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Sparse Multifrontal Rank Revealing QR Factorization
 SIAM J. Matrix Anal. Appl
, 1995
"... We describe an algorithm to compute a rank revealing sparse QR factorization. We augment a basic sparse multifrontal QR factorization with an incremental condition estimator to provide an estimate of the least singular value and vector for each successive column of R. We remove a column from R as ..."
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We describe an algorithm to compute a rank revealing sparse QR factorization. We augment a basic sparse multifrontal QR factorization with an incremental condition estimator to provide an estimate of the least singular value and vector for each successive column of R. We remove a column from R as soon as the condition estimate exceeds a tolerance, using the approximate singular vector to select a suitable column. Removing columns, or pivoting, requires a dynamic data structure and necessarily degrades sparsity. But most of the additional work fits naturally into the multifrontal factorization's use of efficient dense vector kernels, minimizing overall cost. Further, pivoting as soon as possible reduces the cost of pivot selection and data access. We present a theoretical analysis that shows that our use of approximate singular vectors does not degrade the quality of our rankrevealing factorization; we achieve an exponential bound like methods that use exact singular vectors. We prov...
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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. 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...
Dealing with Dense Rows in the Solution of Sparse Linear Least Squares Problems
, 1995
"... Sparse linear least squares problems containing a few relatively dense rows occur frequently in practice. Straightforward solution of these problems could cause catastrophic fill and delivers extremely poor performance. This paper studies a scheme for solving such problems efficiently by handling de ..."
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Sparse linear least squares problems containing a few relatively dense rows occur frequently in practice. Straightforward solution of these problems could cause catastrophic fill and delivers extremely poor performance. This paper studies a scheme for solving such problems efficiently by handling dense rows and sparse rows separately. How a sparse matrix is partitioned into dense rows and sparse rows determines the efficiency of the overall solution process. A new algorithm is proposed to find a partition of a sparse matrix which leads to satisfactory or even optimal performance. Extensive numerical experiments are performed to demonstrate the effectiveness of the proposed scheme. A MATLAB implementation is included. 1 This work was supported in part by the Cornell Theory Center which receives funding from members of its Corporate Research Institute, the National Science Foundation (NSF), the Advanced Research Projects Agency (ARPA), the National Institutes of Health (NIH), New York S...
Parallel Solution Of Sparse Linear Least Squares Problems On DistributedMemory Multiprocessors
"... This paper studies the solution of largescale sparse linear least squares problems on distributedmemory multiprocessors. The method of corrected seminormal equations is considered. New blockoriented parallel algorithms are developed for solving the related sparse triangular systems. The arithmeti ..."
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This paper studies the solution of largescale sparse linear least squares problems on distributedmemory multiprocessors. The method of corrected seminormal equations is considered. New blockoriented parallel algorithms are developed for solving the related sparse triangular systems. The arithmetic and communication complexities of the new algorithms applied to regular grid problems are analyzed. The proposed parallel sparse triangular solution algorithms together with a blockoriented parallel sparse QR factorization algorithm result in a highly efficient blockoriented approach to the parallel solution of sparse linear least squares problems on distributedmemory multiprocessors. Performance of the blockoriented approach is demonstrated empirically through an implementation on an IBM Scalable POWERparallel system SP2. The largest problem solved has over two million rows and more than a quarter million columns. The execution speed for the numerical factorization of this problem ach...