Roots of unity

From Rosetta Code
Revision as of 22:23, 3 February 2008 by rosettacode>Thomas Mertes (Change the output of the Seed7 example)
Task
Roots of unity
You are encouraged to solve this task according to the task description, using any language you may know.

The purpose of this task is to explore working with complex numbers. Given n, find the n-th roots of unity.

Ada

with Ada.Text_Io; use Ada.Text_Io;
with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
with Ada.Numerics.Generic_Complex_Types;
with Ada.Numerics.Generic_Complex_Elementary_Functions;
with Ada.Text_Io.Complex_Io;

procedure Roots_Of_Unity is
   type Real is digits 10;
   package Complex_Type is new Ada.Numerics.Generic_Complex_Types(Real);
   use Complex_Type;
   package Complex_Functions is new Ada.Numerics.Generic_Complex_Elementary_Functions(Complex_Type);
   use Complex_Functions;
   package Complex_Io is new Ada.Text_Io.Complex_Io(Complex_Type);
   Factor : Complex;
begin
   for Root in 2..10 loop
      Put(Item => Root, Width => 4);
      Put(" ");
      for K in 1..Root/2 loop
         Factor := Exp(Real(K) * 2.0 * J * Ada.Numerics.Pi / Real(Root));
         Complex_Io.Put(Item => Factor, Fore => 2, Aft  => 4, Exp  => 0);
      end loop;
      New_Line;
   end loop;
end Roots_Of_Unity;

Output: 

  2 (-1.0000, 0.0000)
  3 (-0.5000, 0.8660)
  4 ( 0.0000, 1.0000)(-1.0000, 0.0000)
  5 ( 0.3090, 0.9511)(-0.8090, 0.5878)
  6 ( 0.5000, 0.8660)(-0.5000, 0.8660)(-1.0000, 0.0000)
  7 ( 0.6235, 0.7818)(-0.2225, 0.9749)(-0.9010, 0.4339)
  8 ( 0.7071, 0.7071)( 0.0000, 1.0000)(-0.7071, 0.7071)(-1.0000, 0.0000)
  9 ( 0.7660, 0.6428)( 0.1736, 0.9848)(-0.5000, 0.8660)(-0.9397, 0.3420)
 10 ( 0.8090, 0.5878)( 0.3090, 0.9511)(-0.3090, 0.9511)(-0.8090, 0.5878)(-1.0000, 0.0000)

ALGOL 68

FORMAT complex fmt=$g(-6,4)"⊥"g(-6,4)$;
FOR root FROM 2 TO 10 DO
  printf(($g(4)$,root));
  FOR n TO root OVER 2 DO
    printf(($xf(complex fmt)$,complex exp( 0 I 2*pi*n/root)))
  OD;
  printf($l$)
OD

Output: 

 +2 -1.000⊥0.0000
 +3 -.5000⊥0.8660
 +4 0.0000⊥1.0000 -1.000⊥0.0000
 +5 0.3090⊥0.9511 -.8090⊥0.5878
 +6 0.5000⊥0.8660 -.5000⊥0.8660 -1.000⊥0.0000
 +7 0.6235⊥0.7818 -.2225⊥0.9749 -.9010⊥0.4339
 +8 0.7071⊥0.7071 0.0000⊥1.0000 -.7071⊥0.7071 -1.000⊥0.0000
 +9 0.7660⊥0.6428 0.1736⊥0.9848 -.5000⊥0.8660 -.9397⊥0.3420
+10 0.8090⊥0.5878 0.3090⊥0.9511 -.3090⊥0.9511 -.8090⊥0.5878 -1.000⊥0.0000

C++

#include <complex>
#include <iostream>
#include <ostream>

double const pi = 4*atan(1);

int main()
{
  for (int n = 2; n <= 10; ++n)
  {
    std::cout << n << ": ";
    for (int k = 0; k < n; ++k)
      std::cout << std::polar(1, 2*pi*k/n) << " ";
    std::cout << std::endl;
  }
}

Forth

Complex numbers are not a native type in Forth, so we calculate the roots by hand. 

: f0. ( f -- )
  fdup 0e 0.001e f~ if fdrop 0e then f. ;
: .roots ( n -- )
  dup 1 do
    pi i 2* 0 d>f f* dup 0 d>f f/          ( F: radians )
    fsincos cr ." real " f0. ." imag " f0.
  loop drop ;

3 set-precision
5 .roots

IDL

For some example n: 

 n = 5
 print,  exp( dcomplex( 0, 2*!pi/n) ) ^ ( 1 + indgen(n) )

Outputs: 

 ( 0.30901696, 0.95105653)( -0.80901704, 0.58778520)( -0.80901693, -0.58778534)( 0.30901713, -0.95105647)( 1.0000000, 1.7484556e-007)

J

   rou=: [: ^ i. * (o. 0j2) % ]

   rou 4
1 0j1 _1 0j_1

   rou 5
1 0.309017j0.951057 _0.809017j0.587785 _0.809017j_0.587785 0.309017j_0.951057

The computation can also be written as a loop, shown here for comparison only. 

rou1=: 3 : 0
 z=. 0 $ r=. ^ o. 0j2 % y [ e=. 1
 for. i.y do.
  z=. z,e
  e=. e*r
 end.
 z
)

