Algorithm 836: COLAMD, a column approximate minimum degree ordering algorithm

ABSTRACT

Two codes are discussed, COLAMD and SYMAMD, that compute approximate minimum degree orderings for sparse matrices in two contexts: (1) sparse partial pivoting, which requires a sparsity preserving column pre-ordering prior to numerical factorization, and (2) sparse Cholesky factorization, which requires a symmetric permutation of both the rows and columns of the matrix being factorized. These orderings are computed by COLAMD and SYMAMD, respectively. The ordering from COLAMD is also suitable for sparse QR factorization, and the factorization of matrices of the form A^TA and AA^T, such as those that arise in least-squares problems and interior point methods for linear programming problems. The two routines are available both in MATLAB and C-callable forms. They appear as built-in routines in MATLAB Version 6.0.

REFERENCES

• 1: Patrick R. Amestoy , Timothy A. Davis , Iain S. Duff, An Approximate Minimum Degree Ordering Algorithm, SIAM Journal on Matrix Analysis and Applications, v.17 n.4, p.886-905, Oct. 1996 [doi>10.1137/S0895479894278952]
• 2: Timothy A. Davis , John R. Gilbert , Stefan I. Larimore , Esmond G. Ng, A column approximate minimum degree ordering algorithm, ACM Transactions on Mathematical Software (TOMS), v.30 n.3, p.353-376, September 2004 [doi>10.1145/1024074.1024079]
• 3: James W. Demmel , Stanley C. Eisenstat , John R. Gilbert , Xiaoye S. Li , Joseph W. H. Liu, A Supernodal Approach to Sparse Partial Pivoting, SIAM Journal on Matrix Analysis and Applications, v.20 n.3, p.720-755, July 1999 [doi>10.1137/S0895479895291765]
• 4: John R. Gilbert , Cleve Moler , Robert Schreiber, Sparse matrices in matlab: design and implementation, SIAM Journal on Matrix Analysis and Applications, v.13 n.1, p.333-356, Jan. 1992 [doi>10.1137/0613024]
• 5: Gilbert, J. R. and Peierls, T. 1988. Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Statist. Comput. 9, 862--874.
• 6: Kermit Sigmon , Timothy A. Davis, MATLAB Primer, CRC Press, Inc., Boca Raton, FL, 2001
• 7: The MathWorks, Inc. 2002. MATLAB External Interfaces/API, Version 6.5.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Sparse, structured, and very large systems (direct and iterative methods)
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Efficiency; Algorithm design and analysis

General Terms:
Algorithms, Experimentation, Performance

Keywords:
Linear equations, ordering methods, sparse nonsymmetric matrices