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(***********************************************************************)
(* *)
(* OCaml *)
(* *)
(* Xavier Leroy, projet Cristal, INRIA Rocquencourt *)
(* *)
(* Copyright 2002 Institut National de Recherche en Informatique et *)
(* en Automatique. All rights reserved. This file is distributed *)
(* under the terms of the GNU Library General Public License, with *)
(* the special exception on linking described in file ../LICENSE. *)
(* *)
(***********************************************************************)
(* $Id$ *)
(* Complex numbers *)
type t = { re: float; im: float }
let zero = { re = 0.0; im = 0.0 }
let one = { re = 1.0; im = 0.0 }
let i = { re = 0.0; im = 1.0 }
let add x y = { re = x.re +. y.re; im = x.im +. y.im }
let sub x y = { re = x.re -. y.re; im = x.im -. y.im }
let neg x = { re = -. x.re; im = -. x.im }
let conj x = { re = x.re; im = -. x.im }
let mul x y = { re = x.re *. y.re -. x.im *. y.im;
im = x.re *. y.im +. x.im *. y.re }
let div x y =
if abs_float y.re >= abs_float y.im then
let r = y.im /. y.re in
let d = y.re +. r *. y.im in
{ re = (x.re +. r *. x.im) /. d;
im = (x.im -. r *. x.re) /. d }
else
let r = y.re /. y.im in
let d = y.im +. r *. y.re in
{ re = (r *. x.re +. x.im) /. d;
im = (r *. x.im -. x.re) /. d }
let inv x = div one x
let norm2 x = x.re *. x.re +. x.im *. x.im
let norm x =
(* Watch out for overflow in computing re^2 + im^2 *)
let r = abs_float x.re and i = abs_float x.im in
if r = 0.0 then i
else if i = 0.0 then r
else if r >= i then
let q = i /. r in r *. sqrt(1.0 +. q *. q)
else
let q = r /. i in i *. sqrt(1.0 +. q *. q)
let arg x = atan2 x.im x.re
let polar n a = { re = cos a *. n; im = sin a *. n }
let sqrt x =
if x.re = 0.0 && x.im = 0.0 then { re = 0.0; im = 0.0 }
else begin
let r = abs_float x.re and i = abs_float x.im in
let w =
if r >= i then begin
let q = i /. r in
sqrt(r) *. sqrt(0.5 *. (1.0 +. sqrt(1.0 +. q *. q)))
end else begin
let q = r /. i in
sqrt(i) *. sqrt(0.5 *. (q +. sqrt(1.0 +. q *. q)))
end in
if x.re >= 0.0
then { re = w; im = 0.5 *. x.im /. w }
else { re = 0.5 *. i /. w; im = if x.im >= 0.0 then w else -. w }
end
let exp x =
let e = exp x.re in { re = e *. cos x.im; im = e *. sin x.im }
let log x = { re = log (norm x); im = atan2 x.im x.re }
let pow x y = exp (mul y (log x))
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