From cd00a15eb1005e9e272521fdac6d34b2c28e19c0 Mon Sep 17 00:00:00 2001 From: enge Date: Thu, 23 Jan 2014 18:00:41 +0000 Subject: algorithms.tex: Minor changes. git-svn-id: svn://scm.gforge.inria.fr/svn/mpc/trunk@1436 211d60ee-9f03-0410-a15a-8952a2c7a4e4 --- doc/algorithms.tex | 15 +++++++++------ 1 file changed, 9 insertions(+), 6 deletions(-) diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 8b27b04..4bf9903 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1911,8 +1911,8 @@ by $\corr {a_0} = a$, $\corr {b_0} = b$, $\corr {a_{n+1}} = \frac {\corr {a_n} + \corr {b_n}}{2}$ and $\corr {b_{n+1}} = \sqrt {\corr {a_n} \corr {b_n}}$. At each step, there is a choice of sign for the square root. -If $\corr {a_n}$ and $\corr {b_n}$ are at an angle different from $0$ -and $\pi$, then +If $\corr {a_n}$ and $\corr {b_n}$ are at an (unoriented) angle +different from $0$ and $\pi$, then they define a two-dimensional pointed cone in the complex plane. Notice that $\corr {a_{n+1}}$ lies in this cone. Following \cite {Cox84} we call \emph {right} the choice that makes @@ -2031,7 +2031,7 @@ computations at a working precision of about $p = N + c B (N, b/a)$. The following discussion provides explicit bounds for all these quantities. -\paragraph {Error propagation.} +\paragraph {Rounding error propagation.} Let $\corr {a_n} = \frac {\corr {a_{n-1}} + \corr {b_{n-1}}}{2}$, @@ -2220,7 +2220,7 @@ we reach \] -\paragraph {Total error and working precision} +\paragraph {Total error and working precision.} Combining with \eqref {eq:propagm} we obtain $\AGM (1, b_0) = \frac {1 + \theta_1}{1 + \theta_2} \appro {a_n} @@ -2244,8 +2244,11 @@ this leads to a relative error bounded by $2^{-N}$. \paragraph {Parameter choice.} -Recall that in our analysis, $k = \max (2, - \Exp (\Re (\appro {a_1} + 1)))$ -is given by the input data. We may then choose a target relative error~$N$, +Recall that in our analysis, $k = \max (2, - \Exp (\Re (\appro {a_1}) + 1))$ +is given by the input data. As we need to know it before choosing the final +precision, it should be computed from an approximation at a tiny precision, +of $2$ bits, say. +We may then choose a target relative error~$N$, which determines the number of iterations~$n = B (N)$ by~\eqref {eq:agmbound}. Then the working precision may be chosen as \[ -- cgit v1.2.1