The Natural Approaches of Shafer-Fink Inequality for Inverse Sine Function

Zhu, Ling

doi:10.3390/math10040647

Cited by 4 publications

(4 citation statements)

References 22 publications

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“…In order to evaluate the new approximated expression for the inverse sine function, we compare it with other approximations given by some researchers via inequalities such as Shafer-Fink inequality and padé's approximants [2,11,13]. As compared to Shafer-Fink's boundaries the approximated expression (2.15) presents relatively good behaviour.…”

Section: Evaluation and Comparison With Other Methodsmentioning

confidence: 99%

“…The approaches for approximation of these functions in the present literature are however often based on sharpening and refining bounds by establishing two-sided inequalities which contain obvious complexity as far as possible application is concerned. For example, for the inequalities involving arcsine function we refer reader to [2,[4][5][6][7][9][10][11][12][13] and the references therein. The arcsine function can also be expressed as a Taylor's series or in terms of hypergeometric function [1,3,8].…”

Section: Introductionmentioning

confidence: 99%

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An Innovative Method for Approximating Arcsine Function

Othman¹,

Bagul²

2022

Preprint

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This paper presents a new method for approximating the classical arcsine function. The proposed approximating methodology is simpler in its approach than other classical approaches and undeniably innovative. It is based on matrix representation besides the basic interpolation to approximate the inverse trigonometric function. It provides an efficient model which allows for reliable and precise calculations. The results are as per our knowledge unseen results in the previous literature.

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Section: Evaluation and Comparison With Other Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Innovative Method for Approximating Arcsine Function

Othman¹,

Bagul²

2022

Preprint

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“…There is interest in upper and lower bounds for arcsine, e.g., [17][18][19][20][21]. The classic upper and lower bounded functions for arcsine are defined by the Shafer-Fink inequality [13]:…”

Section: Published Bounds For Arcsinementioning

confidence: 99%

“…where the lower relative error bound is 2.27 × 10 −3 and the upper relative error bound is 5.61 × 10 −4 . Zhu [21] (Theorem 1), proposed the bounds…”

Section: Published Bounds For Arcsinementioning

confidence: 99%

Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications

Howard

2023

AppliedMath

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Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of 10−16, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals.

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Compressed sensing with smooth L0 constraints for moving force identification from bridge response measurements

Liang,

Hou,

2024

Meas. Sci. Technol.

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Compressed sensing (CS), as an emerging information sampling technique, has been successfully applied in the field of moving force identification (MFI). However, existing MFI compressed sensing models often fail to obtain the optimal sparse solutions and frequently underestimate the amplitude of local impact forces. To effectively address this issue, a new compressed sensing method is proposed for MFI based on smooth L0 norm constraints and bridge response measurements. Firstly, a smooth function is used to approximate the L0 norm, establishing a noise CS reconstruction model for MFI. The introduction of the smoothing function can locally convexify the original MFI problem and enhance the smoothness and differentiability of the objective function, making the optimization problem easier to solve. Subsequently, the Polak-Ribiere-Polyak (PRP) formula is adopted to point the descent direction of the new objective function, and the sparse solution is iteratively advanced through the conjugate gradient algorithm. Finally, the applicability and feasibility of the proposed method is confirmed by numerical simulations and vehicle-bridge interaction tests, respectively. The results show that the proposed method can accurately identify moving forces from limited measurements of bridge responses. Compared with existing methods, it can provide more precise sparse solutions with higher robustness to measurement noises, and address the issue of underestimating on the amplitude of local impact forces, which is expected to enhance the performance and in-situ applicability of MFI.

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The Natural Approaches of Shafer-Fink Inequality for Inverse Sine Function

Abstract: In this paper, we obtain some new natural approaches of Shafer-Fink inequality for arc sine function and the square of arc sine function by using the power series expansions of certain functions, which generalize and strengthen those in the existing literature.

Cited by 4 publications

References 22 publications

An Innovative Method for Approximating Arcsine Function

An Innovative Method for Approximating Arcsine Function

Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications

Compressed sensing with smooth L0 constraints for moving force identification from bridge response measurements

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