From f91821c946dc585eed011764950cc538906bf7fc Mon Sep 17 00:00:00 2001 From: Arkitu Date: Tue, 18 Nov 2025 16:10:54 +0100 Subject: [PATCH] td3 pas finis --- td3/td3.md | 58 ++++++++++++++++++++++++++++++++++++++++++++++++++++ td3/td3.pdf | Bin 0 -> 173950 bytes 2 files changed, 58 insertions(+) create mode 100644 td3/td3.md create mode 100644 td3/td3.pdf diff --git a/td3/td3.md b/td3/td3.md new file mode 100644 index 0000000..6e1deba --- /dev/null +++ b/td3/td3.md @@ -0,0 +1,58 @@ +# Récursivité + +## Exercice 1 + +Formules naïves : + +- $x\times y = \begin{cases} + 0 & \text{si $y=0$}\\ + x & \text{si $y=1$}\\ + x + (x \times (y-1)) & \text{si $y>1$} + \end{cases}$ + +- $a^n = \begin{cases} + 1 & \text{si $a=0$}\\ + a\times a^{n-1} & \text{si $n>0$} + \end{cases}$ + +Formules dichotomiques : + +- $x\times y = \begin{cases} + 0 & \text{si $y=0$}\\ + x & \text{si $y=1$}\\ + 2\times (x \times {y \over 2}) & \text{si $y>1$ et $y$ pair}\\ + x + 2\times (x \times {y - 1 \over 2}) & \text{si $y>1$ et $y$ impair} + \end{cases}$ + +- $a^n = \begin{cases} + 1 & \text{si $a=0$}\\ + (a^{n \over 2})^2 & \text{si $n>0$ et $n$ pair}\\ + a\times (a^{n-1 \over 2})^2 & \text{si $n>0$ et $n$ impair} + \end{cases}$ + +Les formules naïves sont de complexité $O(y)$ et $O(n)$ alors que les formules dichotomiques sont de complexité $O(\log_2(y))$ et $O(\log_2(n))$, ce qui est plus efficace. + +## Exercice 2 + +$\operatorname{pgcd}(a, b) = \begin{cases} +\operatorname{pgcd}(b, a) & \text{si $b>a$}\\ +\operatorname{pgcd}(b, a \mod b) & \text{si $a \mod b > 0$}\\ +b & \text{si $a \mod b = 0$} +\end{cases}$ + +## Exercice 3 + + + +```ocaml +let rec bezout a b = match a mod b with + | 1 -> 1, (a/b) + | r -> + let c, d = bezout b r in + let () = print_int c in + let () = print_newline () in + let () = print_int d in + let () = print_newline () in + c+(d*r), d +;; +``` diff --git a/td3/td3.pdf b/td3/td3.pdf new file mode 100644 index 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