CS 290H: Sparse Matrix Algorithms

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CS 290H Sparse Matrix Algorithms

• John R. Gilbert (gilbert_at_cs.ucsb.edu)
• http//www.cs.ucsb.edu/cs290

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Some examples of sparse matrices

• http//math.nist.gov/MatrixMarket/
• http//www.cs.berkeley.edu/madams/femarket/index.
  html
• http//crd.lbl.gov/xiaoye/SuperLU/SLU-Highlight.g
  if
• http//www.cise.ufl.edu/research/sparse/matrices/

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Link analysis of the web

• Web page vertex
• Link directed edge
• Link matrix Aij 1 if page i links to page j

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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 pA (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 2,000,000,000 by 2,000,000,000.

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A Page Rank Matrix

• Importance ranking of web pages
• Stationary distribution of a Markov chain
• Power method matvec and vector arithmetic
• MatlabP page ranking demo (from SC03) on
  a web crawl of mit.edu (170,000 pages)

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The Landscape of Sparse Axb Solvers
D
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Matrix factorizations for linear equation systems

• Cholesky factorization
• R chol(A)
• (Matlab left-looking column algorithm)
• Nonsymmetric LU with partial pivoting
• L,U,P lu(A)
• (Matlab left-looking, depth-first search,
  symmetric pruning)
• Orthogonal
• Q,R qr(A)
• (Matlab George-Heath algorithm, row-wise Givens
  rotations)

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Graphs and Sparse Matrices Cholesky
factorization
Fill new nonzeros in factor
Symmetric Gaussian elimination for j 1 to n
add edges between js higher-numbered
neighbors
G(A)chordal
G(A)