From 310ebdd32c1d7679c3aa560c61b4a6dafd67aae0 Mon Sep 17 00:00:00 2001 From: thevenyp Date: Wed, 29 Jul 2009 16:38:18 +0000 Subject: doc/algorithms.tex: fix errors and typos. git-svn-id: svn://scm.gforge.inria.fr/svn/mpc/trunk@628 211d60ee-9f03-0410-a15a-8952a2c7a4e4 --- doc/algorithms.tex | 156 ++++++++++++++++++++++++++++++----------------------- 1 file changed, 90 insertions(+), 66 deletions(-) diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 83dd3ea..bee5b3f 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1386,11 +1386,11 @@ of $x_1$ (resp. $x_2$, $y_1$, $y_2$). dB_0(x_1, +0, +0, +0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2) &> 0 \;\text{if}\; x_1 \leq -1 \\ dB_0(x_1, +0, +0, -0)\cdot(\delta_1, \delta_2, \epsilon_1, - \epsilon_2) &> 0 \;\text{if}\; 1 \leq x_1 < 0 \\ + \epsilon_2) &> 0 \;\text{if}\; -1 \leq x_1 < 0 \\ dB_0(x_1, +0, -0, -0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2) &< 0 \;\text{if}\; x_1 \leq -1\\ dB_0(x_1, +0, -0, +0)\cdot(\delta_1, \delta_2, \epsilon_1, - \epsilon_2) &< 0 \;\text{if}\; 1 \leq x_1 < 0 + \epsilon_2) &< 0 \;\text{if}\; -1 \leq x_1 < 0 \end {align*} and the sign of $dB_k(x_1, +0, \rho_1 0, \rho_2 0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2)$ is not constant in all other @@ -1416,15 +1416,14 @@ of $x_1$ (resp. $x_2$, $y_1$, $y_2$). Let $\phi > 0$, then $x_2$ is not null or $x_1 < 0$. Using the expression of the derivative given above, we have \begin{align*} - dB_0(x_1, x_2, +0, +0)\cdot(\delta_1, \delta_2, \epsilon_1, - \epsilon_2) &> 0 \;\text{if}\; |x| \geq 1 \;\text{and}\; x \neq +1+0i\\ - dB_0(x_1, x_2, -0, -0)\cdot(\delta_1, \delta_2, \epsilon_1, - \epsilon_2) &< 0 \;\text{if}\; |x| \geq 1 \;\text{and}\; x \neq +1+0i\\ - dB_0(x_1, x_2, +0, -0)\cdot(\delta_1, \delta_2, \epsilon_1, - \epsilon_2) &> 0 \;\text{if}\; 1 \geq |x| > 0 \;\text{and}\; x \neq +1+0i\\ - dB_0(x_1, x_2, -0, +0)\cdot(\delta_1, \delta_2, \epsilon_1, - \epsilon_2) &< 0 \;\text{if}\; 1 \geq |x| > 0 \;\text{and}\; x - \neq +1+0i + dB_0(x_1, x_2, +0, +0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2) + &> 0 \;\text{if}\; |x| \geq 1 \;\text{and}\; \pi \geq \phi > 0\\ + dB_0(x_1, x_2, -0, -0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2) + &< 0 \;\text{if}\; |x| \geq 1 \;\text{and}\; \pi \geq \phi > 0\\ + dB_0(x_1, x_2, +0, -0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2) + &> 0 \;\text{if}\; 1 \geq |x| > 0 \;\text{and}\; \pi \geq \phi > 0\\ + dB_0(x_1, x_2, -0, +0)\cdot(\delta_1, \delta_2, \epsilon_1, \epsilon_2) + &< 0 \;\text{if}\; 1 \geq |x| > 0 \;\text{and}\; \pi \geq \phi > 0 \end{align*} \item If $y_2 \neq 0$, from \ref {eqn:Bk}, we