who wish to use Maple partly for this assignment, I include a helpful command below.
```
V := Matrix(7, 7, shape = Vandermonde[[tau[0], tau[0], tau[1], tau[1], tau[1], tau[2], tau[2]], confluent = true]);
```
Assignment 1 (worth 5% of your mark). .pdf version , and the LaTeX source , which you will need for question 8. Oh, yes, the .bib file , too.
Assignment 0 (not for credit) Write a short description of your scientific background and interests. Include contact email for inclusion in the course mailing list. Due 17.9.2009.

Tutorials

31.11.2009 FFT and free time for Assignment
24.11.2009 Delay differential equations: Shampine, Thompson and Kierzenka tutorial
10.11.2009 Stiff solvers and BVP solvers (time permitting)
3.11.2009 Residuals by DEVAL, odeexamples
no tutorial 13.10.2009
6.10.2009 Sparse Matrix Stuff GaussSeidelIter.m , JacobiIter.m , eigIM.m , cost.m and Seneca.m.
This new version of Seneca.m and GaussSeidelIter.m can be used to look at Gauss-Seidel iteration as follows. I include a parameterization by a global variable theta (I got tired of changing all the 2.1 etc's to 10)
```
global theta;
theta = 2.1;
n = 100;
b = rand(n,1);
x = b;
for i=1:100,
   x = GaussSeidelIter( b, x );
end;
norm( b - Seneca(x), 1 )
```
yields 0.0025 on my machine.
28.9.2009 svdgraphs in matlab, svdgraphs in Maple

Topics so far

31.11.2009 dft.mw (Maple) Discrete sine transform, dft2.mw (Maple) Discrete cosine transform, dftdem.m (Matlab) fft, powerdem.m (Matlab) periidogram or power spectrum.

14.11.2009 and 26.11.2009 Delay Diiferential Equations. See the help for dde23, the tutorial at the Mathworks, and the Shampine & Thompson 2001 paper


%A Shampine, L.F.
%A Thompson, S.
%D 2001
%I Elsevier Science
%J Applied Numerical Mathematics
%K 
%N 4
%P 441-458
%T Solving DDEs in Matlab
%V 37
%W /cgi-bin/sciserv.pl?collection=journals&journal=01689274&issue=v37i0004&article=441_sdim
%X We have written a program, dde23, to solve delay differential equations (DDEs) with constant delays in Matlab. In this paper we discuss some of its features, including discontinuity tracking, iteration for short delays, and event location. We also develop some theoretical results that underlie the solver, including convergence, error estimation, and the effects of short delays on stability. Some examples illustrate the use of dde23 and show it to be a capable DDE solver that is exceptionally easy to use.
%0 Article
%8 June, 2001
%@ 01689274

See also Larry's current work. In particular, we will use his overlays that he presented at the meeting in the Yucatan: Read1.pdf and Read2.pdf with K.Cooke and P van den Driessche. More: J. M. Heffernan & Robert M. Corless, \Solving some delay differential equations using computer algebra", the Mathematical Scientist 31, no. 1, (2006) pp. 21-34.

17/20.11.2009 Guest lectures by Prof. David Jeffrey: Remote Computing
10/12.11.2009Stiff differential equations, Boundary-Value Problems, Differential-Algebraic Equations, and more.
3/5.11.2009 AM 9561 Numerical Solution of ODE .
27/29.10.2009 AM 9561 Functions and Root-finding
15.10.2009 More on Interpolation and Approximation.pdf and interpolationII.pdf and
13.10.2009 Notes on interpolation and a Maple worksheet
diary from today 8.10.2009
6.10.2009 and 8.10.2009. Sparse and structured linear systems. Moler 2.10 and supplementary material. cost.m and Seneca.m.
29.9.2009 and 1.10.2009 Notes from last year
24.9.2009 Class notes (if you wish a copy, please ask Audrey: don't print yourself, except at home if you like)
22.9.2009 Introduction to the SVD (Singular Value Decomposition): Material from Chapter 2 of Matrix Computations, by Golub and Van Loan.
Matlab commands from Sept 17, 2009

Old course material from 2008 and earlier below this line

Course Lectures 2008 Monday 9:30--11:30 MC204

Final Exam
The final exam will take place in HSB 13, Monday Dec 8, 9:00AM--noon. The computers will allow you to use Matlab but not email or web browsing. After you finish the exam, email will be turned back on and you can send me your results. The tutorial Tuesday December 2 will be a "dry run" of the software, with a prototype question.

Course Text and other online references

On the Midterm

Solutions

The midterm will take place in MC 204, Monday November 3, 2008, from 7:00pm until 10:00pm. Extra time will be available for those who need it (some of the questions take a long time to state, and so if English is your second (or third or fourth) language, maybe it's reasonable to take a little extra time to read them. I believe, though, that the questions themselves are straightforward to do, once you've read them.

Calculators or computers will not be necessary. The exam is closed-book, so no notes or text either.

I hope to have your assignments marked by the end of tomorrow, so you may have them back on the weekend to look at.

