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LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
 ACM Trans. Math. Software
, 1982
"... An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerica ..."
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Cited by 649 (21 self)
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An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable
Lanczos Bidiagonalization With Partial Reorthogonalization
, 1998
"... A partial reorthogonalization procedure (BPRO) for maintaining semiorthogonality among the left and right Lanczos vectors in the Lanczos bidiagonalization (LBD) is presented. The resulting algorithm is mathematically equivalent to the symmetric Lanczos algorithm with partial reorthogonalization (PR ..."
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Cited by 82 (0 self)
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A partial reorthogonalization procedure (BPRO) for maintaining semiorthogonality among the left and right Lanczos vectors in the Lanczos bidiagonalization (LBD) is presented. The resulting algorithm is mathematically equivalent to the symmetric Lanczos algorithm with partial reorthogonalization
Accurate Singular Values of Bidiagonal Matrices
 SIAM J. SCI. STAT. COMPUT
, 1990
"... Computing the singular values of a bidiagonal matrix is the fin al phase of the standard algow rithm for the singular value decomposition of a general matrix. We present a new algorithm hich computes all the singular values of a bidiagonal matrix to high relative accuracy independent of their magni ..."
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Cited by 127 (18 self)
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Computing the singular values of a bidiagonal matrix is the fin al phase of the standard algow rithm for the singular value decomposition of a general matrix. We present a new algorithm hich computes all the singular values of a bidiagonal matrix to high relative accuracy independent
An O(n²) algorithm for the bidiagonal SVD
, 2000
"... The RRR algorithm allows to compute the eigendecomposition of a symmetric tridiagonal matrix T with an O(n²) complexity. This article discusses how this method can be adapted to the bidiagonal SVD B = U \SigmaV T . It turns out that using the RRR algorithm as a black box to compute B T B = V \S ..."
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Cited by 3 (0 self)
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The RRR algorithm allows to compute the eigendecomposition of a symmetric tridiagonal matrix T with an O(n²) complexity. This article discusses how this method can be adapted to the bidiagonal SVD B = U \SigmaV T . It turns out that using the RRR algorithm as a black box to compute B T B = V
Restarted Lanczos Bidiagonalization for the SVD in SLEPc
"... Eigenvalue Problem Computations. They are intended to complement the Users Guide by providing technical details that normal users typically do not need to know but may be of interest for more advanced users. 1 ..."
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Eigenvalue Problem Computations. They are intended to complement the Users Guide by providing technical details that normal users typically do not need to know but may be of interest for more advanced users. 1
A ROBUST AND EFFICIENT PARALLEL SVD SOLVER BASED ON RESTARTED LANCZOS BIDIAGONALIZATION ∗
"... Abstract. Lanczos bidiagonalization is a competitive method for computing a partial singular value decomposition of a large sparse matrix, that is, when only a subset of the singular values and corresponding singular vectors are required. However, a straightforward implementation of the algorithm ha ..."
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Cited by 5 (0 self)
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is to be implemented on a distributedmemory parallel computer, then additional precautions are required so that parallel efficiency is maintained as the number of processors increases. In this paper, we present a Lanczos bidiagonalization procedure implemented in SLEPc, a software library for the solution of large
Augmented implicitly restarted Lanczos bidiagonalization methods
 SIAM J. Sci. Comput
"... Abstract. New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The ..."
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Cited by 30 (9 self)
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Abstract. New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method
Parallel Computation Of Spectral Portraits Of Matrices By Bidiagonalization
, 1995
"... : We describe parallel programs for computation of spectral portraits of matrices on Paragon and Connection Machine 5. The method used consists of bidiagonal reduction of a complex square matrix by unitary Householder transformations and computation of the minimal singular value of the resulting rea ..."
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real bidiagonal matrix by the bisection procedure employing Sturm sequences. The computation of bidiagonal reduction uses the blockcyclic distribution of matrices on a rectangular processor grid in order to get good load balancing. Since the computation of spectral portraits needs to calculate
A BIDIAGONALIZATIONREGULARIZATION PROCEDURE FOR LARGE SCALE DISCRETIZATIONS OF ILLPOSED PROBLEMS*
"... Abstract. In this paper, we consider illposed problems which discretize to linear least squares problems with matrices K of high dimensions. The algorithm proposed uses K only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project th ..."
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the problem onto a subspace for which K is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given. Key words, ill
An Implicit Shift Bidiagonalization Algorithm For IllPosed Systems
 BIT
, 1994
"... . Iterative methods based on Lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large scale discrete illposed linear least squares problems of the form min x kAx \Gamma bk 2 . Methods for regularization in the Krylov subspaces are discussed which use generali ..."
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Cited by 19 (0 self)
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. Iterative methods based on Lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large scale discrete illposed linear least squares problems of the form min x kAx \Gamma bk 2 . Methods for regularization in the Krylov subspaces are discussed which use
Results 1  10
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