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

Medvedev, Dmitry M.; Gray, Stephen K.; Wagner, Albert F.; Minkoff, Michael; Shepard, Ron

doi:10.1088/1742-6596/16/1/033

Cited by 3 publications

(5 citation statements)

References 6 publications

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“…Note that (20) follows more easily from the alternative definition given in Remark 2.2. However, the proof of Proposition 2.4 also shows the validity of (16).…”

Section: General Description Of Spammentioning

confidence: 76%

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On the Subspace Projected Approximate Matrix method

Brandts

Silva

2015

Appl Math

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“…Note that (20) follows more easily from the alternative definition given in Remark 2.2. However, the proof of Proposition 2.4 also shows the validity of (16).…”

Section: General Description Of Spammentioning

confidence: 76%

“…As far as we can tell, also SPAM has been neglected by the numerical linear algebra community. Moreover, even though a number of citations [6], [13], [16], [20], [31] within the Chemistry and Physics communities over the past years demonstrate awareness of its existence, no studies of its mathematical properties seems to exist.…”

Section: Introductionmentioning

confidence: 99%

On the Subspace Projected Approximate Matrix method

Brandts

Silva

2015

Appl Math

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“…In Section 2, we recall the definition and use of the Subspace Projected Approximate matrices [17], leading to their SPAM eigenvalue method -cf. also [4,8,11,14,20]. In Section 3, the selection mechanism based on the subspace-projected approximate matrices is developed and converted into an algorithm.…”

Section: Outlinementioning

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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“…can be retrieved from the inner iteration. Thus, the vectors u and v in the updating procedure (16) are cheaply available.…”

Section: Remark 24mentioning

confidence: 99%

“…As far as we can tell, also SPAM has been neglected by the numerical linear algebra community. Moreover, even though a number of citations [6,14,16,20,33] within the Chemistry and Physics communities over the past years demonstrate awareness of its existence, no studies of its mathematical properties seems to exists.…”

Section: Introductionmentioning

confidence: 99%

On the Subspace Projected Approximate Matrix method

Brandts¹,

Silva²

2011

Preprint

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We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent iterative method to compute a few eigenvalues of a Hermitian matrix A. It falls in the category of inner-outer iteration methods and aims to save on the costs of matrix-vector products with A within its inner iteration. This is done by choosing an approximation A 0 of A, and then, based on both A and A 0 , to define a sequence (A k ) n k=0 of matrices that increasingly better approximate A as the process progresses. Then the matrix A k is used in the kth inner iteration instead of A.In spite of its main idea being refreshingly new and interesting, SPAM has not yet been studied in detail by the numerical linear algebra community. We would like to change this by explaining the method, and to show that for certain special choices for A 0 , SPAM turns out to be mathematically equivalent to known eigenvalue methods. More sophisticated approximations A 0 turn SPAM into a boosted version of Lanczos, whereas it can also be interpreted as an attempt to enhance a certain instance of the preconditioned Jacobi-Davidson method. Numerical experiments are performed that are specifically tailored to illustrate certain aspects of SPAM and its variations. For experiments that test the practical performance of SPAM in comparison with other methods, we refer to other sources. The main conclusion is that SPAM provides a natural transition between the Lanczos method and one-step preconditioned Jacobi-Davidson.

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

Cited by 3 publications

References 6 publications

On the Subspace Projected Approximate Matrix method

On the Subspace Projected Approximate Matrix method

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

On the Subspace Projected Approximate Matrix method

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