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LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

by Christopher C. Paige, Michael A. Saunders - 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 ..."
Abstract - Cited by 653 (21 self) - Add to MetaCart
-gradient algorithms, indicating that I~QR is the most reliable algorithm when A is ill-conditioned. Categories and Subject Descriptors: G.1.2 [Numerical Analysis]: ApprorJmation--least squares approximation; G.1.3 [Numerical Analysis]: Numerical Linear Algebra--linear systems (direct and

Performance of various computers using standard linear equations software

by Jack J. Dongarra , 2009
"... This report compares the performance of different computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from the Earth Simulator to personal computers. ..."
Abstract - Cited by 412 (21 self) - Add to MetaCart
This report compares the performance of different computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from the Earth Simulator to personal computers.

For Most Large Underdetermined Systems of Linear Equations the Minimal ℓ1-norm Solution is also the Sparsest Solution

by David L. Donoho - Comm. Pure Appl. Math , 2004
"... We consider linear equations y = Φα where y is a given vector in R n, Φ is a given n by m matrix with n < m ≤ An, and we wish to solve for α ∈ R m. We suppose that the columns of Φ are normalized to unit ℓ 2 norm 1 and we place uniform measure on such Φ. We prove the existence of ρ = ρ(A) so that ..."
Abstract - Cited by 568 (10 self) - Add to MetaCart
We consider linear equations y = Φα where y is a given vector in R n, Φ is a given n by m matrix with n < m ≤ An, and we wish to solve for α ∈ R m. We suppose that the columns of Φ are normalized to unit ℓ 2 norm 1 and we place uniform measure on such Φ. We prove the existence of ρ = ρ(A) so

Variable iterative methods for nonsymmetric systems of linear equations

by Stanley C. Eisenstat, Howard C. Elmant, Martin H. Schultz - SIAM J. Numer. Anal , 1983
"... Abstract. We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do not ..."
Abstract - Cited by 241 (5 self) - Add to MetaCart
Abstract. We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do

Linear Equations

by Wilfried Schmid, H. Wu , 2008
"... now appears in Section V, Sub-section A, of the Task Group’s report on Conceptual ..."
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now appears in Section V, Sub-section A, of the Task Group’s report on Conceptual

Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit

by David L. Donoho, Yaakov Tsaig, Iddo Drori, Jean-luc Starck , 2006
"... Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NP-hard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our pr ..."
Abstract - Cited by 274 (22 self) - Add to MetaCart
Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NP-hard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our

Decoding by Linear Programming

by Emmanuel J. Candès, Terence Tao , 2004
"... This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to rec ..."
Abstract - Cited by 1399 (16 self) - Add to MetaCart
fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced

Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization,”

by Benjamin Recht , Maryam Fazel , Pablo A Parrilo - SIAM Review, , 2010
"... Abstract The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and col ..."
Abstract - Cited by 562 (20 self) - Add to MetaCart
for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds

Solution of systems of linear equations by minimized iterations

by Cornelius Lanczos - J. Res. Natl. Bur. Stand , 1952
"... A simple algorithm is described which is well adapted to the effective solution of large systems of linear algebraic equations by a succession of well-convergent approximations. 1. ..."
Abstract - Cited by 214 (0 self) - Add to MetaCart
A simple algorithm is described which is well adapted to the effective solution of large systems of linear algebraic equations by a succession of well-convergent approximations. 1.

Parallel Numerical Linear Algebra

by James W. Demmel, Michael T. Heath , Henk A. van der Vorst , 1993
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illust ..."
Abstract - Cited by 773 (23 self) - Add to MetaCart
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
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