Roots of unity: Difference between revisions
m (→{{header|REXX}}: re-instated a glyph for the title.) |
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[1, 1.0i, -1.0, -1.0i] |
[1, 1.0i, -1.0, -1.0i] |
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[1, 0.30901699437494745 + 0.9510565162951535i, -0.8090169943749473 + 0.5877852522924732i, -0.8090169943749475 + -0.587785252292473i, 0.30901699437494723 + -0.9510565162951536i]</pre> |
[1, 0.30901699437494745 + 0.9510565162951535i, -0.8090169943749473 + 0.5877852522924732i, -0.8090169943749475 + -0.587785252292473i, 0.30901699437494723 + -0.9510565162951536i]</pre> |
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=={{header|Lambdatalk}}== |
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<lang scheme> |
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// cleandisp just to display 0 when n < 10^-10 |
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{def cleandisp |
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{lambda {:n} |
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{if {<= {abs :n} 1.e-10} then 0 else :n}}} |
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-> cleandisp |
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{def uroots |
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{lambda {:n} |
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{S.map {{lambda {:n :i} |
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{let { {:theta {/ {* 2 {PI} :i} :n}} |
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} {cons {cleandisp {cos :theta}} |
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{cleandisp {sin :theta}}}}} :n} |
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{S.serie 0 {- :n 1}}} }} |
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-> uroots |
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{S.map {lambda {:i} {hr}i = :i -> {uroots :i}} {S.serie 2 10}} |
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-> i = 2 -> (1 0) (-1 0) i = 3 -> (1 0) (-0.4999999999999998 0.8660254037844388) (-0.5000000000000004 -0.8660254037844384) |
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i = 4 -> (1 0) (0 1) (-1 0) (0 -1) |
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i = 5 -> (1 0) (0.30901699437494745 0.9510565162951535) (-0.8090169943749473 0.5877852522924732) (-0.8090169943749475 -0.587785252292473) (0.30901699437494723 -0.9510565162951536) |
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i = 6 -> (1 0) (0.5000000000000001 0.8660254037844386) (-0.4999999999999998 0.8660254037844388) (-1 0) (-0.5000000000000004 -0.8660254037844384) (0.5 -0.8660254037844386) |
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i = 7 -> (1 0) (0.6234898018587336 0.7818314824680297) (-0.22252093395631434 0.9749279121818236) (-0.900968867902419 0.43388373911755823) (-0.9009688679024191 -0.433883739117558) (-0.2225209339563146 -0.9749279121818235) (0.6234898018587334 -0.7818314824680299) |
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i = 8 -> (1 0) (0.7071067811865476 0.7071067811865475) (0 1) (-0.7071067811865475 0.7071067811865476) (-1 0) (-0.7071067811865477 -0.7071067811865475) (0 -1) (0.7071067811865475 -0.7071067811865477) |
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i = 9 -> (1 0) (0.7660444431189781 0.6427876096865393) (0.17364817766693041 0.984807753012208) (-0.4999999999999998 0.8660254037844388) (-0.9396926207859083 0.3420201433256689) (-0.9396926207859084 -0.34202014332566866) (-0.5000000000000004 -0.8660254037844384) (0.17364817766692997 -0.9848077530122081) (0.7660444431189779 -0.6427876096865396) |
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i = 10 -> (1 0) (0.8090169943749475 0.5877852522924731) (0.30901699437494745 0.9510565162951535) (-0.30901699437494734 0.9510565162951536) (-0.8090169943749473 0.5877852522924732) (-1 0) (-0.8090169943749475 -0.587785252292473) (-0.30901699437494756 -0.9510565162951535) (0.30901699437494723 -0.9510565162951536) (0.8090169943749473 -0.5877852522924732) |
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</lang> |
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=={{header|Liberty BASIC}}== |
=={{header|Liberty BASIC}}== |
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Revision as of 11:10, 22 September 2020
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.
- Task
Given n, find the nth roots of unity.
Ada
<lang ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Float_Text_IO; use Ada.Float_Text_IO; with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types;
procedure Roots_Of_Unity is
Root : Complex;
begin
for N in 2..10 loop
Put_Line ("N =" & Integer'Image (N));
for K in 0..N - 1 loop
Root :=
Compose_From_Polar
( Modulus => 1.0,
Argument => Float (K),
Cycle => Float (N)
);
-- Output
Put (" k =" & Integer'Image (K) & ", ");
if Re (Root) < 0.0 then
Put ("-");
else
Put ("+");
end if;
Put (abs Re (Root), Fore => 1, Exp => 0);
if Im (Root) < 0.0 then
Put ("-");
else
Put ("+");
end if;
Put (abs Im (Root), Fore => 1, Exp => 0);
Put_Line ("i");
end loop;
end loop;
end Roots_Of_Unity;</lang> Ada provides a direct implementation of polar composition of complex numbers x e2πi y. The function Compose_From_Polar is used to compose roots. The third argument of the function is the cycle. Instead of the standard cycle 2π, N is used. Sample output:
N = 2 k = 0, +1.00000+0.00000i k = 1, -1.00000+0.00000i N = 3 k = 0, +1.00000+0.00000i k = 1, -0.50000+0.86603i k = 2, -0.50000-0.86603i N = 4 k = 0, +1.00000+0.00000i k = 1, +0.00000+1.00000i k = 2, -1.00000+0.00000i k = 3, +0.00000-1.00000i N = 5 k = 0, +1.00000+0.00000i k = 1, +0.30902+0.95106i k = 2, -0.80902+0.58779i k = 3, -0.80902-0.58779i k = 4, +0.30902-0.95106i N = 6 k = 0, +1.00000+0.00000i k = 1, +0.50000+0.86603i k = 2, -0.50000+0.86603i k = 3, -1.00000+0.00000i k = 4, -0.50000-0.86603i k = 5, +0.50000-0.86603i N = 7 k = 0, +1.00000+0.00000i k = 1, +0.62349+0.78183i k = 2, -0.22252+0.97493i k = 3, -0.90097+0.43388i k = 4, -0.90097-0.43388i k = 5, -0.22252-0.97493i k = 6, +0.62349-0.78183i N = 8 k = 0, +1.00000+0.00000i k = 1, +0.70711+0.70711i k = 2, +0.00000+1.00000i k = 3, -0.70711+0.70711i k = 4, -1.00000+0.00000i k = 5, -0.70711-0.70711i k = 6, +0.00000-1.00000i k = 7, +0.70711-0.70711i N = 9 k = 0, +1.00000+0.00000i k = 1, +0.76604+0.64279i k = 2, +0.17365+0.98481i k = 3, -0.50000+0.86603i k = 4, -0.93969+0.34202i k = 5, -0.93969-0.34202i k = 6, -0.50000-0.86603i k = 7, +0.17365-0.98481i k = 8, +0.76604-0.64279i N = 10 k = 0, +1.00000+0.00000i k = 1, +0.80902+0.58779i k = 2, +0.30902+0.95106i k = 3, -0.30902+0.95106i k = 4, -0.80902+0.58779i k = 5, -1.00000+0.00000i k = 6, -0.80902-0.58779i k = 7, -0.30902-0.95106i k = 8, +0.30902-0.95106i k = 9, +0.80902-0.58779i
ALGOL 68
<lang algol68>FORMAT complex fmt=$g(-6,4)"⊥"g(-6,4)$; FOR root FROM 2 TO 10 DO
printf(($g(4)$,root)); FOR n FROM 0 TO root-1 DO printf(($xf(complex fmt)$,complex exp( 0 I 2*pi*n/root))) OD; printf($l$)
OD</lang> Output:
+2 1.0000⊥0.0000 -1.000⊥0.0000 +3 1.0000⊥0.0000 -.5000⊥0.8660 -.5000⊥-.8660 +4 1.0000⊥0.0000 0.0000⊥1.0000 -1.000⊥0.0000 -.0000⊥-1.000 +5 1.0000⊥0.0000 0.3090⊥0.9511 -.8090⊥0.5878 -.8090⊥-.5878 0.3090⊥-.9511 +6 1.0000⊥0.0000 0.5000⊥0.8660 -.5000⊥0.8660 -1.000⊥0.0000 -.5000⊥-.8660 0.5000⊥-.8660 +7 1.0000⊥0.0000 0.6235⊥0.7818 -.2225⊥0.9749 -.9010⊥0.4339 -.9010⊥-.4339 -.2225⊥-.9749 0.6235⊥-.7818 +8 1.0000⊥0.0000 0.7071⊥0.7071 0.0000⊥1.0000 -.7071⊥0.7071 -1.000⊥0.0000 -.7071⊥-.7071 -.0000⊥-1.000 0.7071⊥-.7071 +9 1.0000⊥0.0000 0.7660⊥0.6428 0.1736⊥0.9848 -.5000⊥0.8660 -.9397⊥0.3420 -.9397⊥-.3420 -.5000⊥-.8660 0.1736⊥-.9848 0.7660⊥-.6428 +10 1.0000⊥0.0000 0.8090⊥0.5878 0.3090⊥0.9511 -.3090⊥0.9511 -.8090⊥0.5878 -1.000⊥0.0000 -.8090⊥-.5878 -.3090⊥-.9511 0.3090⊥-.9511 0.8090⊥-.5878
AutoHotkey
ahk forum: discussion <lang AutoHotkey>n := 8, a := 8*atan(1)/n Loop %n%
i := A_Index-1, t .= cos(a*i) ((s:=sin(a*i))<0 ? " - i*" . -s : " + i*" . s) "`n"
Msgbox % t</lang>
AWK
<lang AWK>
- syntax: GAWK -f ROOTS_OF_UNITY.AWK
