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Applied Numerical Linear Algebra
 Society for Industrial and Applied Mathematics
, 1997
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate ..."
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Cited by 532 (26 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
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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Cited by 20 (0 self)
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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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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...