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Diffstat (limited to 'doc')
| -rw-r--r-- | doc/algorithms.tex | 34 |
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} |