BEGIN {
pi = 3.1415926
for (n=2; n<=5; n++) {
printf("%d: ",n)
for (root=0; root<=n-1; root++) {
real = cos(2 * pi * root / n)
imag = sin(2 * pi * root / n)
printf("%8.5f %8.5fi",real,imag)
if (root != n-1) { printf(", ") }
}
printf("\n")
}
exit(0)
} </lang>
- Output:
2: 1.00000 0.00000i, -1.00000 0.00000i 3: 1.00000 0.00000i, -0.50000 0.86603i, -0.50000 -0.86603i 4: 1.00000 0.00000i, 0.00000 1.00000i, -1.00000 0.00000i, -0.00000 -1.00000i 5: 1.00000 0.00000i, 0.30902 0.95106i, -0.80902 0.58779i, -0.80902 -0.58779i, 0.30902 -0.95106i
BASIC
For high n's, this may repeat the root of 1 + 0*i. <lang qbasic> CLS
PI = 3.1415926# n = 5 'this can be changed for any desired n angle = 0 'start at angle 0 DO real = COS(angle) 'real axis is the x axis IF (ABS(real) < 10 ^ -5) THEN real = 0 'get rid of annoying sci notation imag = SIN(angle) 'imaginary axis is the y axis IF (ABS(imag) < 10 ^ -5) THEN imag = 0 'get rid of annoying sci notation PRINT real; "+"; imag; "i" 'answer on every line angle = angle + (2 * PI) / n 'all the way around the circle at even intervals LOOP WHILE angle < 2 * PI</lang>
BBC BASIC
<lang bbcbasic> @% = &20408
FOR n% = 2 TO 5
PRINT STR$(n%) ": " ;
FOR root% = 0 TO n%-1
real = COS(2*PI * root% / n%)
imag = SIN(2*PI * root% / n%)
PRINT real imag "i" ;
IF root% <> n%-1 PRINT "," ;
NEXT
PRINT
NEXT n%</lang>
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
C
<lang c>#include <stdio.h>
- include <math.h>
int main() { double a, c, s, PI2 = atan2(1, 1) * 8; int n, i;
for (n = 1; n < 10; n++) for (i = 0; i < n; i++) { c = s = 0; if (!i ) c = 1; else if(n == 4 * i) s = 1; else if(n == 2 * i) c = -1; else if(3 * n == 4 * i) s = -1; else a = i * PI2 / n, c = cos(a), s = sin(a);
if (c) printf("%.2g", c); printf(s == 1 ? "i" : s == -1 ? "-i" : s ? "%+.2gi" : "", s); printf(i == n - 1 ?"\n":", "); }
return 0; }</lang>
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Numerics;
class Program {
static IEnumerable<Complex> RootsOfUnity(int degree)
{
return Enumerable
.Range(0, degree)
.Select(element => Complex.FromPolarCoordinates(1, 2 * Math.PI * element / degree));
}
static void Main()
{
var degree = 3;
foreach (var root in RootsOfUnity(degree))
{
Console.WriteLine(root);
}
}
}</lang> Output:
(1, 0) (-0,5, 0,866025403784439) (-0,5, -0,866025403784438)
C++
<lang cpp>#include <complex>
- include <cmath>
- include <iostream>
double const pi = 4 * std::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;
}
}</lang>
CoffeeScript
Most of the effort here is in formatting the results, and the output is still a bit clumsy. <lang coffeescript># Find the n nth-roots of 1 nth_roots_of_unity = (n) ->
(complex_unit_vector(2*Math.PI*i/n) for i in [1..n])
complex_unit_vector = (rad) ->
new Complex(Math.cos(rad), Math.sin(rad))
class Complex
constructor: (@real, @imag) ->
toString: ->
round_z = (n) ->
if Math.abs(n) < 0.00005 then 0 else n
fmt = (n) -> n.toFixed(3)
real = round_z @real
imag = round_z @imag
s =
if real and imag
"#{fmt real}+#{fmt imag}i"
else if real or !imag
"#{fmt real}"
else
"#{fmt imag}i"
do ->
for n in [2..5]
console.log "---1 to the 1/#{n}"
for root in nth_roots_of_unity n
console.log root.toString()</lang>
output
> coffee nth_roots.coffee ---1 to the 1/2 -1.000 1.000 ---1 to the 1/3 -0.500+0.866i -0.500+-0.866i 1.000 ---1 to the 1/4 1.000i -1.000 -1.000i 1.000 ---1 to the 1/5 0.309+0.951i -0.809+0.588i -0.809+-0.588i 0.309+-0.951i 1.000
Common Lisp
<lang lisp>(defun roots-of-unity (n)
(loop for i below n
collect (cis (* pi (/ (* 2 i) n)))))</lang>
The expression is slightly more complicated than necessary in order to preserve exact rational arithmetic until multiplying by pi. The author of this example is not a floating point expert and not sure whether this is actually useful; if not, the simpler expression is (cis (/ (* 2 pi i) n)).
Crystal
<lang ruby>require "complex"
def roots_of_unity(n)
(0...n).map { |k| (2 * Math::PI * k / n).i.exp }
end
p roots_of_unity(3) </lang> Or alternative <lang ruby> def roots_of_unity(n)
(0...n).map { |k| Complex.new(Math.cos(2 * Math::PI * k / n), Math.sin(2 * Math::PI * k / n)) }
end </lang>
- Output:
[(1+0.0i), (-0.4999999999999998+0.8660254037844387i), (-0.5000000000000004-0.8660254037844384i)]
D
Using std.complex: <lang d>import std.stdio, std.range, std.algorithm, std.complex; import std.math: PI;
auto nthRoots(in int n) pure nothrow {
return n.iota.map!(k => expi(PI * 2 * (k + 1) / n));
}
void main() {
foreach (immutable i; 1 .. 6)
writefln("#%d: [%(%5.2f, %)]", i, i.nthRoots);
}</lang>
- Output:
#1: [ 1.00+ 0.00i] #2: [-1.00+-0.00i, 1.00+ 0.00i] #3: [-0.50+ 0.87i, -0.50+-0.87i, 1.00+ 0.00i] #4: [-0.00+ 1.00i, -1.00+-0.00i, 0.00+-1.00i, 1.00+ 0.00i] #5: [ 0.31+ 0.95i, -0.81+ 0.59i, -0.81+-0.59i, 0.31+-0.95i, 1.00+ 0.00i]
Delphi
<lang Delphi> program Roots_of_unity;
{$APPTYPE CONSOLE}
uses
System.VarCmplx;
function RootOfUnity(degree: integer): Tarray<Variant>; var
k: Integer;
begin
SetLength(result, degree); for k := 0 to degree - 1 do Result[k] := VarComplexFromPolar(1, 2 * pi * k / degree);
end;
const
n = 3;
var
num: Variant;
begin
Writeln('Root of unity from ', n, ':'#10);
for num in RootOfUnity(n) do
Writeln(num);
Readln;
end.</lang>
- Output:
Root of unity from 3: 1 + 0i -0,5 + 0,866025403784438i -0,5 - 0,866025403784439i
EchoLisp
<lang scheme> (define (roots-1 n)
(define theta (// (* 2 PI) n))
(for/list ((i n))
(polar 1. (* theta i))))
(roots-1 2)
→ (1+0i -1+0i)
(roots-1 3)
→ (1+0i -0.4999999999999998+0.8660254037844388i -0.5000000000000004-0.8660254037844384i)
(roots-1 4)
→ (1+0i 0+i -1+0i 0-i)
</lang>
ERRE
<lang> PROGRAM UNITY_ROOTS
! ! for rosettacode.org !
BEGIN
PRINT(CHR$(12);) !CLS
N=5 ! this can be changed for any desired n
ANGLE=0 ! start at ANGLE 0
REPEAT
REAL=COS(ANGLE) ! real axis is the x axis
IF (ABS(REAL)<10^-5) THEN REAL=0 END IF ! get rid of annoying sci notation
IMAG=SIN(ANGLE) ! imaginary axis is the y axis
IF (ABS(IMAG)<10^-5) THEN IMAG=0 END IF ! get rid of annoying sci notation
PRINT(REAL;"+";IMAG;"i") ! answer on every line
ANGLE+=(2*π)/N
! all the way around the circle at even intervals
UNTIL ANGLE>=2*π
END PROGRAM </lang> Note: Adapted from Qbasic version. π is the predefined constant Greek Pi.
Factor
<lang factor>USING: math.functions prettyprint ;
1 3 roots .</lang>
- Output:
{
1.0
C{ -0.4999999999999998 0.8660254037844387 }
C{ -0.5000000000000003 -0.8660254037844384 }
}
Forth
Complex numbers are not a native type in Forth, so we calculate the roots by hand. <lang forth>: 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</lang>
On the other hand, complex numbers are implemented by the FSL.
<lang forth>require fsl-util.fs require fsl/complex.fs
- abs= 1E-12 F~ ;
- clamp-to-0 FDUP 0E0 abs= IF FDROP 0E0 THEN ;
- zclamp-to-0
clamp-to-0 FSWAP clamp-to-0 FSWAP ;
- .roots
1+ 2 DO
I . ." : "
I 0 DO
1E0 2E0 PI F* I S>F F* J S>F F/ polar> zclamp-to-0 z. SPACE
LOOP
CR
LOOP ;
3 SET-PRECISION 5 .roots</lang>
Fortran
Sin/Cos + Scalar Loop
<lang fortran>PROGRAM Roots
COMPLEX :: root INTEGER :: i, n REAL :: angle, pi
pi = 4.0 * ATAN(1.0)
DO n = 2, 7
angle = 0.0
WRITE(*,"(I1,A)", ADVANCE="NO") n,": "
DO i = 1, n
root = CMPLX(COS(angle), SIN(angle))
WRITE(*,"(SP,2F7.4,A)", ADVANCE="NO") root, "j "
angle = angle + (2.0*pi / REAL(n))
END DO
WRITE(*,*)
END DO
END PROGRAM Roots</lang> Output
2: +1.0000+0.0000j -1.0000+0.0000j 3: +1.0000+0.0000j -0.5000+0.8660j -0.5000-0.8660j 4: +1.0000+0.0000j +0.0000+1.0000j -1.0000+0.0000j +0.0000-1.0000j 5: +1.0000+0.0000j +0.3090+0.9511j -0.8090+0.5878j -0.8090-0.5878j +0.3090-0.9511j 6: +1.0000+0.0000j +0.5000+0.8660j -0.5000+0.8660j -1.0000+0.0000j -0.5000-0.8660j +0.5000-0.8660j 7: +1.0000+0.0000j +0.6235+0.7818j -0.2225+0.9749j -0.9010+0.4339j -0.9010-0.4339j -0.2225-0.9749j +0.6235-0.7818j
Exp + Array-valued Statement
<lang fortran>program unity
real, parameter :: pi = 3.141592653589793
complex, parameter :: i = (0, 1)
complex, dimension(0:7-1) :: unit_circle
integer :: n, j
do n = 2, 7
!!!! KEY STEP, does all the calculations in one statement !!!!
