Corless Robert, Gonnet Gaston, Hare D.E.G., Jeffrey D.J., Knuth Donald, On the Lambert W function, Advances in Computational Mathematics 5 (1996), pp. 329--359. See also the documentation of the Maple SHARE Library.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the Lambert function defined implicitly by the equation W (z)e W (z) z, 2) without using any symbolic data. This function is used in many branches of computational physics and in combinatorics. Many interesting differential equations have closed solutions in terms of W . Corless et al. [6] discuss the existence and the analyticity properties of this function. The differentiation of (2) gives dz dW = e W (1 W ) # = z W (1 W ) for z #= 0 # (3) diffalg.tex; 15 09 2000; 3:14; p.16 17 whose inverse dW dz = e W 1 W # = W z 1 1 W # (4) gives a one line code ....

.... 2.0, 9.0, 64.0, 625.0, 7776.0, 117649.0, 2097152, which agrees with the known theoretical values: W (n) 0) n) n 1 . If we insert the formula (4) into any program which calculates numerically W (x) for any x #= 0, for example using the Newton or Haley approximation [6]) we obtain all its derivatives at any point. Can we use the second, apparently cheaper form of (4) which does not use the exponential For z #= 0 naturally yes, provided we knew independently the value of W (z) But lazy algorithms sometimes need some intelligent reformulation in order to ....

Corless Robert, Gonnet Gaston, Hare D.E.G., Jeffrey D.J., Knuth Donald, On the Lambert W function, Advances in Computational Mathematics 5 (1996), pp. 329--359. See also the documentation of the Maple SHARE Library.

....form would be much faster, but this first stage is much more costly. Maple using the equivalent procedure (and reusing all lower order forms) chokes before n = 24. Other recurrence schemes are more suitable. 4.2. LAMBERT FUNCTION We find the Taylor expansion around zero of the Lambert function [7] given by the equation W (z)e W (z) z, 2) without using any symbolic data. This function is used in many branches of computational physics and in combinatorics. The differentiation of (2) gives dz dW = e W (1 W ) # = z W (1 W ) for z #= 0 # (3) whose inverse dW dz = e W ....

....0.0, 1.0, 2.0, 9.0, 64.0, 625.0, 7776.0, 117649.0, 2097152, which agrees with the known theoretical values: W (n) 0) n) n 1 . If we insert the formula (4) into any program which calculates numerically W (x) for any x #= 0, for example using the Newton or Haley approximation [7]) we obtain all its derivatives at any point. Can we use the second, apparently cheaper form of (4) which does not use the exponential For z #= 0 naturally yes, provided we knew independently the value of W (z) Lazy algorithms sometimes need some intelligent re formulation in order to ....

Corless Robert, Gonnet Gaston, Hare D.E.G., Jeffrey D.J., Knuth Donald, On the Lambert W function, Advances in Computational Mathematics 5 (1996), pp. 329--359. See also the documentation of the Maple SHARE Library.

....map (hrm 24) 8.0, 7.9 . 8.0] before plotting the obtained sequence. The example is a bit contrived, but it works in practice without problems. It is almost as efficient as other recurrent formulae. 4. 2 Lambert function We will find the Taylor expansion around zero of the Lambert function [16] given by the equation W (z)e W (z) z; 2) without using any symbolic manipulations. This functions is used in many branches of computational physics, and in combinatorics. We see immediately that dz dW = e W (1 W ) i = z W (1 W ) for z 6= 0 j (3) whose inverse dW dz = e ....

.... 2.0, 9.0, 64.0, 625.0, 7776.0, 117649.0, 2.09715e 006, which agrees with the known theoretical values: W (n) 0) Gamman) n Gamma1 . Accessorily we see that if we plug in the formula (4) into any program which calculates W (x) for example using the Newton or Haley approximation [16]) we obtain all its derivatives at any point. Can we use the second, apparently cheaper form of (4) which does not use the exponential For z 6= 0 obviously yes, provided we knew W using some numerical technique. Lazy algorithms needs sometimes some intelligent pre processing in order to transform ....

R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W Function, Advances in Computational Mathematics 5 (1996), pp. 329--359. See also the documentation of the Maple SHARE Library.

....be dangerous. 3 Applications From the de nition of the lifted exponential and other non polynomial functions we see how to get for free the full Taylor expansion when the rst derivative is recursively known. 3. 1 The Lambert Function A nice case is the de nition of the Lambert W function [6] which ful ls the equation W (z)e W (z) z: 1) Its derivative is easily obtained by inverting dz=dW : dW dz = e W 1 W = W z 1 1 W ; 2) and that gives the following code for the McLaurin sequence of W , knowing that W (0) 0. wl = D 0.0 (exp (negate wl) 1.0 wl) numerically ....

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jerey, D. E. Knuth, On the Lambert W Function, Advances in Comp. Mathematics 5 (1996), pp. 329-359. See also the documentation of the Maple SHARE Library.

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