(*  Newton-Raphson function from Kennedy (p. 11).
*)
;

(* This file does not compile as unrefined SML.
   The "let val dx = f x / f' x" in the body of `newton' will cause SML
   to infer `int' instead of `real' (because its operator overloading is
   pastede on yay).  The fix under SML is simpler than a full signature
   type: change (f, f', x, xacc) to (f : real -> real, f', x, xacc), but
   Stardust doesn't support that syntax -- and even if it did, it would
   inject `real -> real' to `real(NODIM) -> real(NODIM)'!
*)

(*[ val newton : -all d1,d2:dim-
  (* f, a function *)              (real(d1) -> real(d2))                      
  (* f', its derivative *)       * (real(d1) -> real((d1 ^ ~1) * d2))          
  (* x, the initial guess *)     * real(d1)                                    
  (* xacc, relative accuracy *)  * real                                        
                                -> real(d1)
]*)

fun newton (f, f', x, xacc) =
  let val dx = f x / f' x
      val x' = x - dx
  in
      if abs dx / x' < xacc
      then x'
      else newton (f, f', x', xacc)
  end