unit_circle(0:n-1) = exp(2*i*pi/n * (/ (j, j=0, n-1) /) )
write(*,"(i1,a)", advance="no") n, ": "
write(*,"(sp,2f7.4,a)", advance="no") (unit_circle(j), "j ", j = 0, n-1)
write(*,*)
end do
end program unity</lang>
FreeBASIC
<lang freebasic>#define twopi 6.2831853071795864769252867665590057684
dim as uinteger m, n dim as double real, imag, theta input "n? ", n
for m = 0 to n-1
theta = m*twopi/n
real = cos(theta)
imag = sin(theta)
if imag >= 0 then
print using "#.##### + #.##### i"; real; imag
else
print using "#.##### - #.##### i"; real; -imag
end if
next m</lang>
Frink
Calculates the angles in degrees, since Frink will use rational arithmetic (exact) <lang Frink> roots[n] := {
a = makeArray[[n], 0]
alpha = 360/n degrees
theta = 0 degrees
for k = 0 to length[a] - 1
{
a@k = cos[theta] + i sin[theta]
theta = theta + alpha
}
a
} </lang>
- Output:
setPrecision[8] roots[3] [1.0, ( -0.5 + 0.86602540498103642 i ), ( -0.5 - 0.86602540139124295 i )]
FunL
FunL has built-in support for complex numbers. i is predefined to represent the imaginary unit.
<lang funl>import math.{exp, Pi}
def rootsOfUnity( n ) = {exp( 2Pi i k/n ) | k <- 0:n}
println( rootsOfUnity(3) )</lang>
- Output:
{1.0, -0.4999999999999998+0.8660254037844387i, -0.5000000000000004-0.8660254037844385i}
FutureBasic
<lang futurebasic> include "ConsoleWindow"
dim as long n, root dim as double real, imag
for n = 2 to 7 print n;":" ; for root = 0 to n-1 real = cos( 2 * pi * root / n) imag = sin( 2 * pi * root / n) print using "-##.#####"; real;using "-##.#####"; imag; "i"; if root <> n-1 then print ","; next print next </lang> Output:
2: 1.00000 0.00000i, -1.00000 0.00000i 3: 1.00000 0.00000i, -0.50000 0.86603i, -0.50000 -0.86603i 4: 1.00000 0.00000i, 0.00000 1.00000i, -1.00000 0.00000i, -0.00000 -1.00000i 5: 1.00000 0.00000i, 0.30902 0.95106i, -0.80902 0.58779i, -0.80902 -0.58779i, 0.30902 -0.95106i 6: 1.00000 0.00000i, 0.50000 0.86603i, -0.50000 0.86603i, -1.00000 0.00000i, -0.50000 -0.86603i, 0.50000 -0.86603i 7: 1.00000 0.00000i, 0.62349 0.78183i, -0.22252 0.97493i, -0.90097 0.43388i, -0.90097 -0.43388i, -0.22252 -0.97493i, 0.62349 -0.78183i
GAP
<lang gap>roots := n -> List([0 .. n-1], k -> E(n)^k);
r:=roots(7);
- [ 1, E(7), E(7)^2, E(7)^3, E(7)^4, E(7)^5, E(7)^6 ]
List(r, x -> x^7);
- [ 1, 1, 1, 1, 1, 1, 1 ]</lang>
Go
<lang go>package main
import (
"fmt" "math" "math/cmplx"
)
func main() {
for n := 2; n <= 5; n++ {
fmt.Printf("%d roots of 1:\n", n)
for _, r := range roots(n) {
fmt.Printf(" %18.15f\n", r)
}
}
}
func roots(n int) []complex128 {
r := make([]complex128, n)
for i := 0; i < n; i++ {
r[i] = cmplx.Rect(1, 2*math.Pi*float64(i)/float64(n))
}
return r
}</lang> Output:
2 roots of 1: ( 1.000000000000000+0.000000000000000i) (-1.000000000000000+0.000000000000000i) 3 roots of 1: ( 1.000000000000000+0.000000000000000i) (-0.500000000000000+0.866025403784439i) (-0.500000000000000-0.866025403784438i) 4 roots of 1: ( 1.000000000000000+0.000000000000000i) ( 0.000000000000000+1.000000000000000i) (-1.000000000000000+0.000000000000000i) (-0.000000000000000-1.000000000000000i) 5 roots of 1: ( 1.000000000000000+0.000000000000000i) ( 0.309016994374948+0.951056516295154i) (-0.809016994374947+0.587785252292473i) (-0.809016994374947-0.587785252292473i) ( 0.309016994374947-0.951056516295154i)
Groovy
Because the Groovy language does not provide a built-in facility for complex arithmetic, this example relies on the Complex class defined in the Complex numbers example. <lang groovy>/** The following closure creates a list of n evenly-spaced points around the unit circle,
* useful in FFT calculations, among other things */
def rootsOfUnity = { n ->
(0..<n).collect {
Complex.fromPolar(1, 2 * Math.PI * it / n)
}
}</lang> Test program: <lang groovy>def tol = 0.000000001 // tolerance: acceptable "wrongness" to account for rounding error
((1..6) + [16]). each { n ->
println "rootsOfUnity(${n}):"
def rou = rootsOfUnity(n)
rou.each { println it }
assert rou[0] == 1
def actual = n > 1 ? rou[Math.floor(n/2) as int] : rou[0]
def expected = n > 1 ? (n%2 == 0) ? -1 : ~rou[Math.ceil(n/2) as int] : rou[0]
def message = n > 1 ? (n%2 == 0) ? 'middle-most root should be -1' : 'two middle-most roots should be conjugates' :
assert (actual - expected).abs() < tol : message
assert rou.every { (it.rho - 1) < tol } : 'all roots should have magnitude 1'
println()
}</lang> Output:
rootsOfUnity(1): 1.0 rootsOfUnity(2): 1.0 -1.0 + 1.2246467991473532E-16i rootsOfUnity(3): 1.0 -0.4999999998186198 + 0.8660254038891585i -0.5000000003627604 - 0.8660254035749988i rootsOfUnity(4): 1.0 6.123233995736766E-17 + i -1.0 + 1.2246467991473532E-16i -1.8369701987210297E-16 - i rootsOfUnity(5): 1.0 0.30901699437494745 + 0.9510565162951535i -0.8090169943749473 + 0.5877852522924732i -0.8090169943749475 - 0.587785252292473i 0.30901699437494723 - 0.9510565162951536i rootsOfUnity(6): 1.0 0.4999999998186201 + 0.8660254038891584i -0.5000000003627598 + 0.8660254035749991i -1.0 - 6.283181638240517E-10i -0.4999999992744804 - 0.8660254042033175i 0.5000000009068993 - 0.8660254032608401i rootsOfUnity(16): 1.0 0.9238795325112867 + 0.3826834323650898i 0.7071067811865476 + 0.7071067811865475i 0.38268343236508984 + 0.9238795325112867i 6.123233995736766E-17 + i -0.3826834323650897 + 0.9238795325112867i -0.7071067811865475 + 0.7071067811865476i -0.9238795325112867 + 0.3826834323650899i -1.0 + 1.2246467991473532E-16i -0.9238795325112868 - 0.38268343236508967i -0.7071067811865477 - 0.7071067811865475i -0.38268343236509034 - 0.9238795325112865i -1.8369701987210297E-16 - i 0.38268343236509 - 0.9238795325112866i 0.7071067811865474 - 0.7071067811865477i 0.9238795325112865 - 0.3826834323650904i
Haskell
<lang haskell>import Data.Complex (Complex, cis)
rootsOfUnity :: (Enum a, Floating a) => a -> [Complex a] rootsOfUnity n =
[ cis (2 * pi * k / n) | k <- [0 .. n - 1] ]
main :: IO () main = mapM_ print $ rootsOfUnity 3</lang>
- Output:
<lang haskell>1.0 :+ 0.0 (-0.4999999999999998) :+ 0.8660254037844388 (-0.5000000000000004) :+ (-0.8660254037844384)</lang>
Icon and Unicon
<lang icon>procedure main()
roots(10)
end
procedure roots(n)
every n := 2 to 10 do
every writes(n | (str_rep((0 to (n-1)) * 2 * &pi / n)) | "\n")
end
procedure str_rep(k)
return " " || cos(k) || "+" || sin(k) || "i"
end</lang> Notes:
- The The Icon Programming Library implements a complex type but not a polar type
IDL
For some example n: <lang idl>n = 5 print, exp( dcomplex( 0, 2*!dpi/n) ) ^ ( 1 + indgen(n) )</lang> Outputs: <lang idl>( 0.30901699, 0.95105652)( -0.80901699, 0.58778525)( -0.80901699, -0.58778525)( 0.30901699, -0.95105652)( 1.0000000, -1.1102230e-16)</lang>
J
<lang j> rou=: [: ^ 0j2p1 * i. % ]
rou 4
1 0j1 _1 0j_1
rou 5
