September 13, 2004
Computational Algebra, Fall 2004,
Programming Assignment 2
Hand in by September 28 at lecture!
Sparse Matrices, direct algorithms
A short introduction given in lecture notes
The Matlab command
help/ MATLAB/Using Matlab/Mathematics/Sparse matrices
gives an instructive description. Some details are given in Extra tips.
If you are really interested read the original paper:
Gilbert, John R., Cleve Moler, and Robert Schreiber, "Sparse Matrices
in MATLAB: Design and Implementation," SIAM J. Matrix Anal. Appl., Vol. 13,
No. 1, January 1992, pp. 333-356. gilbert92sparse.ps gilbert92sparse.pdf
Test matrices are found in Matrix Market at National
Institute of Standards and Technology (NIST).
1. Simple example: Compare reorderings!
Start with the matrix you get from a finite difference approximation of
the Laplace equation. You get a grid by the command G=numgrid
and a matrix by A=delsq(G)! Look at the matrix with spy(A).
Take an appropriate size, start with a square. Look at the grid matrix
G and see how the points are ordered by columns.
You will get permutation vectors for band width reduction (RCM)
and minimal degree (MMD) by the calls pr=symrcm(A) and pm=symmmd(A).
Substructuring, nested dissection, is not implemented, but there is an option
in numgrid that gives that order for a square. Look at the reordered
matrices with spy and compare to what I showed in the lecture!
Look at the grid G reordered by Reversed Cuthill Mc Kee. You can use the
routine vector2grid as described in Extra tips. Note that it
cannot be used directly on the permutation vector pr, it must have the inverse
permutation. It is simple to get the inverse permutation by doing the call
iv=1:n ; rp(pr)=iv ;
which gives the inverse permutation in the vector rp.
Does your result look like what I showed in the lecture?
The Minimum Degree ordering of G does not look as I have described. Take
a look at it and compare to Nested Dissection, which is precisely as described.
Look also at the matrices with spy, here there is a qualitative similarity
between MMD and ND.
2. When is sparse factorization of advantage?
The sparse factorization will use much less storage space and fewer arithmetic
operations than a full matrix code. On the other hand it needs quite a lot
of book keeping to track at which places fill in occurs. In industrial sparse
matrix codes, one switches to handle a full matrix when a certain percentage
of the elements are nonzero. Look at the LU factors of a matrix ordered with
minimum degree, they are rather dense in the lower right corner!
Now we want to see when it is of advantage to use a sparse matrix code.
Matrix: Takethe matrix A as finite difference matrices over
a square as in previous task.
Right hand side b: Take the vector b as a 1 (one) in one or a set
of contigous positions in the center of the grid G.
Solution x: Plotted over the square it will be a tent like shape with
poles in those positions where b is one. You can use the routine
to get a matrix with the vector x spread over the grid G.
Experiment: Let the number of grid points vary. Make up a table of
the number of nonzeros in the original matrix A and the LU factors for the
different reorderings RCM, MMD and ND. Measure the time needed for
Compare to the space and time requirements for a full matrix code. Just
- Analyse: Finding the permutation
- Factor: Compute the LU factors of the reordered matrix
- Solve: Solve a system for a new right hand side b
- Check solution: Compute r=Ax-b and list ||r||
and you get a full matrix. Be careful, the full matrix may need too much
A note on timing: The Matlab clock ticks very slowly. You might need
to run the shorter operations several times and divide the total time by
the number of repeats.
3. General Sparse Matrices
I havc stored some quite large matrices as Matlab files. Look at them and
see how much fill in there is in Gaussian eleimination with different reorderings.
Note that these matrices are too large to store as full matrices! Do not print
the result of the spy command in the report, it can be very space consuming.
- .m files are executed in Matlab. Just write the file
- .mat files are read in by the command load
- .mtx files are in Matrix Market format and is read with the
command A=mmread('file name');
If you do not have the routine mmread, you may fetch it from Matrix Market.
Take one or more of the matrices:
A finite element stiffness and mass matrix generated by FEMLAB. Will probably
behave like the finite difference matrices of previous task.
- Try a 3 dimensional Laplace generated by delsq3d.m
i. e. for 20*20*40 points.
- A power network for western USA bcspwr
- A VLSI circuit of a clock clock.mat
Only one of the matrices, G, has many elements enough to be interesting,
I think each element stands for a resistor. A VLSI circuit has a tree like
structure with few cycles. This makes the matrices very sparse and algorithms
that are built on level structures like RCM will give very little fill in.
- You may find another interesting matrix on Matrix Market. Get it and
Finished September 13, 2004 by Axel Ruhe