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
<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>
BASIC
For high n's, this may repeat the root of 1 + 0*i.
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
C
<lang c>#include <stdio.h>
- include <math.h>
- define PI 3.1415926
int main (int argc, char *argv[]) {
char sign; int i, n; float rpart, ipart, angle;
for (n = 2; n <= 10; n++)
{
angle = 0.0;
printf("%d: ", n);
for (i = 1; i <= n; i++)
{
rpart = cos(angle);
ipart = sin(angle);
if (ipart < 0)
sign = '-';
else
sign = '+';
printf("%5.4f%cj%5.4f ", rpart, sign, fabs(ipart));
angle = angle + 2.0*PI/(float)n;
}
printf("\n");
}
}</lang>
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>
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)).
D
<lang d>module nthroots ; import std.stdio, std.math ;
creal[] nthroots(int n) {
creal[] res ; for(int k = 1 ; k <= n ; k++) res ~= expi(PI*2*k/n) ; return res ;
} void main() {
for(int i = 1; i <= 8 ; i++)
writefln("%2dth : %5.2f", i, nthroots(i)) ;
}</lang>
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>
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>
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.exp(Complex.I * 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
rootsOfUnity n = [mkPolar 1.0 (2*pi*k/n) | k <- [1..n]]</lang>
Output:
<lang haskell>*Main> rootsOfUnity 3 [(-0.4999999999999998) :+ 0.8660254037844387,
(-0.5000000000000004) :+ (-0.8660254037844384), 1.0 :+ (-2.4492127076447545e-16)]</lang>
Icon and Unicon
Icon
<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
Unicon
This Icon solution works in Unicon.
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>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 } }</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>
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): <lang Mathematica>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</lang> gives back:
Maxima
<lang maxima>solve(1 = x^n, x)</lang>
<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>
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>
Perl
<lang perl>use Math::Complex;
foreach $n (2 .. 10) {
printf "%2d", $n;
foreach $k (0 .. $n-1) {
$ret = cplxe(1, 2 * pi * $k / $n);
$ret->display_format(style => 'cartesian', format => '%.3f');
print " $ret";
}
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
Perl 6
<lang perl6>sub roots_of_unity (Int $n where { $n > 0 }) {
map { exp 2i * pi/$n * $_ }, ^$n
}
printf "% .5f + % .5fi\n", .re, .im for roots_of_unity 10;</lang>
Output:
1.00000 + 0.00000i 0.80902 + 0.58779i 0.30902 + 0.95106i -0.30902 + 0.95106i -0.80902 + 0.58779i -1.00000 + 0.00000i -0.80902 + -0.58779i -0.30902 + -0.95106i 0.30902 + -0.95106i 0.80902 + -0.58779i
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>
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>
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.exp(2j*k*cmath.pi/n)) for k in range(n))
# in Python 2.6+: return (Complex(cmath.rect(1, 2*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 "
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
Hopefully someone will fix the formatting <lang ruby>require 'complex'
for n in 2..10
printf "%2d ", n
puts (0..n-1).map { |k| Complex.polar(1, 2 * Math::PI * k / n) }.join(" ")
end</lang> Output:
2 1.0+0.0i -1.0+1.22460635382238e-16i 3 1.0+0.0i -0.5+0.866025403784439i -0.5-0.866025403784438i 4 1.0+0.0i 6.12303176911189e-17+1.0i -1.0+1.22460635382238e-16i -1.83690953073357e-16-1.0i 5 1.0+0.0i 0.309016994374947+0.951056516295154i -0.809016994374947+0.587785252292473i -0.809016994374948-0.587785252292473i 0.309016994374947-0.951056516295154i 6 1.0+0.0i 0.5+0.866025403784439i -0.5+0.866025403784439i -1.0+1.22460635382238e-16i -0.5-0.866025403784438i 0.5-0.866025403784439i 7 1.0+0.0i 0.623489801858734+0.78183148246803i -0.222520933956314+0.974927912181824i -0.900968867902419+0.433883739117558i -0.900968867902419-0.433883739117558i -0.222520933956315-0.974927912181824i 0.623489801858733-0.78183148246803i 8 1.0+0.0i 0.707106781186548+0.707106781186547i 6.12303176911189e-17+1.0i -0.707106781186547+0.707106781186548i -1.0+1.22460635382238e-16i -0.707106781186548-0.707106781186547i -1.83690953073357e-16-1.0i 0.707106781186547-0.707106781186548i 9 1.0+0.0i 0.766044443118978+0.642787609686539i 0.17364817766693+0.984807753012208i -0.5+0.866025403784439i -0.939692620785908+0.342020143325669i -0.939692620785908-0.342020143325669i -0.5-0.866025403784438i 0.17364817766693-0.984807753012208i 0.766044443118978-0.64278760968654i 10 1.0+0.0i 0.809016994374947+0.587785252292473i 0.309016994374947+0.951056516295154i -0.309016994374947+0.951056516295154i -0.809016994374947+0.587785252292473i -1.0+1.22460635382238e-16i -0.809016994374948-0.587785252292473i -0.309016994374948-0.951056516295154i 0.309016994374947-0.951056516295154i 0.809016994374947-0.587785252292473i
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>
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>
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>>