### 5.1. Simple functions
To create a recursive function we have to use this structure
```Ocaml
(*normal function*)
# let f = expr -> 1) evaluate expr ; 2) create & link f to the result
(*recursiv function*)
# let rec f = expr -> 1) create f ; 2) evaluate expr ; 3) link f to the result
```
**Exmple with factorial function**
```Ocaml
# let rec fact = function
| 0 -> 1
| n -> n * fact(n-1)
# fact 4;;
-: int = 24
```
> ⚠️ CAML have a call stack limite (it will block with stack overflow if the limit is reach)
**Basic expression**
```Ocaml
let rec f params =
if stop condition then
stop expression
else
recursive expression
```
To write a recurvise function we need two things
1) There always is a stop case
2) The parameters of a recursive call are different then its parent call, and go towards the stop condition
### 5.2. Down
```Ocaml
# let rec countdown n =
if n < 0 then
()
else
begin
print_int n ;
print_newline();
countdown(n-1);
end;;
val countdown = int -> unit = ()
# countdown 3;;
3
2
1
- : unit = ()
```
Flowchart of CAML evaluation
```mermaid
flowchart LR
A[ctd 3] --> B[ctd 2] --> C[ctd 1] --> D[ctd 0] --> E[ctd -1] --> F[ ] --> E --> D --> C --> B --> A
```
> ctd is countdown
## 5.3. Several recursive calls
A function that have several recursive calls is defined like this
$$
Fn = {1 -> \text{if } n\eqslantless{1} \brace{Fn+1+Fn+2} }
$$
**Fibonacci functions**
```Ocaml
let rec fibonacci n =
if n = 0 then
0
else if n = 1 then
1
else
fibonacci(n-1) + fibonacci(n-2);;
let rec odd n =
if n = 0 then
false
else
even (n-1);;
and even n =
if n = 0 then
true
else
add (n-1);;
val add : int -> bool =
val even : int -> bool =
```
An accumulator is a variable used to stock temporary the intermediaries results of a recursive operation. In CAML the syntax for the accumulator (`inv`) is: (exemple with reverse_int exercise. See 4.9 - b)
```Ocaml
# let reverse_int =
let rec rev inv = function
| 0 -> inv
| n -> rev (inv*10 + n mod 10)(n/10)
in rev 0 n;;
```
## 4.3. Complexity
Exemple with egypt (4.10) vs multiply (4.6):
```Ocaml
# let multiply a b =
let (a,b) = if a > b then
(a,b) else
(b,a)
in let rec mult = function
| 0 -> 0
| b -> a +mult (b-1)
in mult b;;
```
The best algorithm in term of complexity is the parameter that is constant/linear or logarithmic. If you have an exponential algorithm, you can put it in trash :)
**Exemple with fibonacci algorithm**
```Ocaml
# let rec fibo = function
0|1 -> 1
| n -> fibo (n-1) + fibo(n-2);;
```
| |res|how (for human) ?|How (for function) ?|
|:----:|:----:|:----:|:----:|
|0|1|def|1 call|
|1|1|def|1 call|
|2|2|+|3 calls|
|3|3|+|5 calls|
|4|5|+|9 calls|
|5|8|+|15 calls|
|6|13|+|14 calls|
| | |$0(n)$|$O(2^n)$|
This function is not optimize because the number of calls is growing exponentially.
A good function will be:
```Ocaml
# let fibo n =
let rec fib i fi fi_1 =
if n <= i then
fi
else
fib (i+1)(fi+fi_1)fi
in fib 1 1 1
```