Solution of the transcendental equation wew = x
TL;DR: 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.
Abstract: Submittal of an algorithm for consideration for publication in Communications of the ACM implies unrestricted use of the algorithm within a computer is permissible. Description Purpose. WEW solves the transcendental equation we ~ = x for w, given x > 0, by an iteration that converges much more rapidly than either Newton's method or fixed-point iteration. The user provides x = X. The routine returns w = WEW and the last relative correction e, = EN. Two versions are described here. Version A produces CDC 6600 machine accuracy (48 bits), and the relative error should be approximately eJ. Version B produces at least six significant figures, and the relative error should be approximately eJ. Iteration. Assuming x > 0, we may rewrite the equation defining w as w + log(w) = log(x). (1) For a given approximation w. to w, let w~+l = w~ + 8, be a much better approximation. Substitution into (1) yields 8. +log(1-k-8,,/w.) = log x-log w.,-w. Using the approximation [1] log (1 + 8/w) ~ (Sw + 1/6 83)/ (w 2 + 2/3 8w) and clearing fractions yields the following quadratic equation for 8, : Solving for the root that tends to zero as z.-~ 0 gives 2ZnWn 8n (1 + w,-2/3 z,) + ((1 + w, + 2/3 z,) 2-2z,) ½ \" General permission to republish, but not for profit, an algorithm is granted, provided that reference is made to this publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. Work performed under the auspices of the U.S. Atomic Energy Commission. This has a continued fraction expansion [3] 2WnZn 8n = 2Zn 2(1 W w,)-2(1 + w, + 2/3 z,) 2z, 2(1 + w.-1-2/3 z.) for which the third convergent yields sufficient accuracy. If we ignore the quantity 2/3 z, in the third term, we obtain the iteration formula Initial guesses. For small values of x, the given equation has a series solution due to L. Euler [2]. A Pad~ rational fraction approximation to this series is 1 +7/3x+ 5/6x 2\" As computed from (5), wo(x) < w(x), good to within 5 percent if x = 2.5 and much better for smaller values of x. For larger values of x we may use wo = log(x), (6) which has a maximum relative error no greater than 37 percent for x …
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TL;DR: 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.
Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we
w
. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, 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 containingW.
5,591 citations
TL;DR: In this paper, an extension of sinc interpolation to algebraically decaying functions is presented, where the algebraic order of decay of a function's decay can be estimated everywhere in the horizontal strip of complex plane around the complex plane.
Abstract: An extension of sinc interpolation on $\mathbb{R}$ to the class of
algebraically decaying functions is developed in the paper. Similarly to the
classical sinc interpolation we establish two types of error estimates. First
covers a wider class of functions with the algebraic order of decay on
$\mathbb{R}$. The second type of error estimates governs the case when the
order of function's decay can be estimated everywhere in the horizontal strip
of complex plane around $\mathbb{R}$. The numerical examples are provided.
1,000 citations
TL;DR: In this article, an exact closed-form solution based on Lambert W -function is presented to express the transcendental currentvoltage characteristic containing parasitic power consuming parameters like series and shunt resistances.
449 citations
TL;DR: In this paper, the Lambert W is defined as a transcendental function defined by solutions of the equation W exp(W) = x, and a survey of the literature reveals that, in the case of the principal branch (W-0), the vast majority of W-function applications use, at any given time, only a portion of the branch viz.
228 citations
TL;DR: In this article, Monte Carlo simulation (MCS) and analytical technique are used in this work with a novel utilization of the clearness index probability density function (pdf) to model the solar irradiance using MCS.
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214 citations
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01 Jun 1967
TL;DR: In this article, a convergence theory of positive definite continued fractions is presented. But the convergence theory is not a generalization of the Stieltjes convergence theorem, and the convergence of continued fractions whose partial denominators are equal to unity is not discussed.
Abstract: Part I: Convergence Theory: The continued fraction as a product of linear fractional transformations Convergence theorems Convergence of continued fractions whose partial denominators are equal to unity Introduction to the theory of positive definite continued fractions Some general convergence theorems Stieltjes type continued fractions Extensions of the parabola theorem The value region problem Part II: Function Theory: J-fraction expansions for rational functions Theory of equations J-fraction expansions for power series Matrix theory of continued fractions Continued fractions and definite integrals The moment problem for a finite interval Bounded analytic functions Hausdorff summability The moment problem for an infinite interval The continued fraction of Gauss Stieltjes summability The Pade table Bibliography Index.
1,640 citations