Observations on the assignment

The midterm material is directly from my notes and from Moler's book, though the problems may involve new situations for what you have learned. The topics covered are: (in six multi-part questions in this order)

Floating-point arithmetic
Numerical Linear Algebra
Interpolation and Least-Squares Approximation
Quadrature
Finite Differences
Rootfinding

Tutorials on Matlab and LaTex

Tuesdays 9:30am, North Campus Building 105

Found them...
11.11.2008 Residual Example
28.10.2008 ode solving Three-body problem. orbitodejac.m
21.10.2008 more on rootfinding: basins of attraction.
14.10.2008 Guest Tutorial, Prof. David Jeffrey, Chair, Dept. Applied Mathematics. Office Hours Postponed.
30.9.2008 Kahan on quadrature and my version of Kahan's proof that quadrature is impossible. Also, transcript of the Matlab session 29.9.2008.
23.9.2008 svdgraphs in matlab

Upcoming Lectures

FFT (Moler 8, fourier.pdf) and PDE (Moler 11, pdes.pdf)

Assignments

Assignment 4: Conditioning . Helper files first.m and second.m.
Assignment 3: Rootfinding and ODEs Due 10.11.2008. If you get this today, ask Audrey or Karen for a printout, don't print it yourself

(draft version 2.0) Assignment 2, Due 20.10.2008

Possible transcription error in B coefficients? Two students have reported a potential problem with the compact finite difference question. Here is a (Maple) code fragment: you can check the numbers here.


   c := 2.0 + sqrt(3.0);
   alpha := 1.0/c;

   b := Array(0..n); # same as u
   b[0] := (-(25*c+3)*u[0] + (48*c-10)*u[1] + (18-36*c)*u[2] + (16*c-6)*u[3]
             +(1-3*c)*u[4] )/12/h;
   for k to n-1 do
      b[k] := 3*(u[k+1]-u[k-1])/h;
   end do;
   b[n] := (11*u[n-4] - 58*u[n-3] + 126*u[n-2] - 182*u[n-1] + 103*u[n])/12/h;

Assignment 1, Due 29.9.2008
Assignment 0 (not for credit) Write a short description of your scientific background and interests. Due 15.9.2008.

Topics so far

Large Systems and the Method of Lines pages 114-125 of Shampine, Gladwell and Thompson, "Solving ODES with Matlab, Cambridge 2003. onewaywave.m and its driver:


>> t=0:0.3:8;
>> n=256;
>> x=linspace(0,2*pi,n+1);
>> u0=exp(-100*(x-1).^2);
>> [t,slices]=ode23(@onewaywave,t,u0);
>> mesh(x,t,slices),view(10,70),axis([0,2*pi,0,8,0,5])

10.11.2008 Delay Diiferential Equations. See the help for dde23, the tutorial at the Mathworks, and the Shampine & Gordon 2001 paper


%A Shampine, L.F.
%A Thompson, S.
%D 2001
%I Elsevier Science
%J Applied Numerical Mathematics
%K 
%N 4
%P 441-458
%T Solving DDEs in Matlab
%V 37
%W /cgi-bin/sciserv.pl?collection=journals&journal=01689274&issue=v37i0004&article=441_sdim
%X We have written a program, dde23, to solve delay differential equations (DDEs) with constant delays in Matlab. In this paper we discuss some of its features, including discontinuity tracking, iteration for short delays, and event location. We also develop some theoretical results that underlie the solver, including convergence, error estimation, and the effects of short delays on stability. Some examples illustrate the use of dde23 and show it to be a capable DDE solver that is exceptionally easy to use.
%0 Article
%8 June, 2001
%@ 01689274

See also Larry's current work. For examples in dynamical systems and biology, see Xingfu Zou's page, and in particular the wonderful paper with K.Cooke and P van den Driessche. More: J. M. Heffernan & Robert M. Corless, \Solving some delay differential equations using computer algebra", the Mathematical Scientist 31, no. 1, (2006) pp. 21-34.

3.11.2008 Stiff differential equations, Boundary-Value Problems, Differential-Algebraic Equations, and more.
27.10.2008 AM 9561 Numerical Solution of ODE . Please ask Audrey for a copy rather than printing on Monday: copies are cheaper. She will have them shortly after 9. Feel free to print on your own printer, of course, though if you can avoid it entirely so much the better.
20.10.2008 AM 9561 Functions and Root-finding
6.10.2008. Forgot to post this last week! On conditioning of evaluation and interpolation. Today's planned lecture: Numerical differentiation and finite differences.. See also An image of a talk I gave at CAIMS 2005. Here is the Maple Worksheet used to generate it, and the underlying Maple code. Matlab programs:
29.9.2008. (version 2.0 draft) Interpolation and approximation.
22.9.2008. Sparse and structured linear systems. Moler 2.10 and supplementary material. cost.m and Seneca.m.
15.9.2008. The SVD, least squares, and the solution of linear systems. Chapters 10, 5, and 2 of Moler. See also a Maple worksheet exploring the geometry of the SVD.
The Gauss map
Introduction, Floating-Point, necessity, complexity, stability.
Matlab diary Sept 8, 2008

Material from previous years

David Jeffrey's web page for AM561