The Subspace Projected Approximate Matrix (SPAM) Modification of the Davidson Method

Shepard, Ron; Wagner, Albert F.; Tilson, Jeffrey L.; Minkoff, Michael

doi:10.1006/jcph.2001.6828

Cited by 12 publications

(52 citation statements)

References 24 publications

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“…It emerges that the notion of an approximation associated with a space leads to a connection with the Subspace Projected Approximate Matrix method [17], or SPAM for short. In this subspace iterative method for eigenvalue problems, a sequence (A k ) k≥0 is defined based on an initial approximation A 0 of A.…”

Section: Selecting An Approximation Associated With a Subspacementioning

confidence: 99%

A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Brandts

Silva

2013

E Asian J Appl Math

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Given two n × n matrices A and A0 and a sequence of subspaces with dim the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk(A0 − A)Πk, where Πk is the orthogonal projection on . Consequently, Akν = Aν and ν*Ak = ν*A for all Thus is a sequence of matrices that gradually changes from A0 into An = A. In principle, the definition of may depend on properties of Ak, which can be exploited to try to force Ak+1 to be closer to A in some specific sense. By choosing A0 as a simple approximation of A, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving A. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations Ax = b. In particular, we seek conditions under which the solutions xk of the approximate systems Akxk = b are computable at low computational cost, so the efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence (xk)k≥0 approximates x, by performing some illustrative numerical tests.

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Section: Selecting An Approximation Associated With a Subspacementioning

confidence: 99%

A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Brandts

Silva

2013

E Asian J Appl Math

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“…During the development of the Subspace Projected Approximate Matrix (SPAM) diagonalization method [1], it was necessary to compute bounds of approximate eigenvalues and eigenvectors. This work resulted in the development of a general computational procedure to compute rigorous eigenvalue bounds for general subspace eigenvalue methods.…”

Section: Eigenvalue Boundsmentioning

confidence: 99%

Advanced software for the calculation of thermochemistry, kinetics, and dynamics

Medvedev

Gray

Wagner

et al. 2005

J. Phys.: Conf. Ser.

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Abstract. Recent results are presented for the efficient computation of matrix eigenvalue and eigenvector equations, the fitting of discrete sets of energy points using interpolating moving least squares methods, and time-dependent and time-independent calculations of molecular dynamics and cumulative reaction probabilities. IntroductionThe Born-Oppenheimer separation of the Schrodinger equation for molecules allows the electronic and nuclear motions to be solved in three steps. 1) The solution of the electronic wave function at a discrete set of molecular conformations; 2) the fitting of this discrete set of energy values in order to construct an analytical approximation to the potential energy surface (PES) at all molecular conformations; 3) the use of this analytical PES to solve for the nuclear motion using either timedependent or time-independent formulations to compute molecular energy values, chemical reaction rates, and cumulative reaction probabilities. Our SciDAC project involves the development of technology to address all three of these steps.

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“…The various subspace methods differ in how the individual basis vectors x j are generated, in how the basis vectors are contracted in order to satisfy various resource limitation constraints, and in how preconditioners are used in order to accelerate the convergence of the iterative procedures for the particular eigenpairs of interest. The bounds relations examined in this manuscript apply to all of these various hermitian subspace methods (including the Lanczos [3], Davidson [4], SPAM [5], Generalized Davidson Inverse Iteration [6], JacobiDavidson [7], and Generalized Jacobi-Davidson [8] methods).…”

Section: Introductionmentioning

confidence: 98%