Computational methods of linear algebra

Faddeev, D. K.; Faddeeva, V. N.

doi:10.1007/bf01086544

Cited by 91 publications

(60 citation statements)

References 843 publications

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“…Gradient Descent: For a rank-one problem, a steepest gradient algorithm has been shown to converge globally to an eigenvalue [19]. This eigenvalue is the optimum one if the starting point is not orthogonal to the optimum eigenvector.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of Gradient Descent for Low-Rank Matrix Approximation

Pitaval

Dai

Tirkkonen

2015

IEEE Trans. Inform. Theory

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Abstract-This paper provides a proof of global convergence of gradient search for low-rank matrix approximation. Such approximations have recently been of interest for large scale problems, as well as for dictionary learning for sparse signal representations and matrix completion. The proof is based on the interpretation of the problem as an optimization on the Grassmann manifold and Fubiny-Study distance on this space.

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Section: Introductionmentioning

confidence: 99%

“…Congugate Gradient: The global convergence result of [19] is extended to a generalized RQ conjugate-gradient algorithm for rank one in [21]. Local convergence properties of conjugate-gradient algorithms are discussed in [22], [23], showing faster convergence of conjugate-gradient than steepest descent.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Gradient Descent for Low-Rank Matrix Approximation

Pitaval

Dai

Tirkkonen

2015

IEEE Trans. Inform. Theory

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“…From our trials, it is obvious that in all three cases the rate of convergence of our new algorithm is better or at least as fast as the power [4]. The QR [13] algorithm converges very slowly in the last two cases, when the separation between the eigenvalues is poor.…”

Section: Corollary 3: Any Oepomo's Alternating Sequence Iteration Conmentioning

confidence: 85%

Survey of power, QR, and iterative methods for solution of largest eigenvalue of essentially positive matrices

Oepomo¹

2007

imf

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Abstract. Many of the popular methods for the solution of largest eigenvalue of essentially positive irreducible matrices are surveyed with the hope of finding an efficient method suitable for electromagnetic engineering, radiation problems, system identification problems, and solid mechanics. Eigenvalue computations are both fundamental and ubiquitous in computational science and its fast application areas. Some comparisons between several known algorithms, i.e. Power and QR methods, and earlier theory of Oepomo iterative techniques for solving largest eigenvalue of nonnegative irreducible matrices are presented since there is a continuing demand for new algorithm and library software that efficiently utilize and adapt to new applications.

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“…Following the arguments of Faddeev and Faddeeva [7,8], this Relaxed Monte Carlo method will converge if…”

Section: Relaxed Monte Carlo Methodsmentioning

confidence: 99%

Relaxed Monte Carlo Linear Solver

Tan

Alexandrov

2001

Computational Science — ICCS 2001

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Abstract. The problem of solving systems of linear algebraic equations by parallel Monte Carlo numerical methods is considered. A parallel Monte Carlo method with relaxation is presented. This is a report of a research in progress, showing the effectiveness of this algorithm. Theoretical justification of this algorithm and numerical experiments are presented. The algorithms were implemented on a cluster of workstations using MPI.

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Computational methods of linear algebra

Cited by 91 publications

References 843 publications

Convergence of Gradient Descent for Low-Rank Matrix Approximation

Convergence of Gradient Descent for Low-Rank Matrix Approximation

Survey of power, QR, and iterative methods for solution of largest eigenvalue of essentially positive matrices

Relaxed Monte Carlo Linear Solver

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