diff options
| -rw-r--r-- | doc/algorithms.tex | 33 |
1 files changed, 33 insertions, 0 deletions
diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 775f1a6..32f2254 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1001,6 +1001,39 @@ we find the exact same error estimate \eqref {eq:propmulcomrel} also for the case of division. +\subsubsection {Square root} +Let +\[ +\appro z = \sqrt {\appro {z_1}}. +\] +Write $\corr {z_1} = (1 + \theta_1) \appro {z_1}$ with +$\epsilon_1 = |\theta_1|$, and assume $\epsilon_1 < 1$. +Then $\corr z = (1 + \theta) \appro z$ with +\[ +\theta = \sqrt {1 + \theta_1} - 1 += \frac {1}{2} \theta_1 ++ \sum_{n=2}^\infty \frac {(-1)^{n+1} 1 \cdot 3 \cdots (2 n - 3)}{2^n \, n!} + \theta_1^n +\] +as a Taylor series, and +\[ +\epsilon = |\theta| +\leq +\frac {1}{2} \epsilon_1 ++ \sum_{n=2}^\infty \frac {1 \cdot 3 \cdots (2 n - 3)}{2^n \, n!} +\epsilon_1^n += 1 - \sqrt {1 - \epsilon_1}. +\] +By the mean value theorem, applied to the function $f (x) = \sqrt {1 - x}$, +there is $0 < \xi < \epsilon_1$ with +\begin {equation} +\label {eq:propsqrt} +\epsilon = \frac {1}{2 \sqrt {1 - \xi}} \, \epsilon_1 +\leq \frac {1}{2 \sqrt {1 - \epsilon_1}} \, \epsilon_1. +\end {equation} +For instance $\epsilon \leq \epsilon_1$ for $\epsilon_1 \leq \frac {3}{4}$. + + \subsubsection {Logarithm} |