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| -rw-r--r-- | doc/algorithms.tex | 14 |
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diff --git a/doc/algorithms.tex b/doc/algorithms.tex index e77089f..fc7e060 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1286,6 +1286,20 @@ and for $e \geq 2$: \[ {\rm err}(s) \leq \sum_{k=0}^{K-1} \frac{5}{3} \cdot 2^{-e-p} = \frac{5}{3} K 2^{-e-p}. \] +Assume now that $|\Re z| \leq 2^{-e}$ and $|\Im z| \leq 2^{-e-\delta}$ with +$e \geq 1$ and $\delta \geq 0$. All the above reasoning for the real parts still +holds. For the imaginary parts, we have +$|\Im w| \leq 2^{1-2e-\delta}$, and the bound on the imaginary part of $t$ +becomes $|\Im t| \leq 2^{2k-(2k+1)e-\delta}$. +Now consider that $\varepsilon_k$ is the maximal absolute error on +$\Im t$ only at the end of the loop with index~$k$. +The initial value $\varepsilon_0$ +is multiplied by $2^{-\delta}$ compared to above, +and since all the bounds on the imaginary +parts are multiplied by $2^{-\delta}$ too, the same holds for the recurrence +on $\varepsilon_k$ and their solutions. Thus the final error bound on +$\Im s$ is multiplied by $2^{-\delta}$ too. + \subsection {\texttt {mpc\_pow}} The main issue for the power function is to be able to recognize when the |