Approximations to inverse tangent function

Qiao, Quan-Xi; Chen, Chao-Ping

doi:10.1186/s13660-018-1734-7

Cited by 8 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many new identities and series expansions related to the arctangent function have been reported [7][8][9][10][11][12][13][14][15]. This shows that the discovery of new equations related to the arctangent function as well as their applications remains a very interesting topic.…”

Section: Introductionmentioning

confidence: 96%

A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

et al. 2023

View full text Add to dashboard Cite

In this work, we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes a two-step iteration without surd or complex numbers. The computational test we performed reveals that such a generalization improves the accuracy in computation of the arctangent function by many orders of magnitude with increasing integer M, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of π in the Machin-like formulas.

show abstract

Section: Introductionmentioning

confidence: 96%

A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

et al. 2023

View full text Add to dashboard Cite

show abstract

“…According to [21] any function is computationally efficient if it contains less number of arithmetic and non-arithmetic operations. In this context, among all the bounds mentioned so far in this paper, including those in references [9,11,15,16,19,20,27](in fact, the bounds given in these references contain more operations), the bounds in (1.1) and (1.2) are most computationally efficient.…”

Section: Introductionmentioning

confidence: 96%

“…Further, in 1977-78, Shafer [24,25] obtained the following inequality: ; x > 0 (1.7) was obtained in [27] by L. Zhu. Recently the inequalities (1.3)-(1.7) have been sharpened by many researchers(see for instance, [9,11,15,16,19,20,27] and references therein). Though the bounds for arcsin x x and arctan x…”

Section: Introductionmentioning

confidence: 99%

“…x in (1.3)-(1.7) or in recent papers [9,11,15,16,19,20,27] are sharp they are not computationally as efficient as the bounds in (1.1) and (1.2), in the sense of number of arithmetic/non-arithmetic operations involved in them(see [21]). According to [21] any function is computationally efficient if it contains less number of arithmetic and non-arithmetic operations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Simple efficient bounds for arcsine and arctangent functions

Bagul¹,

Dhaigude

2021

Preprint

View full text Add to dashboard Cite

The aim of this paper is to present new, simple and sufficiently sharp bounds for arcsine and arctangent functions. Some of the bounds are computationally efficient while others are efficient to approximate the integrals Int_{a}^{b} (arcsin x)/x dx and Int_{a}^{b} (arctan x)/x dx. As a matter of interest, several other sharp and generalized inequalities for (arcsin x)/x and (arctan x)/x are also established which are efficient to give some known and other trigonometric inequalities

show abstract

“…An analytical proof that the approximations for arcsine and arccosine, as detailed in Corollary 1, are upper/lower bounds is an unsolved problem.7.2.3. Upper/Lower Bounds for ArctangentAs an example of upper and lower bounds that have been proposed for arctangent, consider the bounds proposed by Qiao and Chen[22] (Theorem 3.1 and Theorem 4.2) for y > 0: 3π 2 y 24 − π 2 + 432 − 24π 2 + π 4 − 12π(12 − π 2 )y + 36π 2 y 2 < atan(y) < 24 − π 2 + 576 − 192π 2 + 16π 4 − 12π(12 − π 2 )y + 36π 2 y 2…”

mentioning

confidence: 99%

Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications

Howard

2023

AppliedMath

View full text Add to dashboard Cite

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.

show abstract

Approximations to inverse tangent function

Abstract: In this paper, we present a sharp Shafer-type inequality for the inverse tangent function. Based on the Padé approximation method, we give approximations to the inverse tangent function. Based on the obtained result, we establish new bounds for arctanx.

Cited by 8 publications

References 13 publications

A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

Simple efficient bounds for arcsine and arctangent functions

Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications

Contact Info

Product

Resources

About