Algorithm 511: CDC 6600 Subroutines IBESS and JBESS for Bessel Functions I
            <sub>υ</sub>
            (x) and J
            <sub>υ</sub>
            (x), x≥0, υ ≥0 [S18]

Amos, Donald E.; Daniel, S.L.; Weston, Matthew

doi:10.1145/355719.355727

Cited by 6 publications

(2 citation statements)

References 1 publication

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“…(9,10) and in Eqs. (11,12) is not common in the literature although equivalent alternatives can be found.…”

Section: -Mathematical Relationsmentioning

confidence: 99%

Calculation of Fresnel Integrals of Real and Complex Arguments up to 28 Significant Digits

Zaghloul,

Alrawas

2023

Preprint

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We present numerical algorithms for efficient and highly accurate computations of Fresnel’s sine and cosine integrals $S\left(z\right)$ and $C\left(z\right)$for real ($z=x$) and complex ($z=x+iy$) arguments. The algorithms are based on series expansion for small values of |z|, expansion in Chebyshev subinterval polynomial approximations for intermediate real values ($y=0$), toggled with sum of half-integer order Bessel functions for complex arguments, together with asymptotic series expansion for large values of |z|. The present algorithms, implemented in a Fortran elemental module, can be run using any of the single, double or quadruple precision arithmetic. Results from the present code have been benchmarked versus comprehensive data tables, generated using Maple, Mathematica and Matlab symbolic toolbox. Compared to a maximum of 16 significant digits in the literature, the present algorithms can calculate $S\left(x\right)$ and C(x) up to 28 significant digits in the range $xϵ[0, {10}^{6}]$ and $S\left(z\right)$ and C(z) to the same accuracy in the domain where $\left|S\right(z\left)\right|$ and $\left|C\left(z\right)\right|$ each is less than the largest finite floating point number in the precision under consideration with $\left|arg z\right|<\frac{\pi }{2}$ or more specifically $3\times {10}^{-4}<\left|\frac{y}{x}\right|<{2\times 10}^{3}$ for $S\left(z\right)$ and ${4\times 10}^{-4}<\left|\frac{y}{x}\right|<{2.2\times 10}^{3}$ for $C\left(z\right)$.

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“…(9,10) and in Eqs. (11,12) is not common in the literature although equivalent alternatives can be found.…”

Section: -Mathematical Relationsmentioning

confidence: 99%

Calculation of Fresnel Integrals of Real and Complex Arguments up to 28 Significant Digits

Zaghloul,

Alrawas

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Hartman and Watson (1974) prove that R(z) is a continuous distribution function on z~O and thus is a monotonically increasing function of z. Amos et al (1977) and Gautschi and Slavik (1978) give computer algorithms for calculating l . (z) for non-negative z and v, and Amos (1974) examines bounds for /.+ 1(z)// .(z).…”

Section: Uniform Distribution For Rmentioning

confidence: 99%

The Circular Structural Model

Anderson

1981

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary This paper deals with consistent estimation of the structural parameters of the circular structural model. Identifiability is proved and the difficulties of dealing with the structural model in practice mentioned. A uniform distribution is then assumed for the angular variables and maximum likelihood estimation of the parameters examined. Also discussed are use of an approximation to the ratio of modified Bessel functions, asymptotic behaviour of the estimators as the ratio of the circle radius to the underlying structural variance parameter increases, and efficiency of other estimators of the parameters. The circular functional model, and the difficulties associated with it, is briefly mentioned.

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Calculation of Fresnel integrals of real and complex arguments up to 28 significant digits

Zaghloul,

Alrawas

2023

Numer Algor

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We present numerical algorithms for efficient and highly accurate computations of Fresnel’s sine and cosine integrals S(z) and C(z) for real (z = x) and complex (z = x + iy) arguments. The algorithms are based on series expansion for small values of |z|, expansion in Chebyshev subinterval polynomial approximations for intermediate real values (y = 0), toggled with sum of half-integer order Bessel functions for complex arguments, together with asymptotic series expansion for large values of |z|. The present algorithms, implemented in a Fortran elemental module, can be run using any of the single, double, or quadruple precision arithmetic. Results from the present code have been benchmarked versus comprehensive data tables, generated using Maple, Mathematica, and Matlab symbolic toolbox. Compared to a maximum of 16 significant digits in the literature, the present algorithms can calculate S(x) and C(x) up to 28 significant digits in the range xϵ[0, 106] and S(z) and C(z) to the same accuracy in the domain where |S(z)| and |C(z)| each is less than the largest finite floating point number in the precision under consideration with $$\mid \mathit{\arg}\ z\mid <\frac{\pi }{2}$$ ∣ arg z ∣ < π 2 or more specifically $$3\times {10}^{-4}<\left|\frac{y}{x}\right|<2\times {10}^3$$ 3 × 10 - 4 < y x < 2 × 10 3 for S(z) and $$4\times {10}^{-4}<\left|\frac{y}{x}\right|<2.2\times {10}^3$$ 4 × 10 - 4 < y x < 2.2 × 10 3 for C(z).

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Algorithm 511: CDC 6600 Subroutines IBESS and JBESS for Bessel Functions I _υ (x) and J _υ (x), x≥0, υ ≥0 [S18]

Cited by 6 publications

References 1 publication

Calculation of Fresnel Integrals of Real and Complex Arguments up to 28 Significant Digits

Calculation of Fresnel Integrals of Real and Complex Arguments up to 28 Significant Digits

The Circular Structural Model

Calculation of Fresnel integrals of real and complex arguments up to 28 significant digits

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Algorithm 511: CDC 6600 Subroutines IBESS and JBESS for Bessel Functions I υ (x) and J υ (x), x≥0, υ ≥0 [S18]

Cited by 6 publications

References 1 publication

Calculation of Fresnel Integrals of Real and Complex Arguments up to 28 Significant Digits

Calculation of Fresnel Integrals of Real and Complex Arguments up to 28 Significant Digits

The Circular Structural Model

Calculation of Fresnel integrals of real and complex arguments up to 28 significant digits

Contact Info

Product

Resources

About

Algorithm 511: CDC 6600 Subroutines IBESS and JBESS for Bessel Functions I _υ (x) and J _υ (x), x≥0, υ ≥0 [S18]