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Diffstat (limited to 'doc/algorithms.tex')
| -rw-r--r-- | doc/algorithms.tex | 23 |
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. |