A new approach for determining the time step when propagating with the Lanczos algorithm

Mohankumar, N.; Carrington, Tucker

doi:10.1016/j.cpc.2010.07.020

Cited by 7 publications

(10 citation statements)

References 26 publications

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“…It is straightforward to obtain a closed-form expression that bounds the right hand side (RHS) of Equation (15), by using Equation (16) and summing a geometric series. This is explained in [8]. We obtain…”

Section: Mohankumar-carrington Derivationmentioning

confidence: 90%

“…In 2010, we (Mohankumar and Carrington-MC) derived a slightly different equation for choosing ∆t [8]. In this paper, we compare the derivations of the Lubich and MC equations and show that ∆t MC is somewhat larger than ∆t L and therefore that using ∆t MC reduces the cost of propagating.…”

Section: Introductionmentioning

confidence: 90%

“…Both derivations begin with a link between an error bound for a Lanczos approximation to a function of a matrix applied to a vector and an error for an approximation of the same function by a polynomial of a certain degree. In [8], we began with a result of Stewart and Leyk [12]. It is very similar to Theorem 2.9 of [9].…”

Section: Common Starting Pointmentioning

confidence: 98%

“…The notation used here is the same as the notation of [8]. m is small enough that loss of orthogonality of the Lanczos vectors is unimportant.…”

Section: Introductionmentioning

confidence: 99%

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A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

Mohankumar¹,

Carrington

2019

Mathematics

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To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk.

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Section: Mohankumar-carrington Derivationmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 90%

Section: Common Starting Pointmentioning

confidence: 98%

“…The notation used here is the same as the notation of [8]. m is small enough that loss of orthogonality of the Lanczos vectors is unimportant.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

Mohankumar¹,

Carrington

2019

Mathematics

Self Cite

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“…Computed effective order of the defect for m = 10 (partly): t 1.0 · 10 0 1.5 · 10 0 2.1 · 10 0 2.9 · 10 0 4.1 · 10 0 5.9 · 10 0 8.3 · 10 0 1.2 · 10 1 1.7 · 10 1 2.3 · 10 1 3. and m = 30: t 3.3 · 10 1 4.7 · 10 1 6.6 · 10 1 9.4 · 10 1 1.3 · 10 2 1.9 · 10 2 2.7 · 10 2 ρ(t) 16.60 13.47 10. 33 7.48 5.15 3.36 2.07…”

Section: The Hermitian Caseunclassified

Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions

2019

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An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of timestepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of ϕ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

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Discretization for a spray deposition model: Criteria for temporal and spatial differencing

Larbi

Salyani

2013

Computers and Electronics in Agriculture

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A new approach for determining the time step when propagating with the Lanczos algorithm

Cited by 7 publications

References 26 publications

A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions

Discretization for a spray deposition model: Criteria for temporal and spatial differencing

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