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FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for ..."
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Cited by 253 (6 self)
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Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool
Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions
, 2009
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing l ..."
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Cited by 62 (4 self)
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Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing
A Multilevel Direct Solver for QuasiPlanar Scatterers Based on the NonUniform Grid Approach
"... We introduce a multilevel direct solver for scattering from quasiplanar objects. The solver relies on the compression of the interactions between distinct domains. The compression is performed in three steps: field compression using the nonuniform grid approach, current compression using the rank ..."
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revealing QR decomposition separating local from interacting currents, and local problem solution based on Schur’s complement. The interacting currents are repeatedly aggregated with the neighboring sections in a multilevel process. The resulting compressed system of equations is solved directly
A Data Locality Optimizing Algorithm
, 1991
"... This paper proposes an algorithm that improves the locality of a loop nest by transforming the code via interchange, reversal, skewing and tiling. The loop transformation algorithm is based on two concepts: a mathematical formulation of reuse and locality, and a loop transformation theory that unifi ..."
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Cited by 804 (16 self)
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that unifies the various transforms as unimodular matrix transformations. The algorithm has been implemented in the SUIF (Stanford University Intermediate Format) compiler, and is successful in optimizing codes such as matrix multiplication, successive overrelaxation (SOR), LU decomposition without pivoting
Symmetric RankRevealing Decompositions
"... We develop algorithms for computing rankrevealing decompositions of symmetric matrices that exploit and inherit the symmetry, and demonstrate which particular decompositions and algorithms are useful for dense and sparse semidefinite and indefinite matrices. The most useful algorithms are based o ..."
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We develop algorithms for computing rankrevealing decompositions of symmetric matrices that exploit and inherit the symmetry, and demonstrate which particular decompositions and algorithms are useful for dense and sparse semidefinite and indefinite matrices. The most useful algorithms are based
DirectionofArrival Estimation Using the RankRevealing URV Decomposition
 In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE
, 1991
"... A algorithm for updating the null space of a matrix is described. The algorithm is based on a new decomposition, called the URV decomposition, which can be updated in O(N 2 ) and serves as an intermediary between the QR decomposition and the singular value decomposition. The URV decomposition is a ..."
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Cited by 7 (3 self)
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A algorithm for updating the null space of a matrix is described. The algorithm is based on a new decomposition, called the URV decomposition, which can be updated in O(N 2 ) and serves as an intermediary between the QR decomposition and the singular value decomposition. The URV decomposition
Computing RankRevealing QR Factorizations of Dense Matrices
 Argonne Preprint ANLMCSP5590196, Argonne National Laboratory
, 1996
"... this paper, and we give only a brief synopsis here. For details, the reader is referred to the code. Test matrices 1 through 5 were designed to exercise column pivoting. Matrix 6 was designed to test the behavior of the condition estimation in the presence of clusters for the smallest singular value ..."
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Cited by 39 (2 self)
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this paper, and we give only a brief synopsis here. For details, the reader is referred to the code. Test matrices 1 through 5 were designed to exercise column pivoting. Matrix 6 was designed to test the behavior of the condition estimation in the presence of clusters for the smallest singular value. For the other cases, we employed the LAPACK matrix generator xLATMS, which generates random symmetric matrices by multiplying a diagonal matrix with prescribed singular values by random orthogonal matrices from the left and right. For the break1 distribution, all singular values are 1.0 except for one. In the arithmetic and geometric distributions, they decay from 1.0 to a specified smallest singular value in an arithmetic and geometric fashion, respectively. In the "reversed" distributions, the order of the diagonal entries was reversed. For test cases 7 though 12, we used xLATMS to generate a matrix of order
Multifrontal multithreaded rankrevealing sparse QR factorization
"... SuiteSparseQR is a sparse QR factorization package based on the multifrontal method. Within each frontal matrix, LAPACK and the multithreaded BLAS enable the method to obtain high performance on multicore architectures. Parallelism across different frontal matrices is handled with Intel’s Threading ..."
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Cited by 15 (2 self)
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structures are found without requiring the formation of the pattern of A T A. Rankdetection is performed within each frontal matrix using Heath’s method, which does not require column pivoting. The resulting sparse QR factorization obtains a substantial fraction of the theoretical peak performance of a
ON RANKREVEALING FACTORISATIONS*
"... Abstract. The problem of finding a rankrevealing QR (RRQR) factorisation of a matrix A consists of permuting the columns of A such that the resulting QR factorisation contains an upper triangular matrix whose linearly dependent columns are separated from the linearly independent ones. In this paper ..."
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Abstract. The problem of finding a rankrevealing QR (RRQR) factorisation of a matrix A consists of permuting the columns of A such that the resulting QR factorisation contains an upper triangular matrix whose linearly dependent columns are separated from the linearly independent ones
Updating a RankRevealing ULV Decomposition
, 1991
"... A ULV decomposition of a matrix A of order n is a decomposition of the form A = ULV^H, where U and V are orthogonal matrices and L is a lower triangular matrix. When A is approximately of rank k, the decomposition is rank revealing if the last n \Gamma k rows of L are small. This paper presents al ..."
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Cited by 56 (4 self)
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algorithms for updating a rankrevealing ULV decomposition. The algorithms run in O(n²) time, and can be implemented on a linear array of processors to run in O(n) time.
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