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ex-2: revised section 2.2.1

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Giù Marcer 2020-05-26 09:58:16 +02:00 committed by rnhmjoj
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notes/sections/2.md
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@ -43,44 +43,50 @@ $$
| \gamma(n_{i+1}) - \gamma | > | \gamma(n_i) - \gamma| | \gamma(n_{i+1}) - \gamma | > | \gamma(n_i) - \gamma|
$$ $$
and $\gamma (n_i)$ was selected as the result (see @tbl:1_results). and $\gamma (n_i)$ was selected as the best result (see @tbl:1_results).
----------------------------------------------- ---------------------------------
n sum $|\gamma(n)-\gamma|$ n $|\gamma(n)-\gamma|$
----------- ------------- --------------------- ----------- ---------------------
\SI{2e1}{} \SI{2.48e-02}{} \SI{2e1}{} \SI{2.48e-02}{}
\SI{2e2}{} \SI{2.50e-03}{} \SI{2e2}{} \SI{2.50e-03}{}
\SI{2e3}{} \SI{2.50e-04}{} \SI{2e3}{} \SI{2.50e-04}{}
\SI{2e4}{} \SI{2.50e-05}{} \SI{2e4}{} \SI{2.50e-05}{}
\SI{2e5}{} \SI{2.50e-06}{} \SI{2e5}{} \SI{2.50e-06}{}
\SI{2e6}{} \SI{2.50e-07}{} \SI{2e6}{} \SI{2.50e-07}{}
\SI{2e7}{} \SI{2.50e-08}{} \SI{2e7}{} \SI{2.50e-08}{}
\SI{2e8}{} \SI{2.50e-09}{} \SI{2e8}{} \SI{2.50e-09}{}
\SI{2e9}{} \SI{2.55e-10}{} \SI{2e9}{} \SI{2.55e-10}{}
\SI{2e10}{} \SI{2.42e-11}{} \SI{2e10}{} \SI{2.42e-11}{}
\SI{2e11}{} \SI{1.44e-08}{} \SI{2e11}{} \SI{1.44e-08}{}
----------------------------------------------- ---------------------------------
Table: Partial results using the definition of $\gamma$ with double Table: Partial results using the definition of $\gamma$ with double
precision. {#tbl:1_results} precision. {#tbl:1_results}
The convergence is logarithmic: to fix the first $d$ decimal places, about The convergence is logarithmic: to fix the first $d$ decimal places, about
$10^d$ terms are needed. The double precision runs out at the $10^d$ terms of the armonic series are needed. The double precision runs out at
10\textsuperscript{th} place, $n=\SI{2e10}{}$. the $10^{\text{th}}$ place, at $n=\SI{2e10}{}$.
Since all the number are given with double precision, there can be at best 15 Since all the number are given with double precision, there can be at best 16
correct digits but only 10 were correctly computed: this means that when the correct digits, since for a double 64 bits are allocated in memory: 1 for the
terms of the series start being smaller than the smallest representable double, sign, 8 for the exponent and 55 for the mantissa:
the sum of all the remaining terms give a number $\propto 10^{-11}$. $$
2^{55} = 10^{d} \thus d = 55 \cdot \log(2) \sim 16.6
$$
Only 10 digits were correctly computed: this means that when the terms of the
series start being smaller than the smallest representable double, the sum of
all the remaining terms gives a number $\propto 10^{-11}$.
Best result in @tbl:first. Best result in @tbl:first.
--------- ----------------------- --------- -----------------------
@ -106,7 +112,7 @@ $$
Varying $M$ from 1 to 100, the best result was obtained for $M = 41$ (see Varying $M$ from 1 to 100, the best result was obtained for $M = 41$ (see
@tbl:second). It went sour: the convergence is worse than using the definition @tbl:second). It went sour: the convergence is worse than using the definition
itself. Only two places were correctly computed (#@tbl:second). itself. Only two places were correctly computed (@tbl:second).
--------- ----------------------- --------- -----------------------
true: 0.57721\ 56649\ 01533 true: 0.57721\ 56649\ 01533