• Here are the practice final and solutions: final_practice.pdf and final_practice_sol.pdf.
• The Final Exam is Wednesday May 6 2009 from 8am to 10am.
  • Please plan to be there 10min before.
  • Two one sided, letter sized, handwritten cheat sheets are allowed.
  • No notes or books are allowed. Calculators are allowed but are not needed.
• For the three of you who will take the Early Final Exam on Monday May 4 2009 from 12:30pm to 2:30pm I will contact you soon with exact room.
• Thank you all for the excellent project presentations!

• The class notes for hyperbolic problems are here: na018.pdf. See also the Leveque book that appears in the references.
• The Matlab examples we saw in class are here: hyper.m and outflow.m.
• There will be half a lecture next Friday.

• Class notes for today are available here: na017.pdf.
• The solutions for HW5 are here: hw5sol.pdf.

Lecture notes for the past week and today are here: na016.pdf. They are concerned with parabolic PDEs (heat equation).

• HW6 is due the last day of classes and is available here: hw6.pdf. Since I will drop the lowest HW grade you can consider this assignment optional if you have done all the other assignments.

Here is a recap of what we have seen in finite elements: na015.pdf.

• Lecture notes for Wed and Friday are here: na014.pdf.
  There is a typo in p103, in the expression for B, B_2,1 should be y₂-y₁.
• HW5 has been extended to next Friday 04/10. This is because there were many misunderstandings about what was asked in the homework.
  Please see this addendum: hw5add.pdf, it should clarify what exactly I am expecting in your solutions.
• HW6 will be given sometime next week.

• Here are solutions for HW4: hw4sol.pdf
• For HW5:
  • P1 is the space of polynomials with degree less than or equal to 1.
  • Please do use sparse matrices for the calculations (the command sparse(n,n)gives a matrix that can be used as zeros(n,n) but that stores only the non-zero components).
  • An essential aspect of FEM is that the stiffness matrix is sparse, making linear system solves much faster. For sparse systems an iterative method like CG would work well. However Matlab’s backslash uses an LU factorization that knows how to take advantage of sparsity. If you want to know what is under the hood check this links: book and talk.
  • The indices in the local to global map i(e,j) may be different than what we saw in the notes because in Matlab it is more convenient to number the nodes and local degrees of freedom starting with one.
• Lecture notes for Monday are here: na013.pdf

• HW5 is due next Friday April 3 and is available here: hw5.pdf. This is a short mostly programming assignment where you need to implement the 1D finite element method.
• Class notes the week are available here: na012.pdf.

Here are some reference numbers for HW4: hw4ref.txt.

• Lecture notes covering the FEM for 1D are here: na011.pdf.
• Please remember to pick a project during the spring break. I will be around during the break if you need help with your HW or your project.

• Your midterm exam has been graded. The solutions that we saw together in class are here: exam001.pdf. Here is a histogram of the exam grades:
  >=50:*
>=60:
>=70:****
>=80:
>=90:*****
• You should receive an email Friday (today) morning with your grade and a partial grade computed with 0.45*(hw average) + 0.55*midterm. The HW average being computed by removing the lowest HW grade. The final grade will take this into account as well. A rough letter grade would be: A=90+, B=80+, C=70+.

• Homework 4 is due Monday March 23 2009 and is here: hw4.pdf.
• The last class notes for eigenvalue algorithms are here: na009.pdf.
• Today we started the Finite Element Method. Class notes are here: na010.pdf. Our goal will be to understand the FEM in 2D and to implement P1 triangular finite elements. We will see enough theory to derive simple error estimates.

Here is the solution to HW3: hw003.pdf.

• The practice midterm is here: exam1_practice.pdf and the solution we did together is here: exam1_practice_sol.pdf.
• The class notes covering Householder reflectors and reduction to Hessenberg form are here: na008.pdf.

• Here is a (detailed) review sheet of what I expect you to know for the exam that we have this coming Fri 02/27 in class: exam1_rev.pdf.
  • Reminder: A practice exam will be solved on Tuesday’s lecture.
  • A calculator is not needed for the exam, but it is allowed.
  • A one-sided, letter sized, handwritten cheat sheet is allowed.
  • No books or other notes are allowed.
• HW3 solutions will be posted on Monday at about 8pm. I will not accept any HW3 after the solutions are posted.

• Here are some numbers for HW3. I asked to use a slightly different method than that of the book for Problems 2 and 3. So here are some numbers for comparison: drva.m, res_drva.txt (for 11.1.3 a,b) and res_drvb.txt (for 11.2.3 a,b,c). These numerical solutions have the same number of accurate digits as the book’s solutions.
• The behavior of spdiags can be tricky if your sub/super diagonals vary. This matlab code:
  

  Q = [ 1 3 0
        2 3 -1
        3 3 -2
        0 3 -3 ];
  A = spdiags(Q,-1:1,4,4);
  

  Computes the matrix with entries:
  

       3    -1     0     0
       1     3    -2     0
       0     2     3    -3
       0     0     3     3
  

  So spdiags uses only the upper part of subdiagonals and the lower part of superdiagonals.
• If you don’t want to use spdiags you can create an n by n all zeros sparse matrix with A=sparse(n,n); and then fill it entry by entry.

