Nth root

Nth root
You are encouraged to solve this task according to the task description, using any language you may know.

Implement the algorithm to compute the principal nth root $\sqrt[n]A$ of a positive real number A, as explained at the Wikipedia page.

Ada

The implementation is generic and supposed to work with any floating-point type. There is no result accuracy argument of Nth_Root, because the iteration is supposed to be monotonically descending to the root when starts at A. Thus it should converge when this condition gets violated, i.e. when x_k+1≥xk.

 
with Ada.Text_IO;  use Ada.Text_IO;
 
procedure Test_Nth_Root is
   generic
      type Real is digits <>;
   function Nth_Root (Value : Real; N : Positive) return Real;
 
   function Nth_Root (Value : Real; N : Positive) return Real is
      type Index is mod 2;
      X : array (Index) of Real := (Value, Value);
      K : Index := 0;
   begin
      loop
         X (K + 1) := ( (Real (N) - 1.0) * X (K) + Value / X (K) ** (N-1) ) / Real (N);
         exit when X (K + 1) >= X (K);
         K := K + 1;
      end loop;
      return X (K + 1);
   end Nth_Root;
 
   function Long_Nth_Root is new Nth_Root (Long_Float);
begin
   Put_Line ("1024.0 10th  =" & Long_Float'Image (Long_Nth_Root (1024.0, 10)));
   Put_Line ("  27.0 3rd   =" & Long_Float'Image (Long_Nth_Root (27.0, 3)));
   Put_Line ("   2.0 2nd   =" & Long_Float'Image (Long_Nth_Root (2.0, 2)));
   Put_Line ("5642.0 125th =" & Long_Float'Image (Long_Nth_Root (5642.0, 125)));
end Test_Nth_Root;
 

Sample output:

1024.0 10th  = 2.00000000000000E+00
  27.0 3rd   = 3.00000000000000E+00
   2.0 2nd   = 1.41421356237310E+00
5642.0 125th = 1.07154759194477E+00

ALGOL 68

Translation of: C

Works with: ALGOL 68 version Standard - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

REAL default p = 0.001;
 
PROC nth root = (INT n, LONG REAL a, p)LONG REAL:
(
  [2]LONG REAL x := (a, a/n);
 
  WHILE ABS(x[2] - x[1]) > p DO
    x := (x[2], ((n-1)*x[2] + a/x[2]**(n-1))/n )
  OD;
  x[2]
);
 
PRIO ROOT = 8;
OP ROOT = (INT n, LONG REAL a)LONG REAL: nth root(n, a, default p);
OP ROOT = (INT n, INT a)LONG REAL: nth root(n, a, default p);
 
main:
(
  printf(($2(" "gl)$,
         nth root(10, LONG 7131.5 ** 10, default p),
         nth root(5, 34, default p)));
  printf(($2(" "gl)$,
         10 ROOT ( LONG 7131.5 ** 10 ),
         5 ROOT 34))
)

Output:

 +7.131500000000000000001144390e  +3
 +2.024397462171090138953733623e  +0
 +7.131500000000000000001144390e  +3
 +2.024397462171090138953733623e  +0

AutoHotkey

p := 0.000001
 
MsgBox, % nthRoot( 10, 7131.5**10, p) "`n"
        . nthRoot(  5, 34.0      , p) "`n"
        . nthRoot(  2, 2         , p) "`n"
        . nthRoot(0.5, 7         , p) "`n"
 
 
;---------------------------------------------------------------------------
nthRoot(n, A, p) { ; http://en.wikipedia.org/wiki/Nth_root_algorithm
;---------------------------------------------------------------------------
    x1 := A
    x2 := A / n
    While Abs(x1 - x2) > p {
        x1 := x2
        x2 := ((n-1)*x2+A/x2**(n-1))/n
    }
    Return, x2
}

Message box shows:

7131.500000
2.024397
1.414214
49.000000

AutoIt

;AutoIt Version: 3.2.10.0
$A=4913
$n=3
$x=20
ConsoleWrite ($n& " root of "& $A & " is " &nth_root_it($A,$n,$x))
ConsoleWrite ($n& " root of "& $A & " is " &nth_root_rec($A,$n,$x))
 
;Iterative
Func nth_root_it($A,$n,$x)
   $x0="0"
   While StringCompare(string($x0),string($x))
      ConsoleWrite ($x&@CRLF)
      $x0=$x
      $x=((($n-1)*$x)+($A/$x^($n-1)))/$n
   WEnd
   Return $x
EndFunc
 
;Recursive
Func nth_root_rec($A,$n,$x)
   ConsoleWrite ($x&@CRLF)
   If $x==((($n-1)*$x)+($A/$x^($n-1)))/$n Then
      Return $x
   EndIf
   Return nth_root_rec($A,$n,((($n-1)*$x)+($A/$x^($n-1)))/$n)
EndFunc

output :

20
17.4275
17.0104009124137
17.0000063582823
17.0000000000024
17
3 root of 4913 is 17

AWK

 
#!/usr/bin/awk -f
BEGIN {
        # test
	print nthroot(8,3)
	print nthroot(16,2)
	print nthroot(16,4)
	print nthroot(125,3)
	print nthroot(3,3)
	print nthroot(3,2)
}
 
function nthroot(y,n) {
        eps = 1e-15;   # relative accuracy
        x   = 1; 
	do {
		d  = ( y / ( x^(n-1) ) - x ) / n ;
		x += d; 
		e = eps*x;   # absolute accuracy	
	} while ( d < -e  || d > e )
 
	return x
}
 

Sample output:

 2
 4 
 2
 5
 1.44225
 1.73205

BASIC

Works with: QBasic

Works with: FreeBASIC

Works with: PowerBASIC

Works with: Visual Basic

This function is fairly generic MS BASIC. It could likely be used in most modern BASICs with little or no change.

FUNCTION RootX (tBase AS DOUBLE, tExp AS DOUBLE, diffLimit AS DOUBLE) AS DOUBLE
    DIM tmp1 AS DOUBLE, tmp2 AS DOUBLE
    ' Initial guess:
    tmp1 = tBase / tExp
    DO
        tmp2 = tmp1
        ' 1# tells compiler that "1" is