Applied Numerical Linear Algebra

Demmel, James

doi:10.1137/1.9781611971446

Cited by 1,990 publications

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“…It utilizes this subspace to approximate the eigenvectors and the corresponding eigenvalues of A. In order to find all the eigenvalues, A is generally reduced to a n × n tridiagonal matrix T [5] because there are efficient algorithms available for finding the eigenvalues of T. The main intuition behind the Lanczos method is that it generates a partial tridiagonal matrix T k (see Algorithm 1) from A where the extremal eigenvalues of T k are the optimal approximations of the extremal eigenvalues of A [8] from the subspace K k .…”

Section: The Lanczos Methodsmentioning

confidence: 99%

“…In the particular case of the symmetric extremal eigenvalue problem, we are only interested in finding either λ max , λ min or both eigenvalues. There are two main families of methods for solving the extremal eigenvalue problem: direct methods and iterative methods [8]. Direct methods compute eigenvalues in one shot, however, they incur a computation cost of Θ(n 3 ) and are more applicable when all the eigenvalues and the corresponding eigenvectors are required.…”

Section: Symmetric Extremal Eigenvalue Problemmentioning

confidence: 99%

“…The symmetric extremal eigenvalue problem is an important scientific computation involving dense linear algebra where one is interested in finding only the extremal eigenvalues of a n × n symmetric matrix. Unlike general eigenvalues computation which has a computational complexity of Θ(n 3 ), the extremal eigenvalues can be computed with much less computational complexity (Θ(n 2 )) [8]. Solving multiple independent extremal eigenvalue problems is common in semidefinite programming (SDP) solvers [6] where one has to find the extremal eigenvalues of more than one symmetric matrix in each iteration and also in real-time eigenvalue based channel sensing for cognitive radios (IEEE 802.22) [7] where one has to sense multiple channels simultaneously.…”

Section: Introductionmentioning

confidence: 99%

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A High Throughput FPGA-Based Implementation of the Lanczos Method for the Symmetric Extremal Eigenvalue Problem

Rafique

Kapre

Constantinides

2012

Lecture Notes in Computer Science

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Abstract. Iterative numerical algorithms with high memory bandwidth requirements but medium-size data sets (matrix size ∼ a few 100s) are highly appropriate for FPGA acceleration. This paper presents a streaming architecture comprising floating-point operators coupled with highbandwidth on-chip memories for the Lanczos method, an iterative algorithm for symmetric eigenvalues computation. We show the Lanczos method can be specialized only for extremal eigenvalues computation and present an architecture which can achieve a sustained single precision floating-point performance of 175 GFLOPs on Virtex6-SX475T for a dense matrix of size 335×335. We perform a quantitative comparison with the parallel implementations of the Lanczos method using optimized Intel MKL and CUBLAS libraries for multi-core and GPU respectively. We find that for a range of matrices the FPGA implementation outperforms both multi-core and GPU; a speed up of 8.2-27.3× (13.4× geo. mean) over an Intel Xeon X5650 and 26.2-116× (52.8× geo. mean) over an Nvidia C2050 when FPGA is solving a single eigenvalue problem whereas a speed up of 41-520× (103× geo.mean) and 131-2220× (408× geo.mean) respectively when it is solving multiple eigenvalue problems.

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Section: The Lanczos Methodsmentioning

confidence: 99%

Section: Symmetric Extremal Eigenvalue Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A High Throughput FPGA-Based Implementation of the Lanczos Method for the Symmetric Extremal Eigenvalue Problem

Rafique

Kapre

Constantinides

2012

Lecture Notes in Computer Science

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“…While different implementation may render different numerical behavior, mathematically 1 the kth approximate solution x k by CG is the optimal one in the sense that the kth approximation error A −1 b − x k satisfies [11,Theorem 6:1] (1.1) Here the superscript "· * " takes conjugate transpose. In practice, x k is computed recursively from x k−1 via short term recurrences [4,7,10,19]. But exactly how it is computed, though extremely crucial in practice, is not important to our analysis here in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, we have the following well known and frequently referenced error bound (see, e.g., [4,10,19,22]):…”

Section: Introductionmentioning

confidence: 99%

On Meinardus' examples for the conjugate gradient method

2008

Math. Comp.

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Abstract. The conjugate gradient (CG) method is widely used to solve a positive definite linear system Ax = b of order N . It is well known that the relative residual of the kth approximate solution by CG (with the initial approximation x 0 = 0) is bounded above by In 1963, Meinardus (Numer. Math., 5 (1963, pp. 14-23) gave an example to achieve this bound for k = N − 1 but without saying anything about all other 1 ≤ k < N − 1. This very example can be used to show that the bound is sharp for any given k by constructing examples to attain the bound, but such examples depend on k and for them the (k + 1)th residual is exactly zero. Therefore it would be interesting to know if there is any example on which the CG relative residuals are comparable to the bound for all 1 ≤ k ≤ N − 1. There are two contributions in this paper:(1) A closed formula for the CG residuals for all 1 ≤ k ≤ N −1 on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of √ 2 of the actual residuals; (2) A complete characterization of extreme positive linear systems for which the kth CG residual achieves the bound is also presented.

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Parallel multigrid solvers for 3D unstructured finite element problems in large deformation elasticity and plasticity

Adams

2000

Int. J. Numer. Meth. Engng.

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Multigrid is a popular solution method for the set of linear algebraic equations that arise from PDEs discretized with the finite element method. The application of multigrid to unstructured grid problems, however, is not well developed. We discuss a method, that uses many of the same techniques as the finite element method itself, to apply standard multigrid algorithms to unstructured finite element problems. We use maximal independent sets (MISs) as a mechanism to automatically coarsen unstructured grids; the inherent flexibility in the selection of an MIS allows for the use of heuristics to improve their effectiveness for a multigrid solver. We present parallel algorithms, based on geometric heuristics, to optimize the quality of MISs and the meshes constructed from them, for use in multigrid solvers for 3D unstructured problems. We discuss parallel issues of our algorithms, multigrid solvers in general, and the parallel finite element application that we have developed to test our solver on challenging problems. We show that our solver, and parallel finite element architecture, does indeed scale well, with test problems in 3D large deformation elasticity and plasticity, with 40 million degree of freedom problem on 240 IBM four‐way SMP PowerPC nodes. Copyright © 2000 John Wiley & Sons, Ltd.

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Applied Numerical Linear Algebra

Cited by 1,990 publications

References 0 publications

A High Throughput FPGA-Based Implementation of the Lanczos Method for the Symmetric Extremal Eigenvalue Problem

A High Throughput FPGA-Based Implementation of the Lanczos Method for the Symmetric Extremal Eigenvalue Problem

On Meinardus' examples for the conjugate gradient method

Parallel multigrid solvers for 3D unstructured finite element problems in large deformation elasticity and plasticity

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