Multicomplex Taylor Series Expansion for Computing High-Order Derivatives

Millwater, Harry; Shirinkam, Sara

doi:10.12732/ijam.v27i4.2

Cited by 11 publications

(5 citation statements)

References 10 publications

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“…For that reason, many special numerical differentiation methods has been developed, e.g. automatic differentiation [9,18,19], complex and multicomplex step differentiation [12,16,17], differentiation rules base on Cauchy integrals [1,2], etc. Thus, computing the derivatives is not necessarily a trivial task and imposes an extra computational cost.…”

Section: Other Advantages Of Fcc Rulesmentioning

confidence: 99%

A comparative study of Filon-type rules for oscillatory integrals

Majidian

2024

J. Numer. Anal. Approx. Theory

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Our aim is to answer the following question: "Among the Filon-type methods for computing oscillatory integrals, which one is the most efficient in practice?". We first discuss why we should seek the answer among the family of Filon-Clenshaw-Curtis rules. A theoretical analysis accompanied by a set of numerical experiments reveals that the plain Filon-Clenshaw-Curtis rules reach a given accuracy faster than the (adaptive) extended Filon-Clenshaw-Curtis rules. The comparison is based on the CPU run-time for certain wave numbers (medium and large).

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Section: Other Advantages Of Fcc Rulesmentioning

confidence: 99%

A comparative study of Filon-type rules for oscillatory integrals

Majidian

2024

J. Numer. Anal. Approx. Theory

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“…. , θ j , using multidual numbers resultant from obtaining the hypercomplex Taylor series expansion of a function is given by [72]:…”

Section: Arbitrary-order Differentiation Using Hypadmentioning

confidence: 99%

Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

et al. 2023

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The calculation of accurate arbitrary-order sensitivities of eigenvalues and eigenvectors is crucial for structural analysis applications, including topology optimization, system identification, finite element model updating, damage detection, and fault diagnosis. Current approaches to obtaining sensitivities for eigenvalues and eigenvectors lack generality, are complicated to implement, prone to numerical errors, and are computationally expensive. In this work, a novel methodology is introduced that uses hypercomplex automatic differentiation (HYPAD) and semi-analytical expressions to obtain arbitrary-order sensitivities for eigenfrequency problems. The new methodology exhibits no sign of truncation nor subtractive cancellation errors regardless of the order of the sensitivity, it is general, and can obtain any high-order sensitivities with the simplicity of first-order computations. A numerical example is presented to verify the accuracy of the method, where the free vibration of a homogeneous cantilever beam is studied. For this problem, up to third-order sensitivities of the eigenvalues and eigenvectors with respect to the material and geometrical parameters were obtained, considering the cases of close and distinct eigenvalues. The results were verified using analytical equations, showing excellent agreement for the eigenvalues and the eigenvectors. The new method promises to facilitate the computation of sensitivities for eigenfrequency problems into routine practice and commercial software.

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“…This section introduces the basic concepts of multicomplex mathematics. More details can be found in Price (1991), Lantoine et al (2012), Millwater and Shirinkam (2014).…”

Section: Multicomplex Mathematicsmentioning

confidence: 99%

Higher-order probabilistic sensitivity calculations using the multicomplex score function method

Garza

Millwater

2016

Probabilistic Engineering Mechanics

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The Score Function method used to compute first order probabilistic sensitivities is extended in this work to arbitrary-order derivatives included mixed partial derivatives through the use of multicomplex mathematics. Multicomplex mathematics provides an effective and convenient numerical means to compute the high-order kernel functions with respect to natural parameters or moments (mean and standard deviation) obviating the need to analytically determine the kernel functions. Using these numerical kernel functions, high-order derivatives of the response moments or the probability-of-failure with respect to the parameters of the input distributions can be obtained. Numerical results indicate that the high-order probabilistic sensitivities converge with respect to the number of samples at the same rate as standard Monte Carlo estimates. Implementation of multicomplex mathematics is facilitated through the use of the Cauchy-Riemann matrices; therefore, the extension of common engineering probability distributions to matrix form is presented. Nomenclature h perturbation size © 2015. This manuscript version is made available under the Elsevier user license

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Multicomplex Taylor Series Expansion for Computing High-Order Derivatives

Cited by 11 publications

References 10 publications

A comparative study of Filon-type rules for oscillatory integrals

A comparative study of Filon-type rules for oscillatory integrals

Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

Higher-order probabilistic sensitivity calculations using the multicomplex score function method

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