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Modifying a Sparse Cholesky Factorization
, 1997
"... Given a sparse symmetric positive definite matrix AA T and an associated sparse Cholesky factorization LL T , we develop sparse techniques for obtaining the new factorization associated with either adding a column to A or deleting a column from A. Our techniques are based on an analysis and mani ..."
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Cited by 42 (14 self)
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Given a sparse symmetric positive definite matrix AA T and an associated sparse Cholesky factorization LL T , we develop sparse techniques for obtaining the new factorization associated with either adding a column to A or deleting a column from A. Our techniques are based on an analysis and manipulation of the underlying graph structure and on ideas of Gill, Golub, Murray, and Saunders for modifying a dense Cholesky factorization. Our algorithm involves a new sparse matrix concept, the multiplicity of an entry in L. The multiplicity is essentially a measure of the number of times an entry is modified during symbolic factorization. We show that our methods extend to the general case where an arbitrary sparse symmetric positive definite matrix is modified. Our methods are optimal in the sense that they take time proportional to the number of nonzero entries in L that change. This work was supported by National Science Foundation grants DMS9404431 and DMS9504974. y davis@cise.uf...
Multiplerank modifications of a sparse Cholesky factorization
 SIAM J. Matrix Anal. Appl
, 2001
"... Abstract. Given a sparse symmetric positive definite matrix AA T and an associated sparse Cholesky factorization LDL T or LL T, we develop sparse techniques for updating the factorization after either adding a collection of columns to A or deleting a collection of columns from A. Our techniques are ..."
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Cited by 11 (7 self)
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Abstract. Given a sparse symmetric positive definite matrix AA T and an associated sparse Cholesky factorization LDL T or LL T, we develop sparse techniques for updating the factorization after either adding a collection of columns to A or deleting a collection of columns from A. Our techniques are based on an analysis and manipulation of the underlying graph structure, using the framework developed in an earlier paper on rank1 modifications [T. A. Davis and W. W. Hager, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606–627]. Computationally, the multiplerank update has better memory traffic and executes much faster than an equivalent series of rank1 updates since the multiplerank update makes one pass through L computing the new entries, while a series of rank1 updates requires multiple passes through L.
Multifrontal Computation with the Orthogonal Factors of Sparse Matrices
 SIAM Journal on Matrix Analysis and Applications
, 1994
"... . This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the righthand side vector b to Q T b. A multifrontalbased method for computing Q T b using Householder factorization is presented ..."
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Cited by 9 (0 self)
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. This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the righthand side vector b to Q T b. A multifrontalbased method for computing Q T b using Householder factorization is presented. A theoretical operation count for the K by K unbordered grid model problem and problems defined on graphs with p nseparators shows that the proposed method requires O(NR ) storage and multiplications to compute Q T b, where NR = O(n log n) is the number of nonzeros of the upper triangular factor R of A. In order to introduce BLAS2 operations, Schreiber and Van Loan's StorageEfficientWY Representation [SIAM J. Sci. Stat. Computing, 10(1989),pp. 5557] is applied for the orthogonal factor Q i of each frontal matrix F i . If this technique is used, the bound on storage increases to O(n(logn) 2 ). Some numerical results for the grid model problems as well as HarwellBoeing problems...
PARALLEL SPARSE: Data Structure and Organization
, 1990
"... PARALLEL SPARSE is an algorithm for the direct solution of general sparse linear systems using Gauss elimination. It is designed for distributed memory machines and has been implemented on the NCUBE7, a hypercube machine with 128 processors. The algorithm is intended to be particularly efficient fo ..."
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PARALLEL SPARSE is an algorithm for the direct solution of general sparse linear systems using Gauss elimination. It is designed for distributed memory machines and has been implemented on the NCUBE7, a hypercube machine with 128 processors. The algorithm is intended to be particularly efficient for linear systems arising from solving partial differential equations using domain decomposition with a nested dissection ordering. PARALLEL SPARSE is part of the Parallel ELLPACK system. This report assumes the reader is familiar with the general approach of parallel sparse and it provides detailed information on three aspects of PARALLEL SPARSE: 1. The data structures used to represent the mamx, the modifications in eliminating unknowns and the dependencies between processors. 2. The data structures that relate the assignment of actual hypercube processors to computational processes. 3. The organization of the codes that run on the hypercube host and on the hypercube nodes: Dynamic data structures are used unlike most other sparse mamx codes. These are more complex but provide better flexibility to handle PDE problems. Much of the complexity seen here compared to traditional other codes is due to the fact that we handle general matrices instead of only symrnemc ones.