fibonacci: hands-on session

lectures/complexity/slides.qmd

@@ -309,9 +309,16 @@ None
 ## Fibonacci series example (3/3)
 
 $$
-\begin{pmatrix}f_{n-1}\\f_{n}\end{pmatrix} = \begin{pmatrix}0 & 1\\1 & 1\end{pmatrix} \begin{pmatrix}f_{n-2}\\f_{n-1}\end{pmatrix}
+\begin{pmatrix}f_{n}\\f_{n+1}\end{pmatrix} = \begin{pmatrix}0 & 1\\1 & 1\end{pmatrix} \begin{pmatrix}f_{n-1}\\f_{n}\end{pmatrix}
 $$
 
+:::{.fragment}
+
+$$
+\begin{pmatrix}f_{n}\\f_{n+1}\end{pmatrix} = \begin{pmatrix}0 & 1\\1 & 1\end{pmatrix}^{n} \begin{pmatrix}f_0\\f_1\end{pmatrix}
+$$
+:::
+
 ## vector
 
 ```{python}
@@ -362,13 +369,16 @@ m * m * V2(0, 1)
 ## power function
 
 ```{python}
-#| echo: true
+#| echo: false
 def power(a: "A", n: int, op: "A,A -> A, assoc" = lambda a, b: a * b):
     assert n > 0
     if n == 1: return a
     if n % 2 == 0: return power(op(a, a), n // 2, op)
     return op(power(op(a, a), n // 2, op), a)
+```
 
+```{python}
+#| echo: true
 power(2, 4)
 ```
 
@@ -380,6 +390,11 @@ power(m, 2) * V2(0, 1)
 ```
 :::
 
+## Hands-on! {.handson}
+
+* Implement that generic `power` function.
+* Do so with a time complexity better than $\mathcal{O}(n)$
+
 ## fib3
 
 ```{python}