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Improved upper bound on the complementary error function

ELECTRONICS LETTERS 2020年第13期56卷 663-665页

作者： Zhang, Zhimin Fang, Jinpan Lin, Jaduo Zhao, Shancheng Xiao, Fengjun Wen, Jinming Beijing Jiaotong Univ Sch Econ & Management Beijing 100044 Peoples R China Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Peoples R China Jinan Univ Coll Cyber Secur Guangdong Key Lab Data Secur & Privacy Preserving Guangzhou 510632 Peoples R China Hangzhou Dianzi Univ Sch Management Hangzhou 310018 Peoples R China Chinese Acad Sci Inst Informat Engn State Key Lab Informat Secur Beijing 100093 Peoples R China

The complementary error function is a fundamental function in statistics and plays a central role in numerous applications. Its upper bound is essential for simplifying many performance analysis in communications. This Letter develops a new upper bound, which is sharper than existing upper bounds, on the complementary error function. Simulation tests are also performed to illustrate the sharpness of the new upper bound.

关键词： error analysis approximation theory computational complexity complementary error function fundamental function upper bounds

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Algorithm and Implementation Details for complementary error function

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IEEE TRANSACTIONS ON COMPUTERS 2017年第7期66卷 1106-1118页

作者： Baikov, Nikita Russian Acad Sci Sci Res Inst Syst Anal Nakhimovsky Prosp 36-1 Moscow 117218 Russia

The algorithm proposed in this article provides high accuracy and requires only double-precision arithmetic. The calculated value obtained by the algorithm is proved to be accurate to 0.785 . ulp as long as the result does not underflow. In the case of subnormal results the error does not exceed 0.875 . ulp. The testing results have also shown that the implementation of the proposed algorithm has reasonable computational time compared to other implementations.

关键词： complementary error function double-precision arithmetic algorithm implementation

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Modality and Limiting Behaviours of Skew Symmetric Distributions

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SANKHYA-SERIES A-MATHEMATICAL STATISTICS AND PROBABILITY 2025年第1期87卷 219-251页

作者： Hitchen, Thomas Nadarajah, Saralees UNIV MANCHESTER DEPT MATH MANCHESTER M13 9PL England

Alruwaili (Stat. Papers, 64, 497-507, 2023) studied modality and limiting behavior of the skew t distribution and its mixtures. In this paper, Alruwaili's work is extended for the skew normal, skew logistic, skew Cauchy, skew Laplace and skew uniform distributions and their mixtures. For each of these distributions and its mixtures, we derive the equation satisfied by the mode and investigate the behaviour of the mode with respect to skew parameters, scale parameters and large values of the skew parameters. We find that the number of modes in the mixtures can be one, two, three or more. The number of modes is dictated by differences between skew parameters. The scale parameters have a minimal effect on locations of the modes.

关键词： complementary error function mixtures probability density function

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Bounds for the generalized Marcum function of the second kind

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RAMANUJAN JOURNAL 2022年第1期58卷 1-21页

作者： Baricz, Arpad Bisht, Nitin Singh, Sanjeev Vijesh, V. Antony Babes Bolyai Univ Dept Econom Cluj Napoca 400591 Romania Obuda Univ Inst Appl Math H-1034 Budapest Hungary Indian Inst Technol Indore Discipline Math Indore 453552 India

In this paper, we focus on the generalized Marcum function of the second kind of order nu > 0, defined by R-nu(a,b) = ca, nu/a(nu-1) integral(infinity)(b) t(nu)e(-t2+a2/2) K nu-1(at)dt, where a > 0, b >= 0, K-nu stands for the modified Bessel function of the second kind, and ca,nu is a constant depending on a and nu such that R-nu(a,0)=1. Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

关键词： Modified Bessel function of the second kind Bounds Survival function complementary error function Generalized Marcum Q-function

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On Moments of the Noncentral Chi Distribution

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SANKHYA-SERIES A-MATHEMATICAL STATISTICS AND PROBABILITY 2023年第1期85卷 803-807页

作者： Nadarajah, Saralees Chan, Stephen Univ Manchester Manchester M13 9PL Lancs England Amer Univ Sharjah Dept Math & Stat Sharjah U Arab Emirates

The mean of a noncentral chi random variable is usually expressed in terms a hypergeometric function. On a 17 pages paper, Lawrence [Sankhyd A. https://***/10.1007/s13171-021-00262-3] derived simpler expressions for the mean, involving the modified Bessel function of the first kind and the error function. We show here that these expressions follow immediately from known relationships between the hypergeometric and modified Bessel / error functions.

关键词： complementary error function error function Modified Bessel function of the first kind

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COMPUTATION OF THE COMPLEX error function USING MODIFIED TRAPEZOIDAL RULES

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2021年第5期59卷 2346-2367页

作者： Al Azah, Mohammad Chandler-Wilde, Simon N. Al Hussein Tech Univ HTU Sch Social & Basic Sci Amman Jordan Univ Reading Dept Math & Stat POB 220 Reading RG6 6AX Berks England

In this paper we propose a method for computing the Faddeeva function w(z) := e(-z2) erfc(-i z) via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel [Math. Cornp., 25 (1971), pp. 339-344] and Hunter and Regan [Math. Comp., 26 (1972), pp. 339-541]. Addressing shortcomings flagged by Weideman [SIAM. T. Numer. Anal., 31 (1994), pp. 1497-1518], we construct approximations which we prove are exponentially convergent as a function of N + 1, the number of quadrature points, obtaining error bounds which show that accuracies of 2 x 10(-15) in the computation of w(z) throughout the complex plane are achieved with N = 11;this is confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where w(z) is nonzero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing w(z) for complex z.

