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March 2013
Parallel Computing: Volume 39 Issue 3, March, 2013

**Publisher:** Elsevier Science Publishers B. V.

As scientific computation continues to scale, it is crucial to use floating-point arithmetic processors as efficiently as possible. Lower precision allows streaming architectures to perform more operations per second and can reduce memory bandwidth pressure on all architectures. However, using a precision that is too low for a given algorithm ...

**Keywords**:
Program analysis, Floating-point, Tools, Correctness, Debugging

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July 2008
ACM Transactions on Mathematical Software (TOMS): Volume 35 Issue 1, July 2008

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 0

Downloads (6 Weeks): 0, Downloads (12 Months): 6, Downloads (Overall): 338

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Eigentest is a package that produces real test matrices with known eigensystems. A test matrix, called an eigenmat, is generated in a factored form, in which the user can specify the eigenvalues and has some control over the condition of the eigenvalues and eigenvectors. An eigenmat A of order n ...

**Keywords**:
Eigensystem, test matrix generation

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June 2005
ACM Transactions on Mathematical Software (TOMS): Volume 31 Issue 2, June 2005

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 11

Downloads (6 Weeks): 3, Downloads (12 Months): 19, Downloads (Overall): 1,044

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In many applications---latent semantic indexing, for example---it is required to obtain a reduced rank approximation to a sparse matrix A . Unfortunately, the approximations based on traditional decompositions, like the singular value and QR decompositions, are not in general sparse. Stewart [(1999), 313--323] has shown how to use a variant ...

**Keywords**:
Gram--Schmidt algorithm, MATLAB, Sparse approximations

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June 2005
SIAM Journal on Matrix Analysis and Applications: Volume 27 Issue 2, 2005

**Publisher:** Society for Industrial and Applied Mathematics

Let the $n\,{\times}\,p$ $(n\geq p)$ matrix $X$ have the QR factorization $X = QR$, where $R$ is an upper triangular matrix of order $p$ and $Q$ is orthonormal. This widely used decomposition has the drawback that $Q$ is not generally sparse even when $X$ is. One cure is to discard ...

**Keywords**:
QR factorization, Gram--Schmidt algorithm, sparse matrix, orthogonalization, rounding-error analysis

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December 2003
Numerische Mathematik: Volume 96 Issue 2, December 2003

**Publisher:** Springer-Verlag New York, Inc.

Let A be a matrix of order n . The properties of the powers A k of A have been extensively studied in the literature. This paper concerns the perturbed powers $${{ P_{{k}} = (A+E_{{k}})(A+E_{{k-1}})\cdots(A+E_{{1}}), }}$$ where the E k are perturbation matrices. We will treat three problems concerning the ...

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December 2003
ACM SIGPLAN Fortran Forum: Volume 22 Issue 3, December 2003

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 3

Downloads (6 Weeks): 0, Downloads (12 Months): 4, Downloads (Overall): 176

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In a note in the Fortran Forum , Markus describes a technique for avoiding memory leaks with derived types. In this note, we show by a simple example that this technique does not work when the object in question is a parameter in nested subprogram invocations. A fix is proposed ...

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February 2002
SIAM Journal on Matrix Analysis and Applications: Volume 24 Issue 2, 2002

**Publisher:** Society for Industrial and Applied Mathematics

In this addendum to an earlier paper by the author, it is shown how to compute a Krylov decomposition corresponding to an arbitrary Rayleigh quotient. This decomposition can be used to restart an Arnoldi process, with a selection of the Ritz vectors corresponding to that Rayleigh quotient.

**Keywords**:
deflation, Krylov sequence, restarting, Arnoldi algorithm, Krylov decomposition, large eigenproblem

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January 2002
SIAM Journal on Scientific Computing: Volume 24 Issue 1, 2002

**Publisher:** Society for Industrial and Applied Mathematics

In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit, after which the current vectors are reorthogonalized. A theorem of Simon [ Linear Algebra Appl ., 61 (1984), pp. 101--132] shows that the Rayleigh quotient---i.e., the ...

