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y=x*e^xの逆関数をランベルトの関数Ｗ(x)と言うらしく
ただこれを初等関数であらわすことはできず、これを使う為には漸化式等を用いて入手するらしい。
この関数はいろいろと使い道がとれそうで便利と感じる。
以下サイトを参考に調べてみました。
There have been quite a few posts recently regarding problems whose
solutions involve the Lambert W-function. I have an iterative algorithm
that I use for computing this function that converges fairly quickly.
First, let's look at the inverse of W(x), w e^w.
Analysis of x e^x
-----------------
For w > 0, w e^w increases monotonically from 0 to infinity. When w is
negative, w e^w is negative. Thus, for x > 0, W(x) is positive and well
defined and increases monotonically.
For w < 0, w e^w reaches a minimum of -1/e at -1. On (-1,0), w e^w
increases monotonically from -1/e to 0. On (-oo,-1), w e^w decreases
monotonically from 0 to -1/e. Thus, on (-1/e,0), W(x) can have one of
two values, one in (-1,0) and another in (-oo,-1). The value in (-1,0)
is called the principal value.
（ここまではグラフ等を書いて理解しました。）
The iteration
-------------
Using Newton's method for solving w e^w yields the following iterative
step for finding W(x):
x e^-w + w^2
w = ------------ [1]
new w + 1
Initial values of w
-------------------
For the principal value when -1/e <= x < 0 and when 0 <= x <= 10, use
w = 0. When x > 10, use w = log(x) - log(log(x)).
For the non-principal value, if x is in [-1/e,-.1], use w = -2; and
if x is in (-.1,0), use w = log(-x) - log(-log(-x)).
Another iteration
-----------------
When x is near -1/e, [1] converges a little slowly. I use a parabolic
iteration method in those cases.
x + 1/e
w = -1 + (w+1) sqrt( ----------- ) [2]
new w e^w + 1/e
Using an initial w of 0 for the principal value, and w = -2 for the
non-principal value.
この説明文でNewton's method とかparabolic iteration method の意味や
プログラムの組み方がチンプンカンです。
このＷ関数を使えるようにプログラムを構成して頂けないでしょうか？
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