• Class notes on Eigenvalues are here: na007.pdf. Today we finished the theory part (9.1).
• The references I am using for eigenvalues are
  • Trefethen and Bau, Numerical Linear Algebra, SIAM 1997
  • Golub and Van Loan, Matrix Computations, 3rd edition, John Hopkins, 1996
• The material we will cover is somewhere between Chapter 9 in your book and Part V of Trefthen and Bau.

• HW3 is here: hw3.pdf and is due Fri 02/20. Please read the hints I wrote, they should make the implementation simpler than in the book.
• The Midterm exam will be on Fri 02/27 and will cover Chapters 5 and 11 (Initial value problems and Boundary value problems). On Tue 02/24 we will do a practice exam. The exam will be closed book and notes.
• Today we started Chapter 9 on Eigenvalues.

• Lecture notes for Boundary value problems (11.1-11.4) are available here: na006.pdf.
• HW2 Solutions are here: hw002.pdf.

• Lecture notes for today are here: na005.pdf, we covered the key notions of A-stability and the region of absolute stability.
• We finished with Initial Value Problems (Chap 5) and on Friday we start Boundary Value Problems (Chap 11).
• Here is a Matlab program that displays the absolute stability region for a multistep method: absstab.m.

• Projects: The project calendar and some project suggestions are here: projects.pdf. Please do not be thrown off by the names of the methods, they are usually simple to understand and implement.
• Solutions to HW1 are here: hwsol001.pdf.
• HW2 has been extended to Fri 02/06. Charlie found a typo in exercise 5.6.2 a (part of problem 4): The true solution should be y(t) = (2*t+1)/(t^2+1) (the denominator in the book has a 2).
• There will be no homework posted this week
• Class notes are here: na004.pdf (covers stability, consistency, systems and higher order equations)

• HW2 is due on Wed 02/04, so that we can make full use of Tuesday’s Q&A session. The problems are available here: hw2.pdf.
• Lecture notes for today are posted here: na003.pdf, and cover part of the section 5.10.

Lecture notes for last week and Monday are here: na002.pdf. They cover 5.6-5.8 (Multistep methods).

• HW1 is due Tue 01/27 and is available here: hw1.pdf. Note: No computer is needed for Problem 2, the solution should be very simple.
• Please contact me as soon as possible if you need access to the Math Computer Lab for Matlab or if you would like to use Octave (the open source alternative).
• Homework policy
  • The preferred method to turn in your HW is in class. You can also send it by email (see below for guidelines), leave it under my office door or in my mailbox. Please email me every time you use any of this alternatives.
  • Typed solutions are not required.
  • You may work in pairs, in that case turn in only one writeup.
• Guidelines for numerical experiments
  • Always include your code and some runs of your program(s). You have to include enough information to convince me that your code is doing what is supposed to be doing. You will be penalized if you don’t, and it makes it harder for me to find potential mistakes.
  • Please present your numerical results in a way that highlights the intended results.
    • For example if you are asked to compare several methods, you could do this with a plot of the actual vs computed values if their discrepancies are large enough. If the errors are too small, then plotting the error (or a table) is probably more meaningful.
    • Keep in mind that semi-log or log-log scales (in Matlab: semilogy, semilogx, loglog ) often make it easier to interpret convergence curves.
    • Your program output should include relevant information. For example if you have a method with adaptive time step, then it is good to know what the time step is at each iteration.
    • The Matlab routine fprintf is very handy. It allows to use C style format strings for your program’s output. It works by first specifying what is to be printed and how. For floating point numbers there are 3 possible formats (in Matlab): %f (floating point as in 1.234 or 0.221), %e (exponent notation as in 1.234e+00 or 2.221e-01) or %g (which lets fprintf choose between %f and %e). One can also specify: the minimum number of characters to be printed and the number of digits after the period (the radix). For example %10.3f tells fprintf to format a floating point number as a float, using 10 characters and with 3 digits after the period. The line feed character is \n. Here is an example:
      

      >> a=pi; b=pi/10000; c=pi*100000;
      >> fprintf('%10.3f %10.3f %10.3f\n',a,b,c)
           3.142      0.000 314159.265
      >> fprintf('%10.3g %10.3g %10.3g\n',a,b,c)
            3.14   0.000314   3.14e+05
      >> fprintf('%10.3e %10.3e %10.3e\n',a,b,c)
       3.142e+00  3.142e-04  3.142e+05
      >> fprintf('a=%10.3e, b=%10.3e c=%10.3e\n',a,b,c)
      a= 3.142e+00, b= 3.142e-04 c= 3.142e+05
      

      
      See help fprintf or doc fprintf for more details.

• I may present algorithms in a slightly different way than in the textbook. Unless specified, use the book’s algorithm because it will make it easier for you to test your code against the examples given in your book.

Lecture notes for the week are available here: na001.pdf, we have covered sections 5.1-5.5.

• Room Change: As of today the classroom changes to:
  • MWF: LCB 323 (across the hall from the fish bowl)
  • T: LCB 322 (the fish bowl)
• Tuesdays will be mostly Q&A sessions, homework (or exam) solutions, demos, etc… If we need time to catch up with the material, a regular lecture will be held.

Class announcements will be posted here.