From d52b9345e935e26971639dcda8970694140f7298 Mon Sep 17 00:00:00 2001
From: enge <enge@211d60ee-9f03-0410-a15a-8952a2c7a4e4>
Date: Thu, 28 Jun 2012 11:46:36 +0000
Subject: 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
---
 doc/algorithms.tex | 34 +++++++++++++++++++++++++++++++++-
 1 file changed, 33 insertions(+), 1 deletion(-)

(limited to 'doc')

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}
-- 
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