let stop_distance epsilon delta max_iter fnext a b n=
  (abs_float fnext < epsilon) && (abs_float(a -. b) < delta);;

let stop_iter epsilon delta max_iter fnext a b n=
  n<max_iter;;

let full_stop epsilon delta max_iter fnext a b n =
  (stop_iter epsilon delta max_iter fnext a b n) &&
    (stop_distance epsilon delta max_iter fnext a b n);;

let next_bissection a b fa fb =
  0.5 *. (a +. b);;

let next_fausse_position_secante a b fa fb =
  a -. (b -. a)/.(fb -. fa)*.fa;;

let next_newton x fx derivee =
  x -. fx /. derivee;;

let research good_enough epsilon delta max_iter next_function f a b =
  let fa = f a and fb = f b in
  let rec iter a b n =
    let next = next_function a b fa fb in
    let fnext = f next in
    if good_enough epsilon delta max_iter fnext a b n then next
    else if fnext>0. then iter a next (n+1)
    else iter next b (n+1)
  in if fa > 0. then iter b a 0
    else if stop_distance epsilon delta max_iter fa a b 0 then a
    else if stop_distance epsilon delta max_iter fb a b 0 then b
    else iter a b 0;;


let bissection ?(good_enough = full_stop)
    ?(epsilon = 1e-8) ?(delta = 1e-8) ?(max_iter = 50)
    f a b =
  research good_enough epsilon delta max_iter next_bissection f a b;;

let fausse_position ?(good_enough = full_stop)
    ?(epsilon = 1e-8) ?(delta = 1e-8) ?(max_iter = 50)
    f a b =
  research good_enough epsilon delta max_iter next_fausse_position_secante f a b;;


let secante ?(good_enough = full_stop)
    ?(epsilon = 1e-8) ?(delta = 1e-8) ?(max_iter = 50)
    f a b =
  let rec iter a b n=
    let fa = f a and fb = f b in
    let next = next_fausse_position_secante a b fa fb in
    let fnext = f next in
    if good_enough epsilon delta max_iter fnext a b n then next
    else iter b next (n+1)
  in iter a b 0;;

let newton ?(good_enough = full_stop)
    ?(epsilon = 1e-8) ?(delta = 1e-8) ?(max_iter = 50)
    f derivee a b =
  let rec iter b n =
    let fb = f b in
    let derivee_in_b = derivee b in
    let next = next_newton b fb derivee_in_b in
    let fnext = f next in
    if good_enough epsilon delta max_iter fnext a b n then next
    else iter next (n+1)
  in let fa = f a and fb = f b in
     if f a > 0. then iter a 0
     else if stop_distance epsilon delta max_iter fa a b 0 then a
     else if stop_distance epsilon delta max_iter fb a b 0 then b
     else iter b 0;;

let f x = x*.x -. 2.;;
let g x = x*.x*.x;;
let h x = x*.x*.x*.x -. 5.;;