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Diffstat (limited to 'doc')
| -rw-r--r-- | doc/algorithms.tex | 25 |
1 files changed, 10 insertions, 15 deletions
diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 84f85c3..775f1a6 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -593,10 +593,8 @@ One readily verifies that \] \end {proof} -Assume now that $\corr {z_1} = (1 + \theta_1) \appro {z_1}$ -with $\epsilon_1 = |\theta_1|$ and -$\corr {z_2} = (1 + \theta_2) \appro {z_2}$ -with $\epsilon_2 = |\theta_2|$ lie in the same quadrant. +Assume now that $\corr {z_n} = (1 + \theta_n) \appro {z_n}$ +with $\epsilon_n = |\theta_n|$ lie in the same quadrant. Then $\corr z = (1 + \theta) \appro z$ with @@ -605,23 +603,16 @@ with {\appro {z_1} + \appro {z_2}}. \] and -\[ +\begin {equation} +\label {eq:propaddrel} \relerror (\appro z) \leq \max (\epsilon_1, \epsilon_2) \frac {|\appro {z_1}| + |\appro {z_2}|}{|\appro {z_1} + \appro {z_2}|} \leq \sqrt 2 \, \max (\epsilon_1, \epsilon_2) -\] -by Lemma~\ref {lm:arithgeom}. - -Defining $c$ as in Proposition~\ref {prop:comrelround}, we obtain -\begin {equation} -\label {eq:addrel} -\relerror (\round (\appro z)) -\leq \sqrt 2 \, \max (\epsilon_1, \epsilon_2) -+ c \left( 1 + \sqrt 2 \, \max (\epsilon_1, \epsilon_2) \right) 2^{1-p}. \end {equation} +by Lemma~\ref {lm:arithgeom}. @@ -767,7 +758,11 @@ A coarser bound may be obtained more easily by considering complex relative errors. Write $\corr {z_n} = (1 + \theta_n) \appro {z_n}$ with $\epsilon_n = | \theta_n |$. Then $\corr z = (1 + \theta) \appro z$ with $\theta = \theta_1 + \theta_2 + \theta_1 \theta_2$ and -$\epsilon = |\theta| \leq \epsilon_1 + \epsilon_2 + \epsilon_1 \epsilon_2$. +\begin {equation} +\label {eq:propmulrel} +\epsilon = \relerror (\appro z) +\leq \epsilon_1 + \epsilon_2 + \epsilon_1 \epsilon_2. +\end {equation} By Proposition~\ref {prop:relerror}, we have $\epsilon_{X, n} \leq k_{X, n} 2^{1-p}$ for $X \in \{ R, I \}$, and by Proposition~\ref {prop:comrelerror}, |