From 587dbf35ddbc864e6c1e44f91829f40d70edcca7 Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Sun, 13 Oct 2024 21:03:55 +0200
Subject: [PATCH] update
---
slides/cours_5.md | 89 ++++++
slides/figs/Float_example.svg | 92 ++++++
slides/figs/Float_example_bare.svg | 430 +++++++++++++++++++++++++++++
3 files changed, 611 insertions(+)
create mode 100644 slides/figs/Float_example.svg
create mode 100644 slides/figs/Float_example_bare.svg
diff --git a/slides/cours_5.md b/slides/cours_5.md
index 026299a..7e3681f 100644
--- a/slides/cours_5.md
+++ b/slides/cours_5.md
@@ -518,4 +518,93 @@ int fib_imp(int n) {
}
```
+# Exponentiation rapide
+
+\Huge L'exponentiation rapide ou indienne
+
+# Exponentiation rapide ou indienne (1/4)
+
+## But: Calculer $x^n$
+
+* Quel est l'algorithmie le plus simple que vous pouvez imaginer?
+
+. . .
+
+```C
+double pow(double x, int n) {
+ if (0 == n) {
+ return 1;
+ }
+ double p = c;
+ for (int i = 1; i < n; ++i) {
+ p = p * x; // x *= x
+ }
+ return x;
+}
+```
+
+* Combien de multiplication et d'assignations en fonction de `n`?
+
+. . .
+
+* `n` assignations et `n` multiplications.
+
+# Exponentiation rapide ou indienne (2/4)
+
+* Proposez un algorithme naïf et récursif
+
+. . .
+
+```C
+double pow(double x, int n) {
+ if (n != 0) {
+ return x * pow(x, n-1);
+ } else {
+ return 1;
+ }
+}
+```
+
+# Exponentiation rapide ou indienne (3/4)
+
+## Exponentiation rapide ou indienne de $x^n$
+
+* Écrivons $n=\sum_{i=0}^{d-1}b_i 2^i,\ b_i=\{0,1\}$ (écriture binaire sur $d$ bits, avec
+$d\sim\log_2(n)$).
+*
+$$
+x^n={x^{2^0}}^{b_0}\cdot {x^{2^1}}^{b_1}\cdots {x^{2^{d-1}}}^{b_{d-1}}.
+$$
+* On a besoin de $d$ calculs pour les $x^{2^i}$.
+* On a besoin de $d$ calculs pour évaluer les produits de tous les termes.
+
+## Combien de calculs en terme de $n$?
+
+. . .
+
+* $n$ est représenté en binaire avec $d$ bits $\Rightarrow d\sim\log_2(n)$.
+* il y a $2\log_2(n)\sim \log_2(n)$ calculs.
+
+# Exponentiation rapide ou indienne (4/4)
+
+## Le vrai algorithme
+
+* Si n est pair: calculer $\left(x^{n/2}\cdot x^{n/2}\right)$,
+* Si n est impair: calculer $x \cdot \left(x^{(n-1)/2}\right)^2=x\cdot x^{n-1}$.
+
+## Exercice: écrire l'algorithme récursif correspondant
+
+. . .
+
+```C
+double pow(double x, int n) {
+ if (0 == n) {
+ return 1;
+ } else if (n % 2 == 0) {
+ return pow(x, n / 2) * pow(x, n/2);
+ } else {
+ return x * pow(x, (n-1));
+ }
+}
+```
diff --git a/slides/figs/Float_example.svg b/slides/figs/Float_example.svg
new file mode 100644
index 0000000..5b34a69
--- /dev/null
+++ b/slides/figs/Float_example.svg
@@ -0,0 +1,92 @@
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+ </g>
+
+ <text x="504" y="48.4" style="text-anchor: start; font-size: 18.7px;">= 0.15625</text>
+</svg>
\ No newline at end of file
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new file mode 100644
index 0000000..c74df6c
--- /dev/null
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--
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