log: alternative implementation that avoids intermediate overflows

     It is probably slower (two calls to log) and should be combined with
     the previous approach.
     Problem of "underflow" (log of number close to 1) not yet solved.


git-svn-id: svn://scm.gforge.inria.fr/svn/mpc/trunk@1202 211d60ee-9f03-0410-a15a-8952a2c7a4e4

1 files changed, 33 insertions, 1 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index 790eca1..edad662 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -3,6 +3,7 @@
 \usepackage[a4paper]{geometry}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
+\usepackage{ae}
 \usepackage{amsmath,amssymb}
 \usepackage{hyperref}
 \usepackage{comment}
@@ -46,7 +47,7 @@
 
 \title {MPC: Algorithms and Error Analysis}
 \author {Andreas Enge \and Philippe Th\'eveny \and Paul Zimmermann}
-\date {Draft; June 27, 2012}
+\date {Draft; June 28, 2012}
 
 \begin {document}
 \maketitle
@@ -454,6 +455,37 @@ For the sine function, a completely analogous argument shows that
 \eqref {eq:proprealcos} also holds.
 
 
+\subsubsection {Logarithm}
+\label {sssec:propreallog}
+
+Let
+\[
+\appro x = \log (1 + \appro {x_1})
+\]
+for $\appro {x_1} > -1$.
+By the mean value theorem, there is a $\xi$ between $x_1$ and $\appro {x_1}$
+such that
+\[
+\error (\appro x) = \frac {1}{1 + \xi} \error (\appro {x_1})
+\leq \frac {1}{1 + \min (x_1, \appro {x_1})} \error (\appro {x_1}).
+\]
+For $x_1 > 0$, this implies
+\begin {eqnarray*}
+\error (\appro x)
+& \leq & \error (\appro {x_1})
+\leq
+k \, 2^{\Exp (\appro {x_1}) - \Exp (\appro x)}
+\, 2^{\Exp (\appro x) - p} \\
+& \leq & 2 \, k \, \frac {\appro {x_1}}{\appro x} \, 2^{\Exp (\appro x) - p} \\
+& \leq & 2 \, k \, \frac {\appro {x_1}}{\appro {x_1} - \appro {x_1}^2/2}
+\, 2^{\Exp (\appro x) - p}
+\end {eqnarray*}
+using $\log (1 + z) \geq z - z^2/2$ for $z > 0$.
+For $0 < x_1 \leq 1$, we have $\appro {x_1}^2/2 \leq \appro {x_1}/2$ and
+\[
+\error (\appro x)
+\leq 4 \, k \, 2^{\Exp (\appro x) - p}.
+\]
 
 
 \subsection {Complex functions}