rootsofunity: use mean value theorem for analysis (suggested by Damien Robert)

git-svn-id: svn://scm.gforge.inria.fr/svn/mpc/trunk@1197 211d60ee-9f03-0410-a15a-8952a2c7a4e4

1 files changed, 9 insertions, 14 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index cee55af..790eca1 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -431,32 +431,27 @@ Let
 \[
 \appro x = \cos {\appro {x_1}}.
 \]
-Using the addition formula for $\cos$,
+By the mean value theorem, there is a $\xi$ between $x_1$ and $\appro {x_1}$
+such that
 \[
-\cos (a + b) = \cos (a) \cos (b) - \sin (a) \sin (b),
+\cos (x_1) - \cos (\appro {x_1}) = -\sin (\xi) (x_1 - \appro {x_1}),
 \]
-we obtain
-\begin {eqnarray*}
+so that
+\[
 \error (\appro x)
-& \leq & |\cos (x)| (1 - \cos (\error (\appro {x_1})))
-+ |\sin (x) \sin (\error (\appro {x_1}))| \\
-& \leq & 2 \error (\appro {x_1})
-\end {eqnarray*}
-since $|\sin (\delta)|$, $1 - \cos (\delta) \leq \delta$
-(one even has $1 - \cos (\delta) \leq \frac {1}{2} \delta^2$,
-but this does not fundamentally improve the error bound).
-
+\leq \error (\appro {x_1}).
+\]
 Taking the exponents into account, one obtains
 \begin {equation}
 \label {eq:proprealcos}
 \error (\appro x)
 \leq
-2 k \, 2^{\Exp (\appro {x_1}) - \Exp (\appro x)}
+k \, 2^{\Exp (\appro {x_1}) - \Exp (\appro x)}
 \, 2^{\Exp (\appro x) - p}.
 \end {equation}
 
 For the sine function, a completely analogous argument shows that
-\eqref {eq:proprealcos} still holds.
+\eqref {eq:proprealcos} also holds.