1 0.309017j0.951057 _0.809017j0.587785 _0.809017j_0.587785 0.309017j_0.951057</lang> The computation can also be written as a loop, shown here for comparison only. <lang j>rou1=: 3 : 0
z=. 0 $ r=. ^ o. 0j2 % y [ e=. 1 for. i.y do. z=. z,e e=. e*r end. z
)</lang>
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. <lang java>import java.util.Locale;
public class Test {
public static void main(String[] a) {
for (int n = 2; n < 6; n++)
unity(n);
}
public static void unity(int n) {
System.out.printf("%n%d: ", 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;
System.out.printf(Locale.US, "(%9f,%9f) ", real, imag);
}
}
}</lang>
2: ( 1.000000, 0.000000) (-1.000000, 0.000000) 3: ( 1.000000, 0.000000) (-0.500000, 0.866025) (-0.500000,-0.866025) 4: ( 1.000000, 0.000000) ( 0.000000, 1.000000) (-1.000000, 0.000000) ( 0.000000,-1.000000) 5: ( 1.000000, 0.000000) ( 0.309017, 0.951057) (-0.809017, 0.587785) (-0.809017,-0.587785) ( 0.309017,-0.951057)
JavaScript
<lang javascript>function Root(angle) { with (Math) { this.r = cos(angle); this.i = sin(angle) } }
Root.prototype.toFixed = function(p) { return this.r.toFixed(p) + (this.i >= 0 ? '+' : ) + this.i.toFixed(p) + 'i' }
function roots(n) { var rs = [], teta = 2*Math.PI/n for (var angle=0, i=0; i<n; angle+=teta, i+=1) rs.push( new Root(angle) ) return rs }
for (var n=2; n<8; n+=1) {
document.write(n, ': ')
var rs=roots(n); for (var i=0; i<rs.length; i+=1) document.write( i ? ', ' : , rs[i].toFixed(5) )
document.write('
')
}
</lang>
- Output:
2: 1.00000+0.00000i, -1.00000+0.00000i 3: 1.00000+0.00000i, -0.50000+0.86603i, -0.50000-0.86603i 4: 1.00000+0.00000i, 0.00000+1.00000i, -1.00000+0.00000i, -0.00000-1.00000i 5: 1.00000+0.00000i, 0.30902+0.95106i, -0.80902+0.58779i, -0.80902-0.58779i, 0.30902-0.95106i 6: 1.00000+0.00000i, 0.50000+0.86603i, -0.50000+0.86603i, -1.00000+0.00000i, -0.50000-0.86603i, 0.50000-0.86603i 7: 1.00000+0.00000i, 0.62349+0.78183i, -0.22252+0.97493i, -0.90097+0.43388i, -0.90097-0.43388i, -0.22252-0.97493i, 0.62349-0.78183i
jq
Using the same example as in the Julia section, and representing x + i*y as [x,y]: <lang jq>def nthroots(n):
(8 * (1|atan)) as $twopi | range(0;n) | (($twopi * .) / n) as $angle | [ ($angle | cos), ($angle | sin) ];
nthroots(10)</lang><lang jq>$ uname -a Darwin Mac-mini 13.3.0 Darwin Kernel Version 13.3.0: Tue Jun 3 21:27:35 PDT 2014; root:xnu-2422.110.17~1/RELEASE_X86_64 x86_64
$ time jq -c -n -f Roots_of_unity.jq [1,0] [0.8090169943749475,0.5877852522924731] [0.30901699437494745,0.9510565162951535] [-0.30901699437494734,0.9510565162951536] [-0.8090169943749473,0.5877852522924732] [-1,1.2246467991473532e-16] [-0.8090169943749475,-0.587785252292473] [-0.30901699437494756,-0.9510565162951535] [0.30901699437494723,-0.9510565162951536] [0.8090169943749473,-0.5877852522924732]
real 0m0.015s user 0m0.004s sys 0m0.004s </lang>
Julia
<lang julia>nthroots(n::Integer) = [ cospi(2k/n)+sinpi(2k/n)im for k = 0:n-1 ]</lang> (One could also use complex exponentials or other formulations.) For example, `nthroots(10)` gives:
10-element Array{Complex{Float64},1}:
1.0+0.0im
0.809017+0.587785im
0.309017+0.951057im
-0.309017+0.951057im
-0.809017+0.587785im
-1.0+0.0im
-0.809017-0.587785im
-0.309017-0.951057im
0.309017-0.951057im
0.809017-0.587785im
Kotlin
<lang scala>import java.lang.Math.*
data class Complex(val r: Double, val i: Double) {
override fun toString() = when {
i == 0.0 -> r.toString()
r == 0.0 -> i.toString() + 'i'
else -> "$r + ${i}i"
}
}
fun unity_roots(n: Number) = (1..n.toInt() - 1).map {
val a = it * 2 * PI / n.toDouble() var r = cos(a); if (abs(r) < 1e-6) r = 0.0 var i = sin(a); if (abs(i) < 1e-6) i = 0.0 Complex(r, i)
}
fun main(args: Array<String>) {
(1..4).forEach { println(listOf(1) + unity_roots(it)) }
println(listOf(1) + unity_roots(5.0))
}</lang>
- Output:
[1] [1, -1.0] [1, -0.4999999999999998 + 0.8660254037844387i, -0.5000000000000004 + -0.8660254037844385i] [1, 1.0i, -1.0, -1.0i] [1, 0.30901699437494745 + 0.9510565162951535i, -0.8090169943749473 + 0.5877852522924732i, -0.8090169943749475 + -0.587785252292473i, 0.30901699437494723 + -0.9510565162951536i]
Lambdatalk
<lang scheme> // cleandisp just to display 0 when n < 10^-10 {def cleandisp
{lambda {:n}
{if {<= {abs :n} 1.e-10} then 0 else :n}}}
-> cleandisp
{def uroots
{lambda {:n}
{S.map {{lambda {:n :i}
{let { {:theta {/ {* 2 {PI} :i} :n}}
} {cons {cleandisp {cos :theta}}
{cleandisp {sin :theta}}}}} :n}
{S.serie 0 {- :n 1}}} }}
-> uroots
{S.map {lambda {:i} {hr}i = :i -> {uroots :i}} {S.serie 2 10}} -> i = 2 -> (1 0) (-1 0) i = 3 -> (1 0) (-0.4999999999999998 0.8660254037844388) (-0.5000000000000004 -0.8660254037844384) i = 4 -> (1 0) (0 1) (-1 0) (0 -1) i = 5 -> (1 0) (0.30901699437494745 0.9510565162951535) (-0.8090169943749473 0.5877852522924732) (-0.8090169943749475 -0.587785252292473) (0.30901699437494723 -0.9510565162951536) i = 6 -> (1 0) (0.5000000000000001 0.8660254037844386) (-0.4999999999999998 0.8660254037844388) (-1 0) (-0.5000000000000004 -0.8660254037844384) (0.5 -0.8660254037844386) i = 7 -> (1 0) (0.6234898018587336 0.7818314824680297) (-0.22252093395631434 0.9749279121818236) (-0.900968867902419 0.43388373911755823) (-0.9009688679024191 -0.433883739117558) (-0.2225209339563146 -0.9749279121818235) (0.6234898018587334 -0.7818314824680299) i = 8 -> (1 0) (0.7071067811865476 0.7071067811865475) (0 1) (-0.7071067811865475 0.7071067811865476) (-1 0) (-0.7071067811865477 -0.7071067811865475) (0 -1) (0.7071067811865475 -0.7071067811865477) i = 9 -> (1 0) (0.7660444431189781 0.6427876096865393) (0.17364817766693041 0.984807753012208) (-0.4999999999999998 0.8660254037844388) (-0.9396926207859083 0.3420201433256689) (-0.9396926207859084 -0.34202014332566866) (-0.5000000000000004 -0.8660254037844384) (0.17364817766692997 -0.9848077530122081) (0.7660444431189779 -0.6427876096865396) i = 10 -> (1 0) (0.8090169943749475 0.5877852522924731) (0.30901699437494745 0.9510565162951535) (-0.30901699437494734 0.9510565162951536) (-0.8090169943749473 0.5877852522924732) (-1 0) (-0.8090169943749475 -0.587785252292473) (-0.30901699437494756 -0.9510565162951535) (0.30901699437494723 -0.9510565162951536) (0.8090169943749473 -0.5877852522924732)
</lang>
Liberty BASIC
<lang lb>WindowWidth =400 WindowHeight =400
'nomainwin
open "N'th Roots of One" for graphics_nsb_nf as #w
- w "trapclose [quit]"
for n =1 To 10
angle =0 #w "font arial 16 bold" print n; "th roots." #w "cls" #w "size 1 ; goto 200 200 ; down ; color lightgray ; circle 150 ; size 10 ; set 200 200 ; size 2" #w "up ; goto 200 0 ; down ; goto 200 400 ; up ; goto 0 200 ; down ; goto 400 200" #w "up ; goto 40 20 ; down ; color black" #w "font arial 6" #w "\"; n; " roots of 1."