Java

Java doesn't have a nice way of dealing with complex numbers, so the real and imaginary parts are calculated separately based on the angle and printed together. There are also checks in this implementation to get rid of extremely small values (< 1.0E-3 where scientific notation sets in for Doubles). Instead, they are simply represented as 0. To remove those checks (for very high n's), remove both if statements. 

public static void unity(int n){
	//all the way around the circle at even intervals
	for(double angle = 0;angle < 2 * Math.PI;angle += (2 * Math.PI) / n){
		double real = Math.cos(angle); //real axis is the x axis
		if(Math.abs(real) < 1.0E-3) real = 0.0; //get rid of annoying sci notation
		double imag = Math.sin(angle); //imaginary axis is the y axis
		if(Math.abs(imag) < 1.0E-3) imag = 0.0; //get rid of annoying sci notation
		System.out.print(real + " + " + imag + "i\t"); //tab-separated answers
	}
}

Perl

Interpreter: Perl 5.8.8 

use Math::Complex;

foreach $n (1 .. 10) {
  printf "%2d ", $n;
  foreach $k (1 .. $n) {
    $ret = exp($k * 2*i * pi / $n);
    $ret->display_format('style' => 'cartesian', 'format' => '%.3f');
    print "($ret)";
  }
  print "\n";
}

Output: 

 1 (1.000-0.000i)
 2 (-1.000+0.000i)(1.000-0.000i)
 3 (-0.500+0.866i)(-0.500-0.866i)(1.000-0.000i)
 4 (0.000+1.000i)(-1.000+0.000i)(-0.000-1.000i)(1.000-0.000i)
 5 (0.309+0.951i)(-0.809+0.588i)(-0.809-0.588i)(0.309-0.951i)(1.000-0.000i)
 6 (0.500+0.866i)(-0.500+0.866i)(-1.000+0.000i)(-0.500-0.866i)(0.500-0.866i)(1.000-0.000i)
 7 (0.623+0.782i)(-0.223+0.975i)(-0.901+0.434i)(-0.901-0.434i)(-0.223-0.975i)(0.623-0.782i)(1.000-0.000i)
 8 (0.707+0.707i)(0.000+1.000i)(-0.707+0.707i)(-1.000+0.000i)(-0.707-0.707i)(-0.000-1.000i)(0.707-0.707i)(1.000-0.000i)
 9 (0.766+0.643i)(0.174+0.985i)(-0.500+0.866i)(-0.940+0.342i)(-0.940-0.342i)(-0.500-0.866i)(0.174-0.985i)(0.766-0.643i)(1.000-0.000i)
10 (0.809+0.588i)(0.309+0.951i)(-0.309+0.951i)(-0.809+0.588i)(-1.000+0.000i)(-0.809-0.588i)(-0.309-0.951i)(0.309-0.951i)(0.809-0.588i)(1.000-0.000i)

Python

Interpreter: Python 2.5

Function nthroots() returns all n-th roots of unity. 

from cmath import exp, pi
def nthroots(n):
    return [exp(k * 2j * pi / n) for k in range(1, n + 1)]

Example: 

>>> f = nthroots
>>> for n in range(1, 4):
...    print map(lambda c: ("%.3f " + ("+" if c.imag > 0 else "-") + " %.3gj") % (c.real, abs(c.imag)), f(n))
['1.000 - 2.45e-016j']
['-1.000 + 1.22e-016j', '1.000 - 2.45e-016j']
['-0.500 + 0.866j', '-0.500 - 0.866j', '1.000 - 2.45e-016j']

Seed7

$ include "seed7_05.s7i";
  include "float.s7i";
  include "complex.s7i";

const proc: main is func
  local
    var integer: n is 0;
    var integer: k is 0;
  begin
    for n range 2 to 10 do
      write(n lpad 2 <& ": ");
      for k range 0 to pred(n) do
        write(polar(1.0, 2.0 * PI * flt(k) / flt(n)) digits 4 lpad 15 <& " ");
      end for;
      writeln;
    end for;
  end func;

Output: 

 2:  1.0000+0.0000i -1.0000+0.0000i
 3:  1.0000+0.0000i -0.5000+0.8660i -0.5000-0.8660i
 4:  1.0000+0.0000i  0.0000+1.0000i -1.0000+0.0000i  0.0000-1.0000i
 5:  1.0000+0.0000i  0.3090+0.9511i -0.8090+0.5878i -0.8090-0.5878i  0.3090-0.9511i
 6:  1.0000+0.0000i  0.5000+0.8660i -0.5000+0.8660i -1.0000+0.0000i -0.5000-0.8660i  0.5000-0.8660i
 7:  1.0000+0.0000i  0.6235+0.7818i -0.2225+0.9749i -0.9010+0.4339i -0.9010-0.4339i -0.2225-0.9749i  0.6235-0.7818i
 8:  1.0000+0.0000i  0.7071+0.7071i  0.0000+1.0000i -0.7071+0.7071i -1.0000+0.0000i -0.7071-0.7071i  0.0000-1.0000i  0.7071-0.7071i
 9:  1.0000+0.0000i  0.7660+0.6428i  0.1736+0.9848i -0.5000+0.8660i -0.9397+0.3420i -0.9397-0.3420i -0.5000-0.8660i  0.1736-0.9848i  0.7660-0.6428i
10:  1.0000+0.0000i  0.8090+0.5878i  0.3090+0.9511i -0.3090+0.9511i -0.8090+0.5878i -1.0000+0.0000i -0.8090-0.5878i -0.3090-0.9511i  0.3090-0.9511i  0.8090-0.5878i
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