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Diffstat (limited to 'doc')
-rw-r--r-- | doc/algorithms.tex | 10 |
1 files changed, 5 insertions, 5 deletions
diff --git a/doc/algorithms.tex b/doc/algorithms.tex index aa9f1aa..669d651 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -2051,8 +2051,8 @@ $k'_I \leq k$, then $\error (\appro x) \leq 2^{p - N} \Ulp (\appro x)$ and $\error (\appro y) \leq 2^{p - N} \Ulp (\appro y)$. -If $\Im(z_1) < 0$, then we can use the fact that $\AGM(1,\bar{z_1}) = -\bar{\AGM(1,z_1)}$, thus the same error analysis applies; +If $\Im(z_1) < 0$, then we can use the fact that $\AGM(1,\bar{z_1})$ is the +conjugate of $\AGM(1,z_1)$, thus the same error analysis applies; and if $\Im(z_1) = 0$, we are computing a real AGM, we can call the corresponding MPFR function. @@ -2061,12 +2061,12 @@ relative error at most $2^{1-p}$. Then we have to replace $\epsilon_0 = 0$ by $\epsilon_0 = 2^{1-p}$ in the above proof. This gives \[ \zeta_1 \leq \epsilon_0 + c (1 + \epsilon_0) 2^{1-p} - \leq (2 + 2^{1-p}) 2^{1-p} \leq \frac{5}{2} 2^{1-p}, \] + \leq (2 + 2^{1-p}) 2^{1-p} \leq \frac{9}{4} 2^{1-p}, \] and \[ \epsilon_1 \leq \zeta_1 + c (1 + \zeta_1) 2^{1-p} - \leq (\frac{5}{2} + 1 + \frac{5}{2} 2^{1-p}) 2^{1-p} + \leq (\frac{9}{4} + 1 + \frac{9}{4} 2^{1-p}) 2^{1-p} \leq 4 \cdot 2^{1-p}, \] -as long as $p \geq 4$. Thus the bound $\epsilon_1 \leq r_1 2^{1-p}$ still +as long as $p \geq 3$. Thus the bound $\epsilon_1 \leq r_1 2^{1-p}$ still holds in that case. \paragraph{The general case.} |