Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2

@article{ChapeauBlondeau2002NumericalEO,
  title={Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2},
  author={François Chapeau-Blondeau and Abdelilah Monir},
  journal={IEEE Trans. Signal Process.},
  year={2002},
  volume={50},
  pages={2160-2165}
}
We address the problem of synthesizing a generalized Gaussian noise with exponent 1/2 by means of a nonlinear memoryless transformation applied to a uniform noise. We show that this transformation is expressable in terms of a special function known under the name of the Lambert W function. We review the main methods for numerical evaluation of the relevant branch of the (multivalued) Lambert W function with controlled accuracy and complement them with an original rational function approximation… Expand
View via Publisher
perso-laris.univ-angers.fr
181 Citations
Highly Influential Citations
8
Background Citations
54
Methods Citations
34
Results Citations
1

181 Citations

Citation Type

Citation Type

Spline approximations of the Lambert W function and application to simulate generalized Gaussian noise with exponent α = 1/2
TLDR
A new procedure to evaluate this special function of the cumulative distribution function in terms of the Lambert W function using spline functions is developed, which is better than the approach of Chapeau-Blondeau and Monir. Expand
Comments on "numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2"
TLDR
Iterative improvement of the W/sub -1/ estimates is discussed, and issues concerning computational efficiency and possible sources of rounding error in fixed precision computational environments are identified. Expand
On the Generation of Gaussian Noise
TLDR
It is shown that the transformation of the Lambert W function required development of a rational approximation as well as a C program and could be performed more easily for a generalized Gaussian noise with any exponent. Expand
New approximations to the principal real-valued branch of the Lambert W-function
TLDR
A simple, global analytic approximation is derived that covers the whole branch of the W-function with a maximum relative error smaller than 5 × 10−3, and machine precision accuracy is reached everywhere with only three steps of a quadratically convergent iterative scheme. Expand
PSEM Approximations for Both Branches of Lambert W Function with Applications
Transcendental functions are a fundamental building block of science and engineering. Among them, a relatively new function denominated as Lambert W is highlighted. The importance of such function… Expand
Precise and fast computation of Lambert W-functions without transcendental function evaluations
  • T. Fukushima
  • Computer Science, Mathematics
  • J. Comput. Appl. Math.
  • 2013
TLDR
A new method to compute the real-valued Lambert W-functions, W"0(z) and W"-"1(z), which requires no call of transcendental functions such as the exponential function or the logarithm. Expand
Efficient procedure to generate generalized Gaussian noise using linear spline tools
TLDR
A simple method to generate generalized Gaussian noises using the inverse transform of cumulative distribution and a new initial value exploiting the close relationship between the incomplete Gamma function and its piecewise linear interpolant is proposed. Expand
Simulating a Generalized Gaussian Noise with Shape Parameter 1/2
This contribution deals with Monte Carlo simulation of generalized Gaussian random variables. Such a parametric family of distributions has been proposed for many applications in science and… Expand
A method of synthesizing generalized Gaussian noise based on spline approximants
Abstract A method of synthesizing generalized Gaussian noise processes based on spline approximation is presented. We formulate an algorithm that computes a sequence of convergent spline approximants… Expand
Numerical computation of inverse complete elliptic integrals of first and second kinds
  • T. Fukushima
  • Computer Science, Mathematics
  • J. Comput. Appl. Math.
  • 2013
TLDR
The developed procedures are so precise that the maximum absolute errors are less than 3-5 machine epsilons, and 30%-40% faster than the evaluation of the integrals themselves by the fastest procedures. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 24 REFERENCES
On the LambertW function
TLDR
A new discussion of the complex branches of W, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W are presented. Expand
Lattice vector quantization of generalized Gaussian sources
TLDR
Algorithm for defining and quantizing to a Z lattice in which the boundary is optimized to the characteristics of GG sources lead to high performance and low complexity for bit rates and dimensions that are of interest in a number of practical coding applications. Expand
Solution of the transcendental equation wew = x
TLDR
WEW solves the transcendental equation the authors ~ = x for w, given x > 0, by an iteration that converges much more rapidly than either Newton's method or fixed-point iteration. Expand
Analysis of Multiresolution Image Denoising Schemes Using Generalized Gaussian and Complexity Priors
TLDR
This paper investigates various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors, and introduces a new family of complexity priors based upon Rissanen's universal prior on integers. Expand
Estimation based on entropy matching for generalized Gaussian PDF modeling
TLDR
A novel method for estimating the shape factor of a generalized Gaussian probability density function that relies on matching the entropy of the modeled distribution with that of the empirical data is presented and assessed. Expand
Nonlinear test statistic to improve signal detection in non-Gaussian noise
TLDR
It is shown that this nonlinear detector can achieve smaller probability of error compared to the linear detector, especially occurs for non-Gaussian noises with heavy tails or a leptokurtic character. Expand
Estimation of shape parameter for generalized Gaussian distributions in subband decompositions of video
TLDR
A simple alternative method to estimate the shape parameter for the generalized Gaussian PDF is proposed that significantly reduces the number of computations by eliminating the need for any statistical goodness-of-fit test. Expand
Real values of the W-function
TLDR
With these approximations an efficient algorithm, consisting of a single iteration of a rapidly converging iteration scheme, gives estimates of <italic>W(x) accurate to at least 16 significant digits (15 digits if double precision is used). Expand
Numerical Recipes in C: The Art of Sci-entific Computing
TLDR
This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes, with over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines. Expand
On the modeling of DCT and subband image data for compression
TLDR
Image subband and discrete cosine transform coefficients are modeled for efficient quantization and noiseless coding and pyramid codes for transform and subband image coding are selected. Expand
...
1
2
3
...