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About 10,000 papers for “Error function”

Efficient Computation of the Complex Error Function

Walter Gautschi · SIAM Journal on Numerical Analysis · 1970

The paper is concerned with the computation of $w(z) = \exp ( - z^2 ){\operatorname{erfc}}( - iz)$ for complex $z = x + iy$ in the first quadrant $Q_1 :x \geqq 0,y \geqq 0$. Using Stieltjes– theory of continued fractions it is first observed that the Laplace continued fraction for $w(z)$, although divergent on the real line, represents $w(z)$ asymptotically for $z \to \infty $ in the sector $ S:{{ - \pi} / 4} < \arg z < {{5\pi } / 4}$. Specifically, the nth convergent approximates $w(z)$ to within an error of $O(z^{ - 2n - 1} 1)$ as $z \to \infty $ in S. A recursive procedure is then developed which permits evaluating $w(z)$ to a prescribed accuracy for any $z \in Q_1 $. The procedure has the property that as $| z |$ becomes sufficiently large, it automatically reduces to the evaluation of the Laplace continued fraction, or, equivalently, to Gauss–Hermite quadrature of $({i / \pi })\int_{ - \infty }^\infty {\exp ( - t^2 ){{dt} / {(z - t)}}} $.

Cited by 272

Zero-error function computation in sensor networks

Hemant Kowshik, P. Rajesh Kumar · 2009

We consider the problem of data harvesting in wireless sensor networks. A designated collector node seeks to compute a function of the sensor measurements. For a directed graph G = (V ,¿) on the sensor nodes, we wish to determine the optimal encoders on each edge which achieve zero-error block computation of the function at the collector node. Our goal is to characterize the rate region in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|¿|</sup> . We start with the two node problem, and determine a necessary and sufficient condition for the encoder that yields the optimal alphabet, from which we then calculate the minimum worst case and average case complexity. We then extend this result to trees and derive a necessary and sufficient condition for the encoder on each edge. The further extension of these results to directed acyclic graphs is not immediate. We provide an outer bound on the rate region by finding the disambiguation requirements for each cut, and describe examples where this outer bound is tight. Finally, we consider a collocated network of nodes with a specified order of transmission. We determine a necessary and sufficient condition for each encoder which is based on the transmissions received, and show that the average case complexity of computing a type-threshold function is ¿(1), in comparison to the worst case complexity of ¿(log n).

Cited by 35

SADIK TRANSFORM OF ERROR FUNCTION (PROBABILITY INTEGRAL)

Sudhanshu Aggarwal*1 & Swarg Deep Sharma2 · Zenodo (CERN European Organization for Nuclear Research) · 2019

The solutions of many advanced engineering problems like Fick’s second law, heat and mass transfer problems, vibrating beams problems contains error and complementary error function. When we use any integral transform to solve these types of problems, it is very necessary to know the integral transform of error function. In this article, we find the Sadik transform of error and complementary error functions. To demonstrate the usefulness of Sadik transform of error function, some numerical applications are considered in application section for solving improper integrals which contain error function. It is pointed out that Sadik transform give the exact solution of improper integral which contains error function without any tedious calculation work.

Cited by 0Open access

ACUITY-LUMINANCE FUNCTION FOR EXTREME REFRACTIVE ERROR*

S. M. Luria, Jo Ann S. Kinney · Optometry and Vision Science · 1970

Visual acuity was measured at various luminances in air while subjects wore negative lenses ranging through 60 diopters. The slopes of the acuity-luminance functions decreased with increasing refractive error. With the more powerful lenses, the slope was similar to that found for scotopic acuity. The loss of acuity with negative lenses was much less than that reported for positive lenses. Acuity was also measured under water without facemasks to ensure that these results could be used to predict performance under such conditions. The simulation in air was apparently successful. It was found that, in the water, acuity was at best one-tenth that in air, but there were marked differences for various groups. Emmetropes suffered the greatest and myopes suffered the least loss. There was no correlation between emmetropic acuity in air and in water.

Cited by 4

Function Hiding Based on Error Correcting Codes

2008

This paper presents an original approach to the problem of function hiding based on Error Correcting Codes and evaluates the security of this approach. The novelty of the technique consists in using Error Correcting Codes to hide functions instead of encrypting data vectors. This protocol mainly deals with the issue of secure evaluation of functions in potentially hostile environments. 1

Cited by 0

ACUITY‐LUMINANCE FUNCTION FOR EXTREME REFRACTIVE ERROR

S. M. Luria, Jo Ann S. Kinney · American journal of optometry and archives of American Academy of Optometry · 1970

ABSTRACT Visual acuity was measured at various luminances in air while subjects wore negative lenses ranging through 60 diopters. The slopes of the acuity‐luminance functions decreased with increasing refractive error. With the more powerful lenses, the slope was similar to that found for scotopic acuity. The loss of acuity with negative lenses was much less than that reported for positive lenses. Acuity was also measured under water without facemasks to ensure that these results could be used to predict performance under such conditions. The simulation in air was apparently successful. It was found that, in the water, acuity was at best one‐tenth that in air. but there were marked differences for various groups. Emmetropes suffered the greatest and myopes suffered the least loss. There was no correlation between emmetropic acuity in air and in water.

Cited by 0

Saddle Points of the Complementary Error Function

Henry E. Fettis, James C. Caslin, Kenneth R. Cramer · Mathematics of Computation · 1973

The first one hundred zeros of the derivative of the function w(z) = e~'' Erfc(-iz) are given, together with an asymptotic formula for estimating the higher zeros.

Cited by 4

Error estimates for radial basis function interpolation

Holger Wendland · Cambridge University Press eBooks · 2004

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 2

Inverse Error Function Trajectories for Image Reconstruction

Rohan Katoch, Beatriz Fusaro, Jun Ueda · 2018

Capturing clear images while a camera is moving fast, is integral to the development of mobile robots that can respond quickly and effectively to visual stimuli. This paper proposes to generate camera trajectories, with position and time constraints, that result in higher reconstructed image quality. The degradation in of an image captured during motion is known as motion blur. Three main methods exist for mitigating the effects of motion blur: (i) controlling optical parameters, (ii) controlling camera motion, and (iii) image reconstruction. Given control of a camera's motion, trajectories can be generated that result in an expected blur kernel or point-spread function. This work compares the motion blur effects and reconstructed image quality of three trajectories: (i) linear, (ii) polynomial, and (iii) inverse error. Where inverse error trajectories result in Gaussian blur kernels. Residence time analysis provides a basis for characterizing the motion blur effects of the trajectories.

Cited by 1

The zeros of the complementary error function

LAFORGIA, Andrea Ivo Antonio, A. ELBERT · Iris (Roma Tre University) · 2008

“We show that the complementary error function,&#13;\nerfc(z) =&#13;\n2&#13;\np&#13;\n&#13;\nZ 1&#13;\nz&#13;\ne−s2&#13;\nds&#13;\nhas no zeros in D = {z: 3&#13;\n4 Arg z 5&#13;\n4}.”

Cited by 0

Multivariate error function based neural network approximations

George A. Anastassiou · Journal of Numerical Analysis and Approximation Theory · 2014

Here we present multivariate quantitative approximations of real and complex valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate quasi-interpolation, Baskakov type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last three types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the Gaussian error special function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Cited by 1Open access

Error function aproximations of interest in Electrochemistry

Romeu [UNESP] Magnani, Gilberto [UNESP] Daun, Assis Vicente [UNESP] Benedetti · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 1996

The error function is present in several equations describing eletrode processes. But, only approximations of this function are used. In this work, these and other approximations are studied and evaluated according to precision.

Cited by 0Open access