Improved <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-exponential and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-trigonometric functions

Cieśliński, Jan L.

doi:10.1016/j.aml.2011.06.009

Cited by 19 publications

(2 citation statements)

References 30 publications

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“…We propose a cautious BFGS update and prove that the method with either a Wolfe-type or an Armijo-type line search converges globally if the function to be minimized has Lipschitz continuous gradients. The q-calculus, particularly known as quantum calculus, has gained a lot of interest in various fields of science, mathematics [24], physics [25], quantum theory [26], statistical mechanics [27] and signal processing [28], etc., where the q-derivative is employed. It is also known as the Jackson derivative, as the concept was first introduced by Jackson [29]; it was further studied in the case of a q-difference equation by Carmichael [30], Mason [31], Adams [32] and Trjitzinsky [33].…”

Section: Introductionmentioning

confidence: 99%

On q-BFGS algorithm for unconstrained optimization problems

et al. 2020

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Variants of the Newton method are very popular for solving unconstrained optimization problems. The study on global convergence of the BFGS method has also made good progress. The q-gradient reduces to its classical version when q approaches 1. In this paper, we propose a quantum-Broyden–Fletcher–Goldfarb–Shanno algorithm where the Hessian is constructed using the q-gradient and descent direction is found at each iteration. The algorithm presented in this paper is implemented by applying the independent parameter q in the Armijo–Wolfe conditions to compute the step length which guarantees that the objective function value decreases. The global convergence is established without the convexity assumption on the objective function. Further, the proposed method is verified by the numerical test problems and the results are depicted through the performance profiles.

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Section: Introductionmentioning

confidence: 99%

On q-BFGS algorithm for unconstrained optimization problems

et al. 2020

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“…With properties connected to this or related functions we refer to [28]. For new definitions of q-exponential and q-trigonometric functions see [15]. Since we are using and proving some asymptotic results, we highlight [24] on this subject, where a complete asymptotic expansion for the q-Pochhammer symbol (or, the infinite q-shifted factorial (z; q) ∞ ) was exhibited.…”

Section: Introductionmentioning

confidence: 99%

On Basic Fourier-Bessel Expansions

Cardoso¹

2018

SIGMA

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When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) q-Bessel function, the corresponding positive zeros j kν and the "shifted" zeros, qj kν , among others, play an essential role. Mixing classical analysis with q-analysis we were able to prove asymptotic relations between those zeros and the "shifted" ones, as well as the asymptotic behavior of the third Jackson q-Bessel function when computed on the "shifted" zeros. A version of a q-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.Using the basic hypergeometric representation [19, p. 4] for r φ s , it is very well known that (1.1) can be written asIn [2] it was shown, under some restrictions, that these functions are the only ones that satisfy a q-analogue of the Hardy result [20] about functions orthogonal with respect to their own zeros.We have the following limit lim q→1 J ν 1 − q 2 x; q = J ν (x), arXiv:1707.05216v3 [math.CA]

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