1. CS 290N / 219: Sparse Matrix Algorithms
John R. Gilbert

2. Systems of linear equations: Ax = b

3. Systems of linear equations: Ax = b
Alice is four years older than Bob.
In three years, Alice will be twice Bob’s age.
How old are Alice and Bob now?

4. Link analysis of the web1 5 2 3 4 6 7 1 2 3 4 7 6 5
Web page = vertex
Link = directed edge
Link matrix: Aij = 1 if page i links to page j

5. Web graph: PageRank (Google) [Brin, Page]
An important page is one that many important pages point to.
Markov process: follow a random link most of the time; otherwise, go to any page at random.
Importance = stationary distribution of Markov process.
Transition matrix is p*A + (1-p)*ones(size(A)), scaled so each column sums to 1.
Importance of page i is the i-th entry in the principal eigenvector of the transition matrix.
But, the matrix is 10,000,000,000 by 10,000,000,000.

6. Mark Adams: Bone Density Modeling

7. Poisson’s equation for temperature

8. The (2-dimensional) model problem
Graph is a regular square grid with n = k^2 vertices.
Corresponds to matrix for regular 2D finite difference mesh.
Gives good intuition for behavior of sparse matrix algorithms on many 2-dimensional physical problems.
There’s also a 3-dimensional model problem.

9. Solving Poisson’s equation for temperature
k = n1/3
For each i from 1 to n, except on the boundaries:
– x(i-k2) – x(i-k) – x(i-1) + 6*x(i) – x(i+1) – x(i+k) – x(i+k2) = 0
n equations in n unknowns: A*x = b
Each row of A has at most 7 nonzeros.

10. Graphs and Sparse Matrices: Cholesky factorization
Fill: new nonzeros in factor 10 1 3 2 4 5 6 7 8 9 10 1 3 2 4 5 6 7 8 9
Symmetric Gaussian elimination:
for j = 1 to n add edges between j’s higher-numbered neighbors
G(A)
G+(A) [chordal]

11. The Landscape of Sparse Ax=b Solvers
Direct
A = LU
Iterative
y’ = Ay
More Robust
More General
Non-
symmetric
Symmetric
positive
definite
More Robust
Less Storage D

12. Complexity of linear solvers
Time to solve model problem (Poisson’s equation) on regular mesh
n1/2 2D 3D
Dense Cholesky:
O(n3 )
Sparse Cholesky:
O(n1.5 )
O(n2 )
CG, exact arithmetic:
CG, no precond:
O(n1.33 )
CG, modified IC:
O(n1.25 )
O(n1.17 )
CG, support trees:
O(n1.20 ) -> O(n1+ )
O(n1.75 ) -> O(n1.31 )
Multigrid:
O(n)

13. Administrivia Course web site: www.cs.ucsb.edu/~gilbert/cs290
Join the (Google) discussion group!! (see web site)
First homework is on the web site, due next Monday
About 5 weekly homeworks, then a final project (implementation experiment, application, or survey paper)
Assigned readings: Davis book, Saad book (online), MGT. (Library reserve; see me for an extra copy of Davis.)