关键词： trapezoidal rule complementary error function Faddeeva function

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functional Inequalities and Bounds for the Generalized Marcum function of the Second Kind

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RESULTS IN MATHEMATICS 2021年第1期76卷 1-28页

作者： Baricz, Arpad Bisht, Nitin Singh, Sanjeev Vijesh, V. Antony Babes Bolyai Univ Dept Econ Cluj Napoca 400591 Romania Obuda Univ Inst Appl Math H-1034 Budapest Hungary Indian Inst Technol Indore Discipline Math Indore 453552 India

In this paper, we consider the generalized Marcum function of the second kind as an analogous function of the so-called generalized Marcum Q-function. We provide the log-convexity (log-concavity) property for its unit complement and improve some of our previous results on it. One of the transformed functions of the generalized Marcum function of the second kind is discussed in details in this paper. This form of the generalized Marcum function of the second kind supplies various important inequalities. We also discuss the Turan type inequality for the generalized Marcum Q-function. Additionally, we provide the bounds for the generalized Marcum function of the second kind as well as for its symmetric difference.

关键词： Modified Bessel function of the second kind bounds log-convexity and geometrical convexity survival function complementary error function generalized Marcum Q-function symmetric difference

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Exploring the potential acceleration of the osmotic dehydration process via pretreatment with CO2-LASER microperforations

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JOURNAL OF FOOD ENGINEERING 2021年 306卷 110610-110610页

作者： Veloso, G. Simpson, R. Nunez, H. Ramirez, C. Almonacid, S. Jaques, A. Univ Tecn Federico Santa Maria Dept Ingn Quim & Ambiental Ave Espana 1680 Valparaiso Chile Ctr Reg Estudios Alimentos Saludables CREAS CONICYT Reg GORE Valparaiso Proyecto R17A10001 Ave Univ 330 Valparaiso Chile

Osmotic dehydration (OD) typically requires a long processing time, so the incorporation of CO2-laser microperforations could reduce it. The aims of this study were 1) to explore the potential acceleration of the OD process coupled with CO2-laser microperforations and 2) to mathematically evaluate the diffusion process through a complementary error function (erfc). Apple slices pretreated with two micropore sizes and two configurations were immersed in a solution of 45. Brix at 40 degrees C for 10 h. The total solid content in the apple slices was measured. A model based on erfc was used to determine the effective diffusion coefficient (D-eff). The results showed a significant increase in D-eff for microperforated samples, especially for those with the largest pores compared with control (R-2 > 0.92). Additionally, pores were able to reduce the processing time by up to 63% for big pores (600 mu m) and between 22% and 45% for small pores (23 mu m).

关键词： Osmotic dehydration CO2-laser microperforations Processing time complementary error function Effective diffusion coefficient

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The generalized Marcum function of the second kind: Monotonicity patterns and tight bounds

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 382卷 113093-113093页

We present a detailed analysis (monotonicity, convexity, recurrence relation, closed form expression and tight bounds) of a new special function which we call as the generalized Marcum function of the second kind, which is an analogous survival (or reliability) function to the so-called generalized Marcum Q-function or the generalized Marcum function of the first kind (the survival function of the non-central chi distribution). The main difference between these two generalized Marcum functions is that they involve the modified Bessel functions of the first and second kind. In this paper we are going to see how far the properties of the generalized Marcum function of the first kind may be extended to apply to the generalized Marcum function of the second kind. These two (linearly independent) special functions have similar properties, however in case of the generalized Marcum function of the second kind we need a little bit more technical analysis in order to prove our main results. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Modified Bessel function of the second kind Bounds Log-convexity and geometrical convexity Survival function complementary error function Generalized Marcum Q-function

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The Fast Physical Optics Method for Calculating the Scattered Field From the Rough Surface

The Fast Physical Optics Method for Calculating the Scattere...

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2020 International Wireless Symposium

作者： Song, Fengli Wu, Yumao Jin, Yaqiu Fudan Univ Key Lab Informat Sci EMW MoE Shanghai 200043 Peoples R China

ISBN: (纸本)9781728167039

In this paper, the fast physical optics (FPO) method for the scattering field of rough surface is proposed. The surface is discretized by quadratic patches. After transforming it into parameter coordinate system, the amplitude and function of physical optical integration are approximated by Lagrange interpolation polynomials. Finally, the closed-form formulas of FPO is obtained by using the complementary error function. Compared with the traditional method, FPO method can greatly reduce the number of patches when it is discrete, and the accuracy is within the acceptable error range. Numerical examples show the effectiveness of the proposed FPO method in accuracy.

关键词： Physical optics quadric surface discretization closed-form formulas complementary error function acceptable error range

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