Error analysis of two algorithms for the computation of the matrix exponential

Codenotti, Bruno; Fassino, Claudia

doi:10.1007/bf02576760

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…This can be done using the following theorem, which is proved for triangular nonnegative matrices in [8, Theorem 2] but is easily extended to general non-negative matrices. A similar normwise bound is also obtained in [6,13].…”

Section: Taylor Series Methodssupporting

confidence: 79%

“…As A d is non-negative, the computation of the Taylor series involves no subtractions and can therefore have entrywise high relative accuracy, modulo possible accumulations of small errors in the summation. This approach has been suggested as a more stable way for computing e A in the context of the transient matrix computations in the continuous Markov chain in [14,15] and more generally in [6,13]. Here, we rigourously examine various numerical issues arising in this process and provide an entrywise error analysis.…”

Section: Taylor Series Methodsmentioning

confidence: 99%

“…From the above discussion, we conclude that the computed exponential has entrywise relative errors in the order of mnκ exp (A)u. We remark that it has been observed in [6,13] that the Taylor series method can produce entrywise high relative accuracy but only normwise error analysis was carried out there. While there are some similarities in the normwise and entrywise analysis, some substantial differences exist such as the need to truncate the series to have entrywise small relative errors (3.5) and the need to compute the quantities involved in the bound accurately.…”

Section: Taylor Series Methodsmentioning

confidence: 99%

“…In §3 we briefly discuss various numerical issues in implementing the Taylor series method combined with a shifting to achieve best entrywise relative accuracy possible. We note that the Taylor series method combined with a shifting has been proposed before as a more accurate method in [6,8,13] as well as in the applied probability community [14,15] and our contributions here are a rigorous truncation criterion and an entrywsie error analysis. The Taylor series method, however, may be expensive as the number of terms required for the series truncation cannot be bounded a priori.…”

Section: Introductionmentioning

confidence: 99%

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Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

Xue

2013

Math. Comp.

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A real square matrix is said to be essentially non-negative if all of its off-diagonal entries are non-negative. It has recently been shown that the exponential of an essentially non-negative matrix is determined entrywise to high relative accuracy by its entries up to a condition number intrinsic to the exponential function (Numer. Math. 110 (2008), 393-403). Thus the smaller entries of the exponential may be computed to the same relative accuracy as the bigger entries. This paper develops algorithms to compute exponentials of essentially non-negative matrices entrywise to high relative accuracy.

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Section: Taylor Series Methodssupporting

confidence: 79%

Section: Taylor Series Methodsmentioning

confidence: 99%

Section: Taylor Series Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

Xue

2013

Math. Comp.

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“…In this section, we develop the main algorithm of this paper, based on aggressive truncation in T m (x) defined in (1.3). To ensure the feasibility of this approach, some a priori estimates of the truncation error are required, which were not available in many traditional Taylor series approaches [7,17,35]. To this end, we begin with establishing these a priori componentwise truncation error estimates.…”

Section: The Aggressively Truncated Taylor Series Methodsmentioning

confidence: 99%

Aggressively Truncated Taylor Series Method for Accurate Computation of Exponentials of Essentially Nonnegative Matrices

Shao

Gao²,

Xue³

2014

SIAM J. Matrix Anal. & Appl.

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Small relative perturbations to the entries of an essentially nonnegative matrix introduce small relative errors to entries of its exponential. It is thus desirable to compute the exponential with high componentwise relative accuracy. Taylor series approximation coupled with scaling and squaring is used to compute the exponential of an essentially nonnegative matrix. An a priori componentwise relative error bound of truncation is established, from which one can choose the degree of Taylor series expansion and the scale factor so that the exponential is computed with desired componentwise relative accuracy. To reduce the computational cost, the degree of the Taylor series expansion is chosen small, while the scale factor is chosen sufficiently large to achieve the desired accuracy. The rounding errors in the squaring stage are not serious as squaring is forward stable for nonnegative matrices. We also establish a posteriori componentwise error bounds and derive a novel interval algorithm for the matrix exponential. Rounding error analysis and numerical experiments demonstrate the efficiency and accuracy of the proposed methods.

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Optimal control of large quantum systems: assessing memory and runtime performance of GRAPE

Lu,

Joshi,

San Dinh

et al. 2024

J. Phys. Commun.

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Gradient Ascent Pulse Engineering (GRAPE) is a popular technique in quantum optimal control, and can be combined with automatic differentiation (AD) to facilitate on-the-fly evaluation of cost-function gradients. We illustrate that the convenience of AD comes at a significant memory cost due to the cumulative storage of a large number of states and propagators. For quantum systems of increasing Hilbert space size, this imposes a significant bottleneck. We revisit the strategy of hard-coding gradients in a scheme that fully avoids propagator storage and significantly reduces memory requirements. Separately, we present improvements to numerical state propagation to enhance runtime performance. We benchmark runtime and memory usage and compare this approach to AD-based implementations, with a focus on pushing towards larger Hilbert space sizes. The results confirm that the AD-free approach facilitates the application of optimal control for large quantum systems which would otherwise be difficult to tackle.

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Error analysis of two algorithms for the computation of the matrix exponential

Cited by 3 publications

References 8 publications

Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

Aggressively Truncated Taylor Series Method for Accurate Computation of Exponentials of Essentially Nonnegative Matrices

Optimal control of large quantum systems: assessing memory and runtime performance of GRAPE

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