**Keywords**:
large eigenproblem, partial reorthogonalization, symmetric matrix, Lanczos method, adjusted Rayleigh quotient

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September 2001

This book is the second volume in a projected five-volume survey of numerical linear algebra and matrix algorithms. This volume treats the numerical solution of dense and large-scale eigenvalue problems with an emphasis on algorithms and the theoretical background required to understand them. Stressing depth over breadth, Professor Stewart treats ...

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March 2001
SIAM Journal on Matrix Analysis and Applications: Volume 23 Issue 3, 2001

**Publisher:** Society for Industrial and Applied Mathematics

Sorensen's implicitly restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve the structure of the Arnoldi decomposition on which the algorithm is based restricts the range of transformations that can be performed on ...

**Keywords**:
large eigenproblem, deflation, Krylov sequence, restarting, Arnoldi algorithm, Krylov decomposition

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January 2000
Computing in Science and Engineering: Volume 2 Issue 1, January 2000

**Publisher:** IEEE Educational Activities Department

A matrix decomposition is a factorization of a matrix into the product of simpler matrices. The introduction of matrix decomposition into numerical linear algebra in the years from 1945 to 1965 revolutionized matrix computations. This article outlines the decompositional approach, comments on its history, and surveys the six most widely ...

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October 1999
SIAM Journal on Matrix Analysis and Applications: Volume 21 Issue 2, Oct.-Jan. 2000

**Publisher:** Society for Industrial and Applied Mathematics

In this paper we present three theorems which give insight into the regularizing properties of MINRES. While our theory does not completely characterize the regularizing behavior of the algorithm, it provides a partial explanation of the observed behavior of the method. Unlike traditional attempts to explain the regularizing properties of ...

**Keywords**:
MINRES, iterative regularization, regularization, ill-posed problem

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February 1999
SIAM Journal on Scientific Computing: Volume 20 Issue 4, July 1999

**Publisher:** Society for Industrial and Applied Mathematics

In this paper we introduce a new decomposition called the pivoted QLP decomposition. It is computed by applying pivoted orthogonal triangularization to the columns of the matrix X in question to get an upper triangular factor R and then applying the same procedure to the rows of R to get ...

**Keywords**:
QLP decomposition, pivoted QR decomposition, rank determination, singular value decomposition

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October 1998
SIAM Journal on Matrix Analysis and Applications: Volume 19 Issue 4, Oct. 1998

**Publisher:** Society for Industrial and Applied Mathematics

This paper treats the problem of triangularizing a matrix by hyperbolic Householder transformations. The stability of this method, which finds application in block updating and fast algorithms for Toeplitz-like matrices, has been analyzed only in special cases. Here we give a general analysis which shows that two distinct implementations of ...

**Keywords**:
pivoting, triangularization, relational stability, hyperbolic transformation

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December 1997
ACM Transactions on Mathematical Software (TOMS): Volume 23 Issue 4, Dec. 1997

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 2

Downloads (6 Weeks): 1, Downloads (12 Months): 10, Downloads (Overall): 704

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SRRT is a Fortran program to calculate an approximate orthonomral basis fr a dominant invariant subspace of a real matrix A by the method of simultaneous iteration. Specifically, given an integer m , SRRIT computes a matrix Q with m orthonormal columns and real quasi-triangular matrix T or order m ...

**Keywords**:
invariant subspace, nonsymmetric eigenvalue problem, project method

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18

October 1997
Proceedings of the second international workshop on Recent advances in total least squares techniques and errors-in-variables modeling

**Publisher:** Society for Industrial and Applied Mathematics

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July 1997
SIAM Journal on Matrix Analysis and Applications: Volume 18 Issue 3, July 1997

**Publisher:** Society for Industrial and Applied Mathematics

This paper gives perturbation analyses for $Q_1$ and $R$ in the QR factorization $A=Q_1R$, $Q_1^TQ_1=I$ for a given real $m\times n$ matrix $A$ of rank $n$ and general perturbations in $A$ which are sufficiently small in norm. The analyses more accurately reflect the sensitivity of the problem than previous such ...

**Keywords**:
QR factorization, condition estimation, matrix equations, perturbation analysis, pivoting

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