have \[ @@ -1505,73 +1504,98 @@ To sum up using the inequalities above and deriving those with negative $x_2$ from them and from the relation $\overline{x}^y = \overline{x^{\overline{y}}}$, we can give the almost complete list of complex powers of numbers (for dyadic complex) that have a determined signed zero -part, the only exception being $x=+1 \pm 0i$ raised to a pure real power which -cannot be treated as we have done here. - -\begin{tabular}{rlcrlrl} - $x^{+0 +0i}$ & $=1 +0i$ &and& - $x^{-0 -0i}$ & $= 1 -0i$ & - if $|x| \geq 1$ &and $x \neq +1 \pm 0i$ \\ - $(x_1 +0i)^{y_1 +0i}$ & $= x_1^{y_1} +0i$ &and& - $(x_1 -0i)^{y_1 -0i}$ & $= x_1^{y_1} -0i$ & - if $x_1 > 1$ &and $y_1 > 0$\\ - $(x_1 \pm 0i)^{\pm0 +0i}$ & $= 1 +0i$ &and& - $(x_1 \pm 0i)^{\pm0 -0i}$ & $= 1 -0i$ & - if $x_1 > 1$\\ - $(x_1 \pm 0i)^{-0 +0i}$ & $= 1 +0i$ &and& - $(x_1 \pm 0i)^{+0 -0i}$ & $= 1 -0i$ & - if $|x_1| > 1$\\ - $(x_1 -0i)^{y_1 +0i}$ & $= x_1^{y_1} +0i$ &and& - $(x_1 +0i)^{y_1 -0i}$ & $= x_1^{y_1} -0i$ & - if $x_1 > 1$ &and $y_1 < 0$ \\ - $(+1 +\sigma_20i)^{y_1 \pm0}$ & - \multicolumn{4}{l}{$=1 +\sigma_2\rho_1 0i$} & - if $x_1=+1$ &and $y_1 \neq 0$ \\ - $x^{+0 -0i}$ & $= 1 +0i$ &and& - $x^{-0 +0i}$ & $= 1 -0i$ & - if $1 \geq |x| > 0$ &and $x \neq +1 \pm 0i$ \\ - $(x_1 +0i)^{y_1 -0i}$ & $= x_1^{y_1} +0i$ &and& - $(x_1 -0i)^{y_1 +0i}$ & $= x_1^{y_1} -0i$ & - if $1 > x_1 > 0$ &and $y_1 > 0$ \\ - $(x_1 +0i)^{+0 -0i}$ & $= 1 +0i$ &and& - $(x_1 -0i)^{+0 +0i}$ & $= 1 -0i$ & +part, the only exception being $x=+1 \pm 0i$ raised to zero power which cannot +be treated as we have done here. + +\begin{tabular}{r@{ $=$ }lr@{ $=$ }ll} + $x^{+0 +0i}$ & $1 +0i$,& + $x^{-0 -0i}$ & $1 -0i$ & + if $|x|>1$ and $x_2>0$\\ + $x^{-0 +0i}$ & $1 +0i$,& + $x^{+0 -0i}$ & $1 -0i$ & + if $|x|>1$ and $x_2<0$\\ + + $(x_1 \pm 0i)^{\pm0 +0i}$ & $1 +0i$,& + $(x_1 \pm 0i)^{\pm0 -0i}$ & $1 -0i$ & + if $x_1>1$\\ + + $(x_1 +0i)^{+0 +0i}$ & $1 +0i$, & + $(x_1 +0i)^{-0 -0i}$ & $1 -0i$ & + if $|x_1|>1$\\ + $(x_1 -0i)^{-0 +0i}$ & $1 +0i$, & + $(x_1 -0i)^{+0 -0i}$ & $1 -0i$ & + if $|x_1|>1$\\ + + $(x_1 +0i)^{y_1 +0i}$ & $x_1^{y_1} +0i$, & + $(x_1 -0i)^{y_1 -0i}$ & $x_1^{y_1} -0i$ & + if $x_1>1$ and $y_1>0$\\ + $(x_1 -0i)^{y_1 +0i}$ & $x_1^{y_1} +0i$, & + $(x_1 +0i)^{y_1 -0i}$ & $x_1^{y_1} -0i$ & + if $x_1>1$ and $y_1<0$\\ + + $(+1 +0i)^{y_1 \pm0i}$ & $1 +0i$, & + $(+1 -0i)^{y_1 \pm0i}$ & $1 -0i$ & + if $y_1>0$\\ + $(+1 -0i)^{y_1 \pm0i}$ & $1 +0i$, & + $(+1 +0i)^{y_1 \pm0i}$ & $1 -0i$ & + if $y_1<0$\\ + + $x^{+0 -0i}$ & $1 +0i$, & + $x^{-0 +0i}$ & $1 -0i$ & + if $1>|x|>0$ and $x_2>0$\\ + $x^{-0 -0i}$ & $1 +0i$, & + $x^{+0 +0i}$ & $1 -0i$ & + if $1>|x|>0$ and $x_2<0$\\ + + $(x_1 \pm0i)^{\pm0 -0i}$ & $1 +0i$, & + $(x_1 \pm0i)^{\pm0 +0i}$ & $1 -0i$ & + if $1 > x_1 > 0$ \\ + + $(x_1 +0i)^{+0 -0i}$ & $1 +0i$, & + $(x_1 -0i)^{+0 +0i}$ & $1 -0i$ & if $1 > |x_1| > 0$ \\ - $(x_1 -0i)^{-0 -0i}$ & $= 1 +0i$ &and& - $(x_1 +0i)^{-0 +0i}$ & $= 1 -0i$ & + $(x_1 -0i)^{-0 -0i}$ & $1 +0i$, & + $(x_1 +0i)^{-0 +0i}$ & $1 -0i$ & if $1 > |x_1| > 0$ \\ - $(x_1 \pm0i)^{\pm0 -0i}$ & $= 1 +0i$ &and& - $(x_1 \pm0i)^{\pm0 +0i}$ & $= 1 -0i$ & - if $1 > x_1 > 0$ \\ - $(x_1 -0i)^{y_1 -0i}$ & $= x_1^{y_1} +0i$ &and& - $(x_1 +0i)^{y_1 +0i}$ & $= x_1^{y_1} -0i$ & - if $1 > x_1 > 0$ &and $y_1 < 0$ \\ - $(\pm 0 +x_2i)^{+0 +0i}$ & $= 1 +0i$ &and& - $(\pm 0 +x_2i)^{-0 -0i}$ & $= 1 -0i$ & + + $(x_1 +0i)^{y_1 -0i}$ & $x_1^{y_1} +0i$, & + $(x_1 -0i)^{y_1 +0i}$ & $x_1^{y_1} -0i$ & + if $1 > x_1 > 0$ and $y_1 > 0$ \\ + $(x_1 -0i)^{y_1 -0i}$ & $x_1^{y_1} +0i$, & + $(x_1 +0i)^{y_1 +0i}$ & $x_1^{y_1} -0i$ & + if $1 > x_1 > 0$ and $y_1 < 0$ \\ + + $(\pm 0 +x_2i)^{+0 +0i}$ & $1 +0i$, & + $(\pm 0 +x_2i)^{-0 -0i}$ & $1 -0i$ & if $x_2 \geq 1$ \\ - $(\pm 0 +x_2i)^{+0 -0i}$ & $= 1 +0i$ &and& - $(\pm 0 +x_2i)^{-0 +0i}$ & $= 1 -0i$ & + $(\pm 0 +x_2i)^{+0 -0i}$ & $1 +0i$, & + $(\pm 0 +x_2i)^{-0 +0i}$ & $1 -0i$ & if $1 \geq x_2 > 0$ \\ - $(\pm 0 +x_2i)^{-0 -0i}$ & $= 1 +0i$ &and& - $(\pm 0 +x_2i)^{+0 +0i}$ & $= 1 -0i$ & + $(\pm 0 +x_2i)^{-0 -0i}$ & $1 +0i$, & + $(\pm 0 +x_2i)^{+0 +0i}$ & $1 -0i$ & if $ 0 > x_2 \geq -1$ \\ - $(\pm 0 +x_2i)^{-0 +0i}$ & $= 1 +0i$ &and& - $(\pm 0 +x_2i)^{+0 -0i}$ & $= 1 -0i$ & + $(\pm 0 +x_2i)^{-0 +0i}$ & $1 +0i$, & + $(\pm 0 +x_2i)^{+0 -0i}$ & $1 -0i$ & if $-1 \geq x_2$ \\ - $(-1 +\sigma_2 0i)^{\rho_1 0 \pm0i}$ & - \multicolumn{4}{l}{$= 1 + \sigma_2 \rho_1 0i$} & - if $x_1=-1$ &and $y_1 = \rho_1 0$ + + $(-1 +0i)^{+0 \pm0i}$ & $1+0i$, & + $(-1 +0i)^{-0 \pm0i}$ & $1-0i$ \\ + $(-1 -0i)^{-0 \pm0i}$ & $1+0i$, & + $(-1 -0i)^{+0 \pm0i}$ & $1-0i$ \\ \end{tabular} -So when $x^y$ is a pure real number, a compatible pattern is: +So when $x^y$ is a pure real number, the following pattern is compatible with +the determined cases: -\begin{tabular}{ll} - $x^y = x_1^{y_1} + \rho_2 0$ & if $|x| > 1$\\ - $x^y = 1 + \sigma_2 \rho_1 0$ & if $|x| = 1$\\ - $x^y = x_1^{y_1} - \rho_2 0$ & if $|x| < 1$ +\begin{tabular}{rl} + $x^y = x_1^{y_1} + \rho_2 0i$ & if $|x| > 1$\\ + $x^y = 1 + \sigma_2 \rho_1 0i$ & if $|x| = 1$\\ + $x^y = x_1^{y_1} - \rho_2 0i$ & if $|x| < 1$\\ + $x^y = x_1^{y_1} + \sigma_2 \rho_1 0i$ & if $y_1 \neq 0$ \end{tabular} where $\sigma_2$ (resp $\rho_1$, $\rho_2$) is the sign of $x_2$ (resp. $y_1$, -$y_2$). +$y_2$) and with the convention $0^0=+1$. \bibliographystyle{acm} \bibliography{algorithms} -- cgit v1.2.1