Generalized logarithm

As a generalized logarithm and generalized exponential be special functions designated which have similar growth characteristics and relationships to each other have such logarithm and exponential function and certain functional equations are defined iteratively starting from one interval to the real axis.

They were introduced in 1986 by Charles William Clenshaw , Daniel W. Lozier, Frank WJ Olver and Peter R. Turner, although there are predecessors in literature. The main application is in floating point arithmetic .

definition

A generalized exponential function fulfills the following three conditions:

${\displaystyle \phi (x+1)=e^{\phi (x)}}$, For ${\displaystyle -1<x<\infty }$

${\displaystyle \phi (0)=0}$

${\displaystyle \phi (x)}$ is strictly increasing for ${\displaystyle 0\leq x\leq 1}$

As usual, this is the usual exponential function (and in the following it is the natural logarithm, Euler's number). ${\displaystyle e^{x}}$ ${\displaystyle \ln x}$ ${\displaystyle e}$

${\displaystyle \phi (x)}$is strictly monotonically increasing from to if from to increasing and thus has an inverse to , the associated generalized logarithm . ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle x}$ ${\displaystyle -1}$ ${\displaystyle \infty }$ ${\displaystyle (-\infty ,\infty )}$ ${\displaystyle \psi (x)}$

The following applies to the generalized logarithm : ${\displaystyle \psi (x)}$

${\displaystyle \psi (e^{x})=1+\psi (x)}$, For ${\displaystyle -\infty <x<\infty }$

${\displaystyle \psi (0)=0}$

${\displaystyle \psi (x)}$ is strictly increasing for ${\displaystyle 0\leq x\leq 1}$

The values at the integer points are equal: , , etc. As with the gamma function , the complete function of the values to be constructed at the integer points. ${\displaystyle \phi (1)=1}$ ${\displaystyle \phi (2)=e}$ ${\displaystyle \phi (3)=e^{e}}$

The solution is not clear, but depends on the choice of growth in the interval . The easiest choice is to pretend: ${\displaystyle [0,1]}$

${\displaystyle \phi (x)=x}$in the interval . ${\displaystyle [0,1]}$

This also corresponds to the main applications in floating point arithmetic (see below). Then follows:

${\displaystyle \phi (x)=\ln(x+1)}$, For ${\displaystyle -1<x\leq 0}$

${\displaystyle \phi (x)=e^{x-1}}$, For ${\displaystyle 1\leq x\leq 2}$

and generally after -fold iteration: ${\displaystyle l}$

${\displaystyle \phi (x)=e^{e^{e^{...^{e^{x-l}}}}}}$, For ${\displaystyle l\leq x\leq l+1}$

Analog for the logarithm:

${\displaystyle \psi (x)=x}$, For ${\displaystyle 0\leq x\leq 1}$

${\displaystyle \psi (x)=e^{x}-1}$, For ${\displaystyle -\infty <x\leq 0}$

${\displaystyle \psi (x)=1+\ln x}$, For ${\displaystyle 1\leq x\leq e}$

and generally by -fold iteration (with the times the iteration of common logarithm): ${\displaystyle l}$ ${\displaystyle \ln ^{l}x}$ ${\displaystyle l}$

${\displaystyle \psi (x)=l+\ln ^{l}x}$, For ${\displaystyle l\leq x}$

and one that is determined by . ${\displaystyle l}$ ${\displaystyle 0\leq \ln ^{l}x<1}$

The first derivative of is continuous at , the second derivative has a jump from to (corresponding to the other integer places). ${\displaystyle \phi (x)}$ ${\displaystyle x=1}$ ${\displaystyle 0}$ ${\displaystyle 1}$

application

The functions find application in a representation of real numbers for computer precision arithmetic called level index arithmetic (LI), which was introduced by Clenshaw and Olver in 1984. In floating point arithmetic, a compromise has to be found between precision and the ability to represent very large numbers. In the LI, numbers are represented by iterating the exponential function, with the degree of iteration being referred to as level . ${\displaystyle l}$

${\displaystyle x=e^{e^{e^{...^{e^{f}}}}}}$

The exponent is the index. Example:: is displayed as ${\displaystyle 0\leq f<1}$ ${\displaystyle x=1234567=e^{e^{e^{0.9711308}}}}$

${\displaystyle x=l+f=3+0.9711308=3.9711308}$.

literature

• CW Clenshaw, DW Lozier, FWJ Olver, PR Turner: Generalized exponential and logarithmic functions . In: Computers & Mathematics with Applications . tape 12 , no. 5-6 , 1986, pp. 1091-1101 , doi : 10.1016 / 0898-1221 (86) 90233-6 .
• CW Clenshaw, FWJ Olver, PR Turner: Level-index arithmetic: An introductory survey , in: Turner (Ed.), Numerical Analysis and Parallel Processing. Lecture Notes in Mathematics. 1397, 1989, pp. 95-168.

• Hellmuth Kneser : Real analytical solutions of the equation and related functional equations ${\displaystyle \varphi (\varphi (x))=e^{x}}$ . In: J. Reine Angew. Math . tape 187 , 1950, ISSN  0075-4102 , p. 56-67 ( uni-goettingen.de ).

Individual evidence

1. ↑ Generalized Logarithms and Exponentials. Retrieved June 6, 2018 . 
2. ↑ Peter Walker: Infinitely Differentiable Generalized Logarithmic and Exponential Functions . In: Mathematics of Computation . tape 54 , no. 196 , 1991, pp. 723-733 ( eretrandre.org [PDF]).
3. ↑ Clenshaw et al. a., Computers & Mathematics with Applications, Volume 12, 1986, p. 1091
4. ↑ Clenshaw, Olver, Beyond floating point arithmetic , Journal of the Association for Computing Machinery, Volume 31, 1984, pp. 319-328