for i = 1 To n
x = cos( Radian( angle))
y = sin( Radian( angle))
print using( "##", i); ": ( " + using( "##.######", x);_
" +i *" +using( "##.######", y); ") or e^( i *"; i -1; " *2 *Pi/ "; n; ")"
#w "color "; 255 *i /n; " 0 "; 256 -255 *i /n
#w "up ; goto 200 200"
#w "down ; goto "; 200 +150 *x; " "; 200 -150 *y
#w "up ; goto "; 200 +165 *x; " "; 200 -165 *y
#w "\"; str$( i)
#w "up"
angle =angle +360 /n
next i
timer 500, [on] wait [on] timer 0
next n
wait
[quit]
close #w
end
function Radian( theta)
Radian =theta *3.1415926535 /180
end function</lang>
Lua
Complex numbers from the Lua implementation on the complex numbers page. <lang lua>--defines addition, subtraction, negation, multiplication, division, conjugation, norms, and a conversion to strgs. complex = setmetatable({ __add = function(u, v) return complex(u.real + v.real, u.imag + v.imag) end, __sub = function(u, v) return complex(u.real - v.real, u.imag - v.imag) end, __mul = function(u, v) return complex(u.real * v.real - u.imag * v.imag, u.real * v.imag + u.imag * v.real) end, __div = function(u, v) return u * complex(v.real / v.norm, -v.imag / v.norm) end, __unm = function(u) return complex(-u.real, -u.imag) end, __concat = function(u, v)
if type(u) == "table" then return u.real .. " + " .. u.imag .. "i" .. v
elseif type(u) == "string" or type(u) == "number" then return u .. v.real .. " + " .. v.imag .. "i" end end, __index = function(u, index)
local operations = {
norm = function(u) return u.real ^ 2 + u.imag ^ 2 end,
conj = function(u) return complex(u.real, -u.imag) end,
}
return operations[index] and operations[index](u)
end, __newindex = function() error() end }, { __call = function(z, realpart, imagpart) return setmetatable({real = realpart, imag = imagpart}, complex) end } ) n = io.read() + 0 val = complex(math.cos(2*math.pi / n), math.sin(2*math.pi / n)) root = complex(1, 0) for i = 1, n do
root = root * val print(root .. "")
end</lang>
Maple
<lang Maple>RootsOfUnity := proc( n )
solve(z^n = 1, z);
end proc:</lang> <lang Maple>for i from 2 to 6 do
printf( "%d: %a\n", i, [ RootsOfUnity(i) ] );
end do;</lang> Output: <lang Maple>2: [1, -1] 3: [1, -1/2-1/2*I*3^(1/2), -1/2+1/2*I*3^(1/2)] 4: [1, -1, I, -I] 5: [1, 1/4*5^(1/2)-1/4+1/4*I*2^(1/2)*(5+5^(1/2))^(1/2), -1/4*5^(1/2)-1/4+1/4*I*2^(1/2)*(5-5^(1/2))^(1/2), -1/4*5^(1/2)-1/4-1/4*I*2^(1/2)*(5-5^(1/2))^(1/2), 1/4*5^(1/2)-1/4-1/4*I*2^(1/2)*(5+5^(1/2))^(1/2)] 6: [1, -1, 1/2*(-2-2*I*3^(1/2))^(1/2), -1/2*(-2-2*I*3^(1/2))^(1/2), 1/2*(-2+2*I*3^(1/2))^(1/2), -1/2*(-2+2*I*3^(1/2))^(1/2)]</lang>
Mathematica
Setting this up in Mathematica is easy, because it already handles complex numbers: <lang Mathematica>RootsUnity[nthroot_Integer?Positive] := Table[Exp[2 Pi I i/nthroot], {i, 0, nthroot - 1}]</lang> Note that Mathematica will keep the expression as exact as possible. Simplifications can be made to more known (trigonometric) functions by using the function ExpToTrig. If only a numerical approximation is necessary the function N will transform the exact result to a numerical approximation. Examples (exact not simplified, exact simplified, approximated):
RootsUnity[2] RootsUnity[3] RootsUnity[4] RootsUnity[5] RootsUnity[2]//ExpToTrig RootsUnity[3]//ExpToTrig RootsUnity[4]//ExpToTrig RootsUnity[5]//ExpToTrig RootsUnity[2]//N RootsUnity[3]//N RootsUnity[4]//N RootsUnity[5]//N
gives back:
MATLAB
<lang MATLAB>function z = rootsOfUnity(n)
assert(n >= 1,'n >= 1'); z = roots([1 zeros(1,n-1) -1]);
end</lang> Sample Output: <lang MATLAB>>> rootsOfUnity(3)
ans =
-0.500000000000000 + 0.866025403784439i -0.500000000000000 - 0.866025403784439i 1.000000000000000 </lang>
Maxima
<lang maxima>solve(1 = x^n, x)</lang> Demonstration: <lang maxima>for n:1 thru 5 do display(solve(1 = x^n, x));</lang> Output: <lang maxima>solve(1 = x, x) = [x = 1] solve(1 = x^2, x) = [x = -1, x = 1] solve(1 = x^3, x) = [x = (sqrt(3)*%i-1)/2, x = -(sqrt(3)*%i+1)/2, x = 1] solve(1 = x^4, x) = [x = %i, x = -1, x = -%i, x = 1] solve(1 = x^5, x) = [x = %e^((2*%i*%pi)/5), x = %e^((4*%i*%pi)/5), x = %e^(-(4*%i*%pi)/5), x = %e^(-(2*%i*%pi)/5), x = 1]</lang>
MiniScript
<lang MiniScript> complexRoots = function(n)
result = []
for i in range(0, n-1)
real = cos(2*pi * i/n)
if abs(real) < 1e-6 then real = 0
imag = sin(2*pi * i/n)
if abs(imag) < 1e-6 then imag = 0
result.push real + " " + "+" * (imag>=0) + imag + "i"
end for
return result
end function
for i in range(2,5)
print i + ": " + complexRoots(i).join(", ")
end for</lang>
- Output:
2: 1 +0i, -1 +0i 3: 1 +0i, -0.5 +0.866025i, -0.5 -0.866025i 4: 1 +0i, 0 +1i, -1 +0i, 0 -1i 5: 1 +0i, 0.309017 +0.951057i, -0.809017 +0.587785i, -0.809017 -0.587785i, 0.309017 -0.951057i
МК-61/52
<lang>П0 0 П1 ИП1 sin ИП1 cos С/П 2 пи
- ИП0 / ИП1 + П1 БП 03</lang>
Nim
<lang nim>import complex, math
proc rect(r, phi: float): Complex = (r * cos(phi), sin(phi))
proc croots(n): seq[Complex] =
result = @[] if n <= 0: return for k in 0 .. < n: result.add rect(1, 2 * k.float * Pi / n.float)
for nr in 2..10:
echo nr, " ", croots(nr)</lang>
Output:
2 @[(1.0, 0.0), (-1.0, 1.224646799147353e-16)] 3 @[(1.0, 0.0), (-0.4999999999999998, 0.8660254037844387), (-0.5000000000000004, -0.8660254037844384)] 4 @[(1.0, 0.0), (6.123233995736766e-17, 1.0), (-1.0, 1.224646799147353e-16), (-1.83697019872103e-16, -1.0)] 5 @[(1.0, 0.0), (0.3090169943749475, 0.9510565162951535), (-0.8090169943749473, 0.5877852522924732), (-0.8090169943749476, -0.587785252292473), (0.3090169943749472, -0.9510565162951536)] 6 @[(1.0, 0.0), (0.5000000000000001, 0.8660254037844386), (-0.4999999999999998, 0.8660254037844387), (-1.0, 1.224646799147353e-16), (-0.5000000000000004, -0.8660254037844384), (0.5000000000000001, -0.8660254037844386)] 7 @[(1.0, 0.0), (0.6234898018587336, 0.7818314824680298), (-0.2225209339563143, 0.9749279121818236), (-0.900968867902419, 0.4338837391175582), (-0.9009688679024191, -0.433883739117558), (-0.2225209339563146, -0.9749279121818236), (0.6234898018587334, -0.7818314824680299)] 8 @[(1.0, 0.0), (0.7071067811865476, 0.7071067811865475), (6.123233995736766e-17, 1.0), (-0.7071067811865475, 0.7071067811865476), (-1.0, 1.224646799147353e-16), (-0.7071067811865477, -0.7071067811865475), (-1.83697019872103e-16, -1.0), (0.7071067811865474, -0.7071067811865477)] 9 @[(1.0, 0.0), (0.766044443118978, 0.6427876096865393), (0.1736481776669304, 0.984807753012208), (-0.4999999999999998, 0.8660254037844387), (-0.9396926207859083, 0.3420201433256689), (-0.9396926207859084, -0.3420201433256687), (-0.5000000000000004, -0.8660254037844384), (0.17364817766693, -0.9848077530122081), (0.7660444431189778, -0.6427876096865396)] 10 @[(1.0, 0.0), (0.8090169943749475, 0.5877852522924731), (0.3090169943749475, 0.9510565162951535), (-0.3090169943749473, 0.9510565162951536), (-0.8090169943749473, 0.5877852522924732), (-1.0, 1.224646799147353e-16), (-0.8090169943749476, -0.587785252292473), (-0.3090169943749476, -0.9510565162951535), (0.3090169943749472, -0.9510565162951536), (0.8090169943749473, -0.5877852522924734)]
OCaml
<lang ocaml>open Complex
let pi = 4. *. atan 1.
let () =
for n = 1 to 10 do
Printf.printf "%2d " n;
for k = 1 to n do
let ret = polar 1. (2. *. pi *. float_of_int k /. float_of_int n) in
Printf.printf "(%f + %f i)" ret.re ret.im
done;
print_newline ()
done</lang>
Octave
<lang octave>for j = 2 : 10
printf("*** %d\n", j);
for n = 1 : j
disp(exp(2i*pi*n/j));
endfor
disp("");
endfor</lang>
OoRexx
<lang oorexx>/*REXX program computes the K roots of unity (which include complex roots).*/ parse Version v Say v parse arg n frac . /*get optional arguments from the C.L. */ if n== then n=1 /*Not specified? Then use the default.*/ if frac= then frac=5 /* " " " " " " */ start=abs(n) /*assume only one K is wanted. */ if n<0 then start=1 /*Negative? Then use a range of K's. */
/*display unity roots for a range, or */
do k=start to abs(n) /* just for one K. */
say right(k 'roots of unity',40,"-") /*display a pretty separator with title*/
do angle=0 by 360/k for k /*compute the angle for each root. */
rp=adjust(rxCalcCos(angle,,'D')) /*compute real part via COS function.*/
if left(rp,1)\=='-' then rp=" "rp /*not negative? Then pad with a blank.*/
ip=adjust(rxCalcSin(angle,,'D')) /*compute imaginary part via SIN funct.*/
if left(ip,1)\=='-' then ip="+"ip /*Not negative? Then pad with + char.*/
if ip=0 then say rp /*Only real part? Ignore imaginary part*/
else say left(rp,frac+4)ip'i' /*show the real & imaginary part*/
end /*angle*/
end /*k*/
exit /*stick a fork in it, we're all done. */ /*----------------------------------------------------------------------------*/ adjust: parse arg x; near0='1e-' || (digits()-digits()%10) /*compute small #*/
if abs(x)<near0 then x=0 /*if near zero, then assume zero.*/
return format(x,,frac)/1 /*fraction digits past dec point.*/
- requires rxMath library</lang>
- Output:
D:\>rexx nrootoo 5 REXX-ooRexx_4.2.0(MT)_64-bit 6.04 22 Feb 2014 ------------------------5 roots of unity 1 0.30902 +0.95106i -0.80902 +0.58779i -0.80902 -0.58779i 0.30902 -0.95106i
PARI/GP
<lang parigp>vector(n,k,exp(2*Pi*I*k/n))</lang>
sqrtn() can give the first n'th root, from which the others by multiplying or powering.
