Rational Chebyshev approximations for the error function

Cody, W. J.

doi:10.1090/s0025-5718-1969-0247736-4

Cited by 221 publications

(70 citation statements)

References 2 publications

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“…Using Eqs. [13] and [14], we have calculated the respective time dependencies for various values of t a /t d . The upper curve in Fig.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…(b) Relaxation of perturbations in the surfactant adsorption, p (t), and surface tension, σ p (t), calculated with the help of Eq. [13]. Each curve corresponds to a fixed value of the ratio t a /t d .…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…For this reason, Eq. [13] is suitable for both numerical computations and asymptotic analysis. Similar analytical expression for the subsurface concentration can be deduced from Eqs.…”

Section: Adsorption Time Vs Diffusion Timementioning

confidence: 99%

“…Similar analytical expression for the subsurface concentration can be deduced from Eqs. [4] and [13]:…”

Section: Adsorption Time Vs Diffusion Timementioning

confidence: 99%

See 3 more Smart Citations

Adsorption Relaxation for Nonionic Surfactants under Mixed Barrier-Diffusion and Micellization-Diffusion Control

Danov

Valkovska

Kralchevsky

2002

Journal of Colloid and Interface Science

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“…Using Eqs. [13] and [14], we have calculated the respective time dependencies for various values of t a /t d . The upper curve in Fig.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…For this reason, Eq. [13] is suitable for both numerical computations and asymptotic analysis. Similar analytical expression for the subsurface concentration can be deduced from Eqs.…”

Section: Adsorption Time Vs Diffusion Timementioning

confidence: 99%

“…Similar analytical expression for the subsurface concentration can be deduced from Eqs. [4] and [13]:…”

Section: Adsorption Time Vs Diffusion Timementioning

confidence: 99%

See 2 more Smart Citations

Adsorption Relaxation for Nonionic Surfactants under Mixed Barrier-Diffusion and Micellization-Diffusion Control

Danov

Valkovska

Kralchevsky

2002

Journal of Colloid and Interface Science

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“…The code for the numerical evaluation of eftc(x) in MLF2 was written by Cody (1969). The following recursion relation is used to calculate higher-order parabolic cylinder functions.…”

Section: Appendix a Derivations And Implementation A 1 Implementatiomentioning

confidence: 99%

Improved Structure Refinement Through Maximum Likelihood

Pannu¹,

Read²

1996

Acta Crystallogr A Found Crystallogr

346

274

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When crystal structures of proteins or small molecules are used to address questions of scientific relevance, the accuracy and precision of the atomic coordinates are crucial. Accordingly, the atomic model is generally improved by refining it to improve agreement with the observed diffraction data. The refinement of crystal structures is conventionally based on least-squares methods but such procedures are handicapped since conditions necessary for the use of the least-squares target are not satisfied. It is proposed here that refinement should be based on maximum likelihood and two maximum-likelihood targets have been implemented in the program XPLOR. Preliminary tests with protein structures give dramatic results. Compared to leastsquares refinement, maximum-likelihood refinement can achieve more than twice the improvement in average phase error. The resulting electron-density maps are correspondingly clearer and suffer less from model bias.

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Chebyshev subinterval polynomial approximations for continuous distribution functions

Tsai

Moskowitz

1989

Naval Research Logistics

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An algorithm for constructing a three‐subinterval approximation for any continous distribution function is presented in which the Chebyshev criterion is used, or equivalently, the maximum absolute error (MAE) is minimized. The resulting approximation of this algorithm for the standard normal distribution function provides a guideline for constructing the simple approximation formulas proposed by Shah [13]. Furthermore, the above algorithm is extended to more accurate computer applications, by constructing a four‐polynomial approximation for a distribution function. The resulting approximation for the standard normal distribution function is at least as accurate as, faster, and more efficient than the six‐polynomial approximation proposed by Milton and Hotchkiss [11] and modified by Milton [10].

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Rational Chebyshev approximations for the error function

Abstract: This note presents nearly-best rational approximations ftir the functions erf (i) and erfc (x), with maximal relative errors ranging down to between 6

Cited by 221 publications

References 2 publications

Adsorption Relaxation for Nonionic Surfactants under Mixed Barrier-Diffusion and Micellization-Diffusion Control

Adsorption Relaxation for Nonionic Surfactants under Mixed Barrier-Diffusion and Micellization-Diffusion Control

Improved Structure Refinement Through Maximum Likelihood

Chebyshev subinterval polynomial approximations for continuous distribution functions

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