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Diffstat (limited to 'doc')
| -rw-r--r-- | doc/algorithms.tex | 38 |
1 files changed, 37 insertions, 1 deletions
diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 6ca7454..cee55af 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -46,7 +46,7 @@ \title {MPC: Algorithms and Error Analysis} \author {Andreas Enge \and Philippe Th\'eveny \and Paul Zimmermann} -\date {Draft; June 16, 2010} +\date {Draft; June 27, 2012} \begin {document} \maketitle @@ -424,6 +424,42 @@ Then by \cite[\S1.7]{MPFRAlgorithms}, \end {equation} +\subsubsection {Cosine and sine} +\label {sssec:proprealcossin} + +Let +\[ +\appro x = \cos {\appro {x_1}}. +\] +Using the addition formula for $\cos$, +\[ +\cos (a + b) = \cos (a) \cos (b) - \sin (a) \sin (b), +\] +we obtain +\begin {eqnarray*} +\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). + +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)} +\, 2^{\Exp (\appro x) - p}. +\end {equation} + +For the sine function, a completely analogous argument shows that +\eqref {eq:proprealcos} still holds. + + + \subsection {Complex functions} |