Best bounds for the Lambert W functions
نویسندگان
چکیده
منابع مشابه
The Lambert W Function
has a countably infinite number of solutions, which are denoted by Wk(z) for integers k. Each choice of k specifies a branch of the Lambert W function. By convention, only the branches k = 0 (called the principal branch) and k = −1 are real-valued for any z; the range of every other branch excludes the real axis, although the range of W1(z) includes (−∞,−1/e] in its closure. Only W0(z) contains...
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where lnk z = ln z+2πik, and ln z is the principal branch of natural logarithm [14]. This paper considers only the principal branch k = 0, which is the branch that maps the positive real axis onto itself, and therefore we shall usually abbreviate W0 as W herein. Many functions of W are members of a number of function classes, specifically, the classes of Stieltjes functions, Pick functions and ...
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The Lambert W(x) function and its possible applications in physics are presented. The actual numerical implementation in C++ consists of Halley’s and Fritsch’s iterations with initial approximations based on branch-point expansion, asymptotic series, rational fits, and continued-logarithm recursion.
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We give a uniform treatment of several series expansions for the Lambert W function, leading to an innnite family of new series. We also discuss standardization, complex branches, a family of arbitrary-order iterative methods for computation of W, and give a theorem showing how to correctly solve another simple and frequently occurring nonlinear equation in terms of W and the unwinding number.
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An algorithm is proposed for computing primary matrix Lambert W functions of a square matrix A, which are solutions of the matrix equation WeW = A. The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton’s method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton’s method for the La...
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ژورنال
عنوان ژورنال: Journal of Mathematical Inequalities
سال: 2020
ISSN: 1846-579X
DOI: 10.7153/jmi-2020-14-80