<lang parigp>nth_roots(n) = my(z);sqrtn(1,n,&z); vector(n,i, z^i);</lang>
Both the above give floating point complex numbers even when a root could be exact, like -1 or fourth root I.
quadgen() can be used for an exact 6th root. (Quads cannot be mixed with ordinary complex numbers, and they always print as w.)
<lang parigp>sixth_root = quadgen(-3); /* 6th root of unity, exact */ vector(6,n, sixth_root^n) /* all the 6'th roots */</lang>
Pascal
<lang pascal>Program Roots;
var
root: record // poor man's complex type. r: real; i: real; end; i, n: integer; angle: real;
begin
for n := 2 to 7 do
begin
angle := 0.0;
write(n, ': ');
for i := 1 to n do
begin
root.r := cos(angle);
root.i := sin(angle);
write(root.r:8:5, root.i:8:5, 'i ');
angle := angle + (2.0 * pi / n);
end;
writeln;
end;
end.</lang> Output:
2: 1.00000 0.00000i -1.00000 0.00000i 3: 1.00000 0.00000i -0.50000 0.86603i -0.50000-0.86603i 4: 1.00000 0.00000i 0.00000 1.00000i -1.00000 0.00000i -0.00000-1.00000i 5: 1.00000 0.00000i 0.30902 0.95106i -0.80902 0.58779i -0.80902-0.58779i 0.30902-0.95106i 6: 1.00000 0.00000i 0.50000 0.86603i -0.50000 0.86603i -1.00000-0.00000i -0.50000-0.86603i 0.50000-0.86603i 7: 1.00000 0.00000i 0.62349 0.78183i -0.22252 0.97493i -0.90097 0.43388i -0.90097-0.43388i -0.22252-0.97493i 0.62349-0.78183i
Perl
The root() function returns a list of the N many N'th roots of any complex Z, in this case 1.
<lang perl>use Math::Complex;
foreach my $n (2 .. 10) {
printf "%2d", $n;
my @roots = root(1,$n);
foreach my $root (@roots) {
$root->display_format(style => 'cartesian', format => '%.3f');
print " $root";
}
print "\n";
}</lang> Output:
2 1.000 -1.000+0.000i 3 1.000 -0.500+0.866i -0.500-0.866i 4 1.000 0.000+1.000i -1.000+0.000i -0.000-1.000i 5 1.000 0.309+0.951i -0.809+0.588i -0.809-0.588i 0.309-0.951i 6 1.000 0.500+0.866i -0.500+0.866i -1.000+0.000i -0.500-0.866i 0.500-0.866i 7 1.000 0.623+0.782i -0.223+0.975i -0.901+0.434i -0.901-0.434i -0.223-0.975i 0.623-0.782i 8 1.000 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 9 1.000 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 10 1.000 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
Phix
<lang Phix>for n=2 to 10 do
printf(1,"%2d:",n)
for root=0 to n-1 do
atom real = cos(2*PI*root/n)
atom imag = sin(2*PI*root/n)
printf(1,"%s %6.3f %6.3fi",{iff(root?",":""),real,imag})
end for
printf(1,"\n")
end for</lang>
2: 1.000 0.000i, -1.000 0.000i 3: 1.000 0.000i, -0.500 0.866i, -0.500 -0.866i 4: 1.000 0.000i, 0.000 1.000i, -1.000 0.000i, -0.000 -1.000i 5: 1.000 0.000i, 0.309 0.951i, -0.809 0.588i, -0.809 -0.588i, 0.309 -0.951i 6: 1.000 0.000i, 0.500 0.866i, -0.500 0.866i, -1.000 0.000i, -0.500 -0.866i, 0.500 -0.866i 7: 1.000 0.000i, 0.623 0.782i, -0.223 0.975i, -0.901 0.434i, -0.901 -0.434i, -0.223 -0.975i, 0.623 -0.782i 8: 1.000 0.000i, 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 9: 1.000 0.000i, 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 10: 1.000 0.000i, 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
PicoLisp
<lang PicoLisp>(load "@lib/math.l")
(for N (range 2 10)
(let Angle 0.0
(prin N ": ")
(for I N
(let Ipart (sin Angle)
(prin
(round (cos Angle) 4)
(if (lt0 Ipart) "-" "+")
"j"
(round (abs Ipart) 4)
" " ) )
(inc 'Angle (*/ 2 pi N)) )
(prinl) ) )</lang>
PL/I
<lang PL/I>complex_roots:
procedure (N); declare N fixed binary nonassignable; declare x float, c fixed decimal (10,8) complex; declare twopi float initial ((4*asin(1.0)));
do x = 0 to twopi by twopi/N;
c = complex(cos(x), sin(x));
put skip list (c);
end;
end complex_roots;
1.00000000+0.00000000I 0.80901700+0.58778524I 0.30901697+0.95105654I -0.30901703+0.95105648I -0.80901706+0.58778518I -1.00000000-0.00000008I -0.80901694-0.58778536I -0.30901709-0.95105648I 0.30901712-0.95105648I 0.80901724-0.58778494I </lang>
Prolog
Solves the roots of unity symbolically, as complex powers of e. <lang Prolog> roots(N, Rs) :-
succ(Pn, N), numlist(0, Pn, Ks), maplist(root(N), Ks, Rs).
root(N, M, R2) :-
R0 is (2*M) rdiv N, % multiple of PI (R0 > 1 -> R1 is R0 - 2; R1 = R0), % adjust for principal values cis(R1, R2).
cis(0, 1) :- !. cis(1, -1) :- !. cis(1 rdiv 2, i) :- !. cis(-1 rdiv 2, -i) :- !. cis(-1 rdiv Q, exp(-i*pi/Q)) :- !. cis(1 rdiv Q, exp(i*pi/Q)) :- !. cis(P rdiv Q, exp(P*i*pi/Q)). </lang>
- Output:
?- roots(2,X).
X = [1, -1].
?- roots(3,X).
X = [1, exp(2*i*pi/3), exp(-2*i*pi/3)].
?- roots(4,X).
X = [1, i, -1, -i].
?- roots(5,X).
X = [1, exp(2*i*pi/5), exp(4*i*pi/5), exp(-4*i*pi/5), exp(-2*i*pi/5)].
?- roots(8,X), forall(member(A,X), format("~w~n", A)).
1
exp(i*pi/4)
i
exp(3*i*pi/4)
-1
exp(-3*i*pi/4)
-i
exp(-i*pi/4)
X = [1, exp(i*pi/4), i, exp(3*i*pi/4), -1, exp(... * ... * pi/4), -i, exp(... / ...)].
PureBasic
<lang Purebasic>OpenConsole() For n = 2 To 10
angle = 0 PrintN(Str(n)) For i = 1 To n x.f = Cos(Radian(angle)) y.f = Sin(Radian(angle)) PrintN( Str(i) + ": " + StrF(x, 6) + " / " + StrF(y, 6)) angle = angle + (360 / n) Next
Next Input()</lang>
Python
<lang python>import cmath
class Complex(complex):
def __repr__(self):
rp = '%7.5f' % self.real if not self.pureImag() else
ip = '%7.5fj' % self.imag if not self.pureReal() else
conj = if (
self.pureImag() or self.pureReal() or self.imag < 0.0
) else '+'
return '0.0' if (
self.pureImag() and self.pureReal()
) else rp + conj + ip
def pureImag(self):
return abs(self.real) < 0.000005
def pureReal(self):
return abs(self.imag) < 0.000005
def croots(n):
if n <= 0:
return None
return (Complex(cmath.rect(1, 2 * k * cmath.pi / n)) for k in range(n))
# in pre-Python 2.6:
# return (Complex(cmath.exp(2j*k*cmath.pi/n)) for k in range(n))
for nr in range(2, 11):
print(nr, list(croots(nr)))</lang>
- Output:
2 [1.00000, -1.00000] 3 [1.00000, -0.50000+0.86603j, -0.50000-0.86603j] 4 [1.00000, 1.00000j, -1.00000, -1.00000j] 5 [1.00000, 0.30902+0.95106j, -0.80902+0.58779j, -0.80902-0.58779j, 0.30902-0.95106j] 6 [1.00000, 0.50000+0.86603j, -0.50000+0.86603j, -1.00000, -0.50000-0.86603j, 0.50000-0.86603j] 7 [1.00000, 0.62349+0.78183j, -0.22252+0.97493j, -0.90097+0.43388j, -0.90097-0.43388j, -0.22252-0.97493j, 0.62349-0.78183j] 8 [1.00000, 0.70711+0.70711j, 1.00000j, -0.70711+0.70711j, -1.00000, -0.70711-0.70711j, -1.00000j, 0.70711-0.70711j] 9 [1.00000, 0.76604+0.64279j, 0.17365+0.98481j, -0.50000+0.86603j, -0.93969+0.34202j, -0.93969-0.34202j, -0.50000-0.86603j, 0.17365-0.98481j, 0.76604-0.64279j] 10 [1.00000, 0.80902+0.58779j, 0.30902+0.95106j, -0.30902+0.95106j, -0.80902+0.58779j, -1.00000, -0.80902-0.58779j, -0.30902-0.95106j, 0.30902-0.95106j, 0.80902-0.58779j]
R
<lang R>for(j in 2:10) {
r <- sprintf("%d: ", j)
for(n in 1:j) {
r <- paste(r, format(exp(2i*pi*n/j), digits=4), ifelse(n<j, ",", ""))
}
print(r)
}</lang> Output:
[1] "2: -1+0i , 1-0i " [1] "3: -0.5+0.866i , -0.5-0.866i , 1-0i " [1] "4: 0+1i , -1+0i , 0-1i , 1-0i " [1] "5: 0.309+0.9511i , -0.809+0.5878i , -0.809-0.5878i , 0.309-0.9511i , 1-0i " [1] "6: 0.5+0.866i , -0.5+0.866i , -1+0i , -0.5-0.866i , 0.5-0.866i , 1-0i " [1] "7: 0.6235+0.7818i , -0.2225+0.9749i , -0.901+0.4339i , -0.901-0.4339i , -0.2225-0.9749i , 0.6235-0.7818i , 1-0i " [1] "8: 0.7071+0.7071i , 0+1i , -0.7071+0.7071i , -1+0i , -0.7071-0.7071i , 0-1i , 0.7071-0.7071i , 1-0i " [1] "9: 0.766+0.6428i , 0.1736+0.9848i , -0.5+0.866i , -0.9397+0.342i , -0.9397-0.342i , -0.5-0.866i , 0.1736-0.9848i , 0.766-0.6428i , 1-0i " [1] "10: 0.809+0.5878i , 0.309+0.9511i , -0.309+0.9511i , -0.809+0.5878i , -1+0i , -0.809-0.5878i , -0.309-0.9511i , 0.309-0.9511i , 0.809-0.5878i , 1-0i "
Racket
<lang Racket>#lang racket
(define (roots-of-unity n)
(for/list ([k n]) (make-polar 1 (* k (/ (* 2 pi) n)))))</lang>
Will produce a list of roots, for example:
> (for ([r (roots-of-unity 3)]) (displayln r)) 1 -0.4999999999999998+0.8660254037844388i -0.5000000000000004-0.8660254037844384i
Raku
(formerly Perl 6) Raku has a built-in function cis which returns a unitary complex number given its phase. Raku also defines the tau = 2*pi constant. Thus the k-th n-root of unity can simply be written cis(k*τ/n).
