An Approximate Minimum Degree Ordering Algorithm

An approximate minimum degree (AMD) ordering algorithm for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.

[1]  Iain S. Duff,et al.  Parallel implementation of multifrontal schemes , 1986, Parallel Comput..

[2]  J. W. Walker,et al.  Direct solutions of sparse network equations by optimally ordered triangular factorization , 1967 .

[3]  S DuffIain,et al.  An Unsymmetric-Pattern Multifrontal Method for Sparse LU Factorization , 1997 .

[4]  Patrick Amestoy,et al.  Vectorization of a Multiprocessor Multifrontal Code , 1989, Int. J. High Perform. Comput. Appl..

[5]  Alan George,et al.  The Design of a User Interface for a Sparse Matrix Package , 1979, TOMS.

[6]  Iain S. Duff,et al.  On Algorithms for Obtaining a Maximum Transversal , 1981, TOMS.

[7]  Alan George,et al.  The Evolution of the Minimum Degree Ordering Algorithm , 1989, SIAM Rev..

[8]  D. Rose.  A GRAPH-THEORETIC STUDY OF THE NUMERICAL SOLUTION OF SPARSE POSITIVE DEFINITE SYSTEMS OF LINEAR EQUATIONS , 1972 .

[9]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[10]  Timothy A. Davis,et al.  Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization , 1991 .

[11]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[12]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[13]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[14]  I. Duff,et al.  A Comparison of Sparsity Orderings for Obtaining a Pivotal Sequence in Gaussian Elimination , 1974 .

[15]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[16]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[17]  Alan George,et al.  A Fast Implementation of the Minimum Degree Algorithm Using Quotient Graphs , 1980, TOMS.

[18]  Joseph W. H. Liu,et al.  Modification of the minimum-degree algorithm by multiple elimination , 1985, TOMS.

[19]  John G. Lewis,et al.  Sparse matrix test problems , 1982, SGNM.

[20]  Cleve Ashcraft,et al.  Compressed Graphs and the Minimum Degree Algorithm , 1995, SIAM J. Sci. Comput..

[21]  Stanley C. Eisenstat,et al.  Yale sparse matrix package I: The symmetric codes , 1982 .

[22]  H. Markowitz.  The Elimination form of the Inverse and its Application to Linear Programming , 1957 .

[23]  Iain S. Duff,et al.  Users' guide for the Harwell-Boeing sparse matrix collection (Release 1) , 1992 .

[24]  Iain S. Duff,et al.  The Multifrontal Solution of Unsymmetric Sets of Linear Equations , 1984 .

[25]  A. George,et al.  A Minimal Storage Implementation of the Minimum Degree Algorithm , 1980 .

[26]  Iain S. Duff,et al.  MA27 -- A set of Fortran subroutines for solving sparse symmetric sets of linear equations , 1982 .

[27]  A. George,et al.  On the application of the minimum degree algorithm to finite element systems , 1978 .

[28]  Stanley C. Eisenstat,et al.  Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination , 1981 .