A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

Ekström, Sven‐Erik; Vassalos, Paris

doi:10.1007/s11075-021-01130-9

Cited by 4 publications

(18 citation statements)

References 23 publications

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“…The time-covariance matrix T , eq , is a real symmetric Toeplitz matrixeach descending diagonal of T from left to right is constant, thereby it is fully defined by its first row. Toeplitz matrices arise often − in theory of weakly stationary processes, theory of signal processing, and information theory applications in statistical physics.…”

Section: Computational Detailsmentioning

confidence: 99%

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

Komarova

2023

J. Chem. Theory Comput.

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Unraveling the density matrix of a non-stationary quantum state as an explicit function of a few observables provides a complementary view of quantum dynamics. We have recently developed a practical way to identify the minimal set of the dominant observables that govern the quantal dynamics even in the case of strong non-adiabatic effects and large anharmonicity [Komarova et al., J. Chem. Phys. 155, 204110 (2021)]. Fast convergence in the number of the dominant contributions is achieved when instead of the density matrix we describe the time-evolution of the surprisal, the logarithm of the density operator. In the present work, we illustrate the efficiency of the proposed approach using an example of the early time dynamics in pyrazine in a Hilbert space accounting for up to four vibrational normal modes, {Q 10a, Q 6a, Q 1, and Q 9a}, and two coupled electronic states, the optically dark B 1 3 u ( n π * ) and the bright B 1 2 u ( π π * ) states. Dynamics in four-dimensional (4D) configurational space involve 19,600 vibronic eigenstates. Our results reveal that the rate of the ultrafast population decay as well as the shape of the nuclear wave packets in 2D, accounting only for {Q 10a,Q 6a} normal modes, are accurately captured with only six dominant time-independent observables in the surprisal. Extension of the dynamics to 3D and 4D vibrational subspace requires only five additional constraints. The time-evolution of a quantum state in 4D vibrational space on two electronic states is thus compacted to only 11 time-dependent coefficients of these observables.

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Section: Computational Detailsmentioning

confidence: 99%

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

Komarova

2023

J. Chem. Theory Comput.

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“…( 10), is a real symmetric Toeplitz matrix -each descending diagonal of from left to right is constant, thereby it is fully defined by its first row. Toeplitz matrices arise often [64][65][66][67][68] in theory of weakly stationary processes, theory of signal processing and information theory applications in statistical physics.…”

Section: ˆ( ) Ln ( )mentioning

confidence: 99%

Density matrix via few dominant observables for the ultrafast non-radiative decay in pyrazine

Komarova

2022

Preprint

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Unraveling of the density matrix of a non-stationary quantum state as a function of a few observables provides a complementary view of quantum dynamics. Here we pay special attention to the entangled electron-nuclear motion. We have recently developed a practical way to identify the minimal set of the dominant observables that govern the quantal dynamics even in the case of strong non-adiabatic effects and large anharmonicity [Komarova et al., J. Chem. Phys. 155, 204110 (2021)]. Fast convergence in the number of the dominant contributions is achieved when instead of the density matrix we describe the time-evolution of the surprisal, the logarithm of the density operator. In the present work, we illustrate the efficiency of the proposed approach using an example of the early time dynamics in pyrazine in a Hilbert space accounting for up to four vibrational normal modes, {Q10a, Q6a, Q1 , Q9a}, and two coupled electronic states, the optically dark and the bright states. Dynamics in four-dimensional configurational space involve 19600 vibronic eigenstates. Our results reveal that the rate of the ultrafast population decay as well as the shape of the nuclear wave packets in 2D, accounting only for {Q10a, Q6a} normal modes, are accurately captured with only six dominant time-independent observables in the surprisal. Extension of the dynamics to three- and four-dimensional vibrational subspace requires only five additional constraints. The time-evolution of a quantum state in 4D vibrational space on two electronic states is thus compacted to only 11 time-dependent coefficients of these observables.

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“…In a recent work 16 the cases of interest were those in which

{\left\{{T}_n(f)\right\}}_n{\nsim}_{\lambda }f

and the eigenvalues of

{T}_n(f)

are real for all

n

. In such a setting, often the matrix sequence

{\left\{{T}_n(f)\right\}}_n

is such that there exists a real‐valued function

\mathfrak{f}

satisfying

{\left\{{T}_n(f)\right\}}_n{\sim}_{\lambda}\mathfrak{f}

, with the eigenvalues of

{T}_n(f)

admitting an asymptotic expansion of the same type as considered in previous theoretical 17,18 and numerical 19–21 works.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Böttcher et al 15 proposed a novel numerical algorithm to approximate

\Lambda (a)

with high accuracy and moderate computational cost. Nevertheless, in this article, we will rely on the algorithm proposed by Ekström and Vassalos, 16 because it produces an expression for

\mathfrak{f}

instead of its range. With this connection we introduce the class

𝒯

as the collection of all generating functions

f\in {L}^1\left(\left[-\pi, \pi \right]\right)

satisfying one of the following conditions,

f

belongs to the Tilli class

𝒯 𝒞

and the essential range

\mathit{\mathcal{ER}}(f)

is connected, the limiting set

\Lambda (f)

has one nonclosed analytic arc only. …”

Section: Introductionmentioning

confidence: 99%

“…We call methods that numerically approximate this asymptotic expansion matrix‐less if, after the respective precomputing phase, the eigenvalue computation does not need to work in any way with the related matrix. Recent literature had studied the real‐valued , 16,19 preconditioned , 20 banded block , 21 parallel , 22 and B‐spline IgA 23 settings. Böttcher et al , 15 besides the rich and crystal clear theoretical study, presented an algorithm in the spirit of that of Beam and Warming 24 …”

Section: Introductionmentioning

confidence: 99%

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Matrix‐less methods for the spectral approximation of large non‐Hermitian Toeplitz matrices: A concise theoretical analysis and a numerical study

Bogoya,

Ekström,

Serra‐Capizzano

et al. 2024

Numerical Linear Algebra App

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It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.

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A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

Cited by 4 publications

References 23 publications

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

Density matrix via few dominant observables for the ultrafast non-radiative decay in pyrazine

Matrix‐less methods for the spectral approximation of large non‐Hermitian Toeplitz matrices: A concise theoretical analysis and a numerical study

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