MESSAGE TO THE CLASS I have read the projects and I am very satisfied -- I think you will be satisfied with course grades. If you don't need the copy I have, I will keep it. (Hongmei can pick hers up except pages 1-2.) I wrote small notes on the cover pages and Brett will have those any afternoon in 2-235. THANK YOU for taking the course. Gilbert Strang
Link to paper on "Approximate and Incomplete Factorizations" http://www.library.uu.nl/digiarchief/dip/dispute/2001-0621-115821/proc.pdf
Is there a simple way to count the fill-in in elimination ?
Run chol (or lu) and use "nnz" (number of non-zeros).
>> % Generate discrete Laplacian operator >> A=delsq(numgrid('S',100)); >> % Choleskey factorize >> R=chol(A); >> nnz(R) ans = 941289 >> % As above, but permute rows & columns by approximate minimum degree >> p=symamd(A); >> R=chol(A(p,p)); >> nnz(R) ans = 184917
Does min degree just take nodes a row at a time (if it breaks ties
If that is N^3 -- and would sparse backslash do it ?? --
Yes it does, it is a "greedy algorithm." The approximate one might be a little different, but in principle it does the same thing. Sparse backslash does this automatically. If symmetric, then it uses "symmmd" (symmetric minimum degree), then "chol." If non-symmetric, "colamd" (column approximate minimum degree), then umfpack.
this is about as fast as possible except for Fast Poisson Solvers ?
Well, it is probably as fast as possible for Gaussian elimination. But iterative solvers do better, I think CG with Incomplete Choleskey should be N^(2.5). Then of course fast solvers, FFT-based and multigrid.
Homework for Monday April 11 (start now please!!) from text
pages 403-411 and also Section 3.6 of the "new
book". Answer and turn in problems 1,2,3 in that Section 3.6
Develop a code for INCOMPLETE LU and try it on the 2D matrix for increasing N. You can make it a general code or specialize it to these particular matrices (and decide which entries to keep in the approximate factors L and U). Try on A2D u = random right side. You could use eig to find the spectral radius of the preconditioned matrix B. That matrix is I - inv(P)A2D where P ≈ LU. Of course if P = A2D then B = 0 and convergence in 1 step.
Read sections 3.6 and 3.7 (later 7.2 and 7.4) from the new book 3.6 3.7 7.2 7.4
Quiz on Friday March 11: Exact solutions and finite difference approximations to the equations discussed from Sections 6.4-6.6. Please bring printout of code and exact and approximate solutions at t=1 to the heat equation ut = uxx with u(x,0) = \delta(x). Choose any difference method and any stable \Delta t and \Delta x: grade does not depend on order of accuracy !
M. Benzi, G. H. Golub and J. Liesen, "Numerical solution of
saddle point problems", Acta Numerica, 14 (2005), pp 1-137
You can download the final version from http://www.mathcs.emory.edu/~benzi/Web_papers/bgl04.pdf
Lecturer: Gilbert Strang, email@example.com. Office: 2-240. Office hours: Mon 3-4, Wed 2-3.
Graders: Zhou Zhang (firstname.lastname@example.org) and Fangyun Yang (email@example.com). Office: 2-251. Office hour: Tues 11:30-12:30.
Textbook: Introduction to Applied Mathematics by Gilbert Strang
Due IN CLASS
WED 2/9 FRI 2/11
(stapled if possible)
PLEASE **PRINT YOUR NAME**
6.4: 1-5, 13-15, 22, 29;
6.5: 1-2, 17
FRIDAY FEB 25 MON FEB 28
6.5: 16 and 19 and (Find a stable difference method for
ut = uxxx)
6.6: 1, 2, 5, 9
ALSO SOLVE ut + uux = 0 to t = 4 with u(x,0) = 1 and 0,
also 0 and 1, as discussed in class (Lax-Friedrichs method).
Class MATLAB codes:
Exams: 2 or 3 mid-term exams. Open books and notes.
Homework: One problem set due in class each week, except exam weeks.
Grading: Exams 50%, project 30%, homework 20%.