<lang perl6>constant n = 10; for ^n -> \k {
say cis(k*τ/n);
}</lang>
- Output:
1+0i 0.809016994374947+0.587785252292473i 0.309016994374947+0.951056516295154i -0.309016994374947+0.951056516295154i -0.809016994374947+0.587785252292473i -1+1.22464679914735e-16i -0.809016994374948-0.587785252292473i -0.309016994374948-0.951056516295154i 0.309016994374947-0.951056516295154i 0.809016994374947-0.587785252292473i
REXX
REXX doesn't have complex arithmetic, so the (real) values of cos and sin of multiples of 2 pi radians (divided by K) are used.
Also, REXX doesn't have the pi constant defined, nor a sin or cos function, so they are included below within the REXX program.
Note: this REXX version only displays 5 significant digits past the decimal point, but this can be overridden by specifying the 2nd argument when invoking the REXX program. (See the value of the REXX variable frac, 4th line). <lang rexx>/*REXX program computes the K roots of unity (which usually includes complex roots).*/ numeric digits length( pi() ) - length(.) /*use number of decimal digits in pi. */ parse arg n frac . /*get optional arguments from the C.L. */ if n== | n=="," then n= 1 /*Not specified? Then use the default.*/ if frac= | frac=="," then frac= 5 /* " " " " " " */
start= abs(n) /*assume only one K is wanted. */
if n<0 then start= 1 /*Negative? Then use a range of K's. */
pi2= pi + pi /*obtain the value of pi doubled. */
do #=start to abs(n) /*show unity roots (for a range or 1 K.*/
say right(# 'roots of unity', 40, "─") ' (showing' frac "fractional decimal digits)"
do angle=0 by pi2/# for # /*compute the angle for each root. */
rp= adj( cos(angle) ) /*compute real part via COS function.*/
if left(rp, 1) \== '-' then rp= " "rp /*not negative? Then pad with a blank.*/
ip= adj( sin(angle) ) /*compute imaginary part via SIN funct.*/
if left(ip, 1) \== '-' then ip= "+"ip /*Not negative? Then pad with + char.*/
if ip=0 then say rp /*Only real part? Ignore imaginary part*/
else say left(rp, frac+4)ip'i' /*display the real and imaginary part. */
end /*angle*/
end /*#*/
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ pi: pi=3.141592653589793238462643383279502884197169399375105820974944592307816; return pi r2r: return arg(1) // ( pi() + pi ) /*reduce #radians: -2pi ─► +2pi radians*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ adj: parse arg x; near0= '1e-' || (digits() * 9 % 10) /*compute a tiny number.*/
if abs(x) < near0 then x= 0 /*if it's near zero, then assume zero.*/
return format(x, , frac) / 1 /*fraction digits past decimal point. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; x= r2r(x); a= abs(x); numeric fuzz min(9, digits()-9)
if a=pi/3 then return .5; if a=pi/2 | a=pi*2 then return 0
if a=pi then return -1; if a=pi*2/3 then return -.5; z= 1; _= 1; $x= x * x
do k=2 by 2 until p=z; p=z; _= -_ * $x / (k*(k-1)); z= z + _; end; return z
/*──────────────────────────────────────────────────────────────────────────────────────*/ sin: procedure; parse arg x; x= r2r(x); numeric fuzz min(5, digits()-3)
if abs(x)=pi then return 0; z= x; _= x; $x= x * x
do k=2 by 2 until p=z; p=z; _= -_ * $x / (k*(k+1)); z= z + _; end; return z</lang>
- output when using the input of: 5
────────────────────────5 roots of unity (showing 5 fractional decimal digits) 1 0.30902 +0.95106i -0.80902 +0.58779i -0.80902 -0.58779i 0.30902 -0.95106i
- output when using the input of: 10 36
───────────────────────10 roots of unity (showing 36 fractional decimal digits) 1 0.809016994374947424102293417182819059 +0.587785252292473129168705954639072769i 0.309016994374947424102293417182819059 +0.951056516295153572116439333379382143i -0.309016994374947424102293417182819059 +0.951056516295153572116439333379382143i -0.809016994374947424102293417182819059 +0.587785252292473129168705954639072769i -1 -0.809016994374947424102293417182819059 -0.587785252292473129168705954639072769i -0.309016994374947424102293417182819059 -0.951056516295153572116439333379382143i 0.309016994374947424102293417182819059 -0.951056516295153572116439333379382143i 0.809016994374947424102293417182819059 -0.587785252292473129168705954639072769i
- output when using the input of: -12
(Shown at five-sixths size.)
────────────────────────1 roots of unity (showing 5 fractional decimal digits) 1 ────────────────────────2 roots of unity (showing 5 fractional decimal digits) 1 -1 ────────────────────────3 roots of unity (showing 5 fractional decimal digits) 1 -0.5 +0.86603i -0.5 -0.86603i ────────────────────────4 roots of unity (showing 5 fractional decimal digits) 1 0 +1i -1 0 -1i ────────────────────────5 roots of unity (showing 5 fractional decimal digits) 1 0.30902 +0.95106i -0.80902 +0.58779i -0.80902 -0.58779i 0.30902 -0.95106i ────────────────────────6 roots of unity (showing 5 fractional decimal digits) 1 0.5 +0.86603i -0.5 +0.86603i -1 -0.5 -0.86603i 0.5 -0.86603i ────────────────────────7 roots of unity (showing 5 fractional decimal digits) 1 0.62349 +0.78183i -0.22252 +0.97493i -0.90097 +0.43388i -0.90097 -0.43388i -0.22252 -0.97493i 0.62349 -0.78183i ────────────────────────8 roots of unity (showing 5 fractional decimal digits) 1 0.70711 +0.70711i 0 +1i -0.70711 +0.70711i -1 -0.70711 -0.70711i 0 -1i 0.70711 -0.70711i ────────────────────────9 roots of unity (showing 5 fractional decimal digits) 1 0.76604 +0.64279i 0.17365 +0.98481i -0.5 +0.86603i -0.93969 +0.34202i -0.93969 -0.34202i -0.5 -0.86603i 0.17365 -0.98481i 0.76604 -0.64279i ───────────────────────10 roots of unity (showing 5 fractional decimal digits) 1 0.80902 +0.58779i 0.30902 +0.95106i -0.30902 +0.95106i -0.80902 +0.58779i -1 -0.80902 -0.58779i -0.30902 -0.95106i 0.30902 -0.95106i 0.80902 -0.58779i ───────────────────────11 roots of unity (showing 5 fractional decimal digits) 1 0.84125 +0.54064i 0.41542 +0.90963i -0.14231 +0.98982i -0.65486 +0.75575i -0.95949 +0.28173i -0.95949 -0.28173i -0.65486 -0.75575i -0.14231 -0.98982i 0.41542 -0.90963i 0.84125 -0.54064i ───────────────────────12 roots of unity (showing 5 fractional decimal digits) 1 0.86603 +0.5i 0.5 +0.86603i 0 +1i -0.5 +0.86603i -0.86603 +0.5i -1 -0.86603 -0.5i -0.5 -0.86603i 0 -1i 0.5 -0.86603i 0.86603 -0.5i
Ring
<lang ring> decimals(4) for n = 2 to 5
see string(n) + " : "
for root = 0 to n-1
real = cos(2*3.14 * root / n)
imag = sin(2*3.14 * root / n)
see "" + real + " " + imag + "i"
if root != n-1 see ", " ok
next
see nl
next </lang>
RLaB
RLaB can find the n-roots of unity by solving the polynomial equation
It uses the solver polyroots. Interested user is recommended to check the rlabplus manual for details on the solver and the parameters that tune the solver performance. <lang RLaB>// specify polynomial >> n = 10; >> a = zeros(1,n+1); a[1] = 1; a[n+1] = -1; >> polyroots(a)
radius roots success
>> polyroots(a).roots
-0.309016994 + 0.951056516i
-0.809016994 + 0.587785252i
-1 + 5.95570041e-23i
-0.809016994 - 0.587785252i
-0.309016994 - 0.951056516i
0.309016994 - 0.951056516i
0.809016994 - 0.587785252i
1 + 0i
0.809016994 + 0.587785252i
0.309016994 + 0.951056516i</lang>
Ruby
<lang ruby>def roots_of_unity(n)
(0...n).map {|k| Complex.polar(1, 2 * Math::PI * k / n)}
end
p roots_of_unity(3)</lang>
- Output:
[(1+0.0i), (-0.4999999999999998+0.8660254037844387i), (-0.5000000000000004-0.8660254037844384i)]
Run BASIC
<lang runbasic>PI = 3.1415926535 FOR n = 2 TO 5
PRINT n;":" ;
FOR root = 0 TO n-1
real = COS(2*PI * root / n)
imag = SIN(2*PI * root / n)
PRINT using("-##.#####",real);using("-##.#####",imag);"i";
IF root <> n-1 then PRINT "," ;
NEXT
PRINT
NEXT </lang> Output:
2: 1.00000 0.00000i, -1.00000 0.00000i 3: 1.00000 0.00000i, -0.50000 0.86603i, -0.50000 -0.86603i 4: 1.00000 0.00000i, 0.00000 1.00000i, -1.00000 0.00000i, 0.00000 -1.00000i 5: 1.00000 0.00000i, 0.30902 0.95106i, -0.80902 0.58779i, -0.80902 -0.58779i, 0.30902 -0.95106i
Rust
Here we demonstrate initialization from polar complex coordinate, radius 1, e^πi/n, and raising the resulting complex number to the power 2k for k in 0..n-1, which generates approximate roots (see the Mathematica answer for a nice display of exact vs approximate). This code will require adding the num crate to one's rust project, typically in Cargo.toml [dependencies] \n num="0.2.0"; <lang C>use num::Complex; fn main() {
let n = 8;
let z = Complex::from_polar(&1.0,&(1.0*std::f64::consts::PI/n as f64));
for k in 0..=n-1 {
println!("e^{:2}πi/{} ≈ {:>14.3}",2*k,n,z.powf(2.0*k as f64));
}
}</lang>
e^ 0πi/8 ≈ 1.000+0.000i e^ 2πi/8 ≈ 0.707+0.707i e^ 4πi/8 ≈ 0.000+1.000i e^ 6πi/8 ≈ -0.707+0.707i e^ 8πi/8 ≈ -1.000+0.000i e^10πi/8 ≈ -0.707-0.707i e^12πi/8 ≈ -0.000-1.000i e^14πi/8 ≈ 0.707-0.707i
Scala
Using Complex class from task Arithmetic/Complex. <lang scala>def rootsOfUnity(n:Int)=for(k <- 0 until n) yield Complex.fromPolar(1.0, 2*math.Pi*k/n)</lang> Usage:
rootsOfUnity(3) foreach println 1.0+0.0i -0.4999999999999998+0.8660254037844387i -0.5000000000000004-0.8660254037844385i
Scheme
<lang scheme>(define pi (* 4 (atan 1)))
(do ((n 2 (+ n 1)))
((> n 10))
(display n)
(do ((k 0 (+ k 1)))
((>= k n))
(display " ")
(display (make-polar 1 (* 2 pi (/ k n)))))
(newline))</lang>
Seed7
<lang 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;</lang>
Output: <lang seed7>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</lang>
Sidef
<lang ruby>func roots_of_unity(n) {
n.of { |j|
exp(2i * Num.pi / n * j)
}
}
roots_of_unity(5).each { |c|
printf("%+.5f%+.5fi\n", c.reals)
}</lang>
- Output:
+1.00000+0.00000i +0.30902+0.95106i -0.80902+0.58779i -0.80902-0.58779i +0.30902-0.95106i
Sparkling
<lang sparkling>function unity_roots(n) { // nth-root(1) = cos(2 * k * pi / n) + i * sin(2 * k * pi / n) return map(range(n), function(idx, k) { return { "re": cos(2 * k * M_PI / n), "im": sin(2 * k * M_PI / n) }; }); }
// pirnt 6th roots of unity foreach(unity_roots(6), function(k, v) { printf("%.3f%+.3fi\n", v.re, v.im); });</lang>
Stata
<lang stata>n=7 exp(2i*pi()/n*(0::n-1))
1 +-----------------------------+ 1 | 1 | 2 | .623489802 + .781831482i | 3 | -.222520934 + .974927912i | 4 | -.900968868 + .433883739i | 5 | -.900968868 - .433883739i | 6 | -.222520934 - .974927912i | 7 | .623489802 - .781831482i | +-----------------------------+</lang>
Tcl
<lang Tcl>package require Tcl 8.5 namespace import tcl::mathfunc::*
set pi 3.14159265 for {set n 2} {$n <= 10} {incr n} {
set angle 0.0
set row $n:
for {set i 1} {$i <= $n} {incr i} {
lappend row [format %5.4f%+5.4fi [cos $angle] [sin $angle]]
set angle [expr {$angle + 2*$pi/$n}]
}
puts $row
}</lang>
TI-89 BASIC
<lang ti89b>cZeros(x^n - 1, x)</lang> For n=3 in exact mode, the results are <lang ti89b>{-1/2+√(3)/2*i, -1/2-√(3)/2*i, 1}</lang>
Ursala
The roots function takes a number n to the nth root of -1, squares it, and iteratively makes a list of its first n powers (oblivious to roundoff error). Complex functions cpow and mul are used, which are called from the host system's standard C library. <lang Ursala>#import std
- import nat
- import flo
roots = ~&htxPC+ c..mul:-0^*DlSiiDlStK9\iota c..mul@iiX+ c..cpow/-1.+ div/1.+ float
- cast %jLL
tests = roots* <1,2,3,4,5,6></lang> The output is a list of lists of complex numbers.
<
<1.000e+00-2.449e-16j>,
<
1.000e+00-2.449e-16j,
-1.000e+00+1.225e-16j>,
<
1.000e+00-8.327e-16j,
-5.000e-01+8.660e-01j,
-5.000e-01-8.660e-01j>,
<
1.000e+00-8.882e-16j,
2.220e-16+1.000e+00j,
-1.000e+00+4.441e-16j,
-6.661e-16-1.000e+00j>,
<
1.000e+00-5.551e-17j,
3.090e-01+9.511e-01j,
-8.090e-01+5.878e-01j,
-8.090e-01-5.878e-01j,
3.090e-01-9.511e-01j>,
<
1.000e+00-1.221e-15j,
5.000e-01+8.660e-01j,
-5.000e-01+8.660e-01j,
-1.000e+00+6.106e-16j,
-5.000e-01-8.660e-01j,
5.000e-01-8.660e-01j>>
VBA
<lang vb>Public Sub roots_of_unity()
For n = 2 To 9
Debug.Print n; "th roots of 1:"
For r00t = 0 To n - 1
Debug.Print " Root "; r00t & ": "; WorksheetFunction.Complex(Cos(2 * WorksheetFunction.Pi() * r00t / n), _
Sin(2 * WorksheetFunction.Pi() * r00t / n))
Next r00t
Debug.Print
Next n
End Sub</lang>
- Output:
2 th roots of 1: Root 0: 1 Root 1: -1+1.22460635382238E-16i 3 th roots of 1: Root 0: 1 Root 1: -0.5+0.866025403784439i Root 2: -0.5-0.866025403784438i 4 th roots of 1: Root 0: 1 Root 1: 6.12303176911189E-17+i Root 2: -1+1.22460635382238E-16i Root 3: -1.83690953073357E-16-i 5 th roots of 1: Root 0: 1 Root 1: 0.309016994374947+0.951056516295154i Root 2: -0.809016994374947+0.587785252292473i Root 3: -0.809016994374948-0.587785252292473i Root 4: 0.309016994374947-0.951056516295154i 6 th roots of 1: Root 0: 1 Root 1: 0.5+0.866025403784439i Root 2: -0.5+0.866025403784439i Root 3: -1+1.22460635382238E-16i Root 4: -0.5-0.866025403784438i Root 5: 0.5-0.866025403784439i 7 th roots of 1: Root 0: 1 Root 1: 0.623489801858734+0.78183148246803i Root 2: -0.222520933956314+0.974927912181824i Root 3: -0.900968867902419+0.433883739117558i Root 4: -0.900968867902419-0.433883739117558i Root 5: -0.222520933956315-0.974927912181824i Root 6: 0.623489801858733-0.78183148246803i 8 th roots of 1: Root 0: 1 Root 1: 0.707106781186548+0.707106781186547i Root 2: 6.12303176911189E-17+i Root 3: -0.707106781186547+0.707106781186548i Root 4: -1+1.22460635382238E-16i Root 5: -0.707106781186548-0.707106781186547i Root 6: -1.83690953073357E-16-i Root 7: 0.707106781186547-0.707106781186548i 9 th roots of 1: Root 0: 1 Root 1: 0.766044443118978+0.642787609686539i Root 2: 0.17364817766693+0.984807753012208i Root 3: -0.5+0.866025403784439i Root 4: -0.939692620785908+0.342020143325669i Root 5: -0.939692620785908-0.342020143325669i Root 6: -0.5-0.866025403784438i Root 7: 0.17364817766693-0.984807753012208i Root 8: 0.766044443118978-0.64278760968654i
zkl
<lang zkl>PI2:=(0.0).pi*2; foreach n,i in ([1..9],n){
c:=s:=0;
if(not i) c = 1;
else if(n==4*i) s = 1;
else if(n==2*i) c = -1;
else if(3*n==4*i) s = -1;
else a,c,s:=PI2*i/n,a.cos(),a.sin();
if(c) print("%.2g".fmt(c));
print( (s==1 and "i") or (s==-1 and "-i" or (s and "%+.2gi" or"")).fmt(s));
print( (i==n-1) and "\n" or ", ");
}</lang>
- Output:
1 1, -1 1, -0.5+0.87i, -0.5-0.87i 1, i, -1, -i 1, 0.31+0.95i, -0.81+0.59i, -0.81-0.59i, 0.31-0.95i 1, 0.5+0.87i, -0.5+0.87i, -1, -0.5-0.87i, 0.5-0.87i 1, 0.62+0.78i, -0.22+0.97i, -0.9+0.43i, -0.9-0.43i, -0.22-0.97i, 0.62-0.78i 1, 0.71+0.71i, i, -0.71+0.71i, -1, -0.71-0.71i, -i, 0.71-0.71i 1, 0.77+0.64i, 0.17+0.98i, -0.5+0.87i, -0.94+0.34i, -0.94-0.34i, -0.5-0.87i, 0.17-0.98i, 0.77-0.64i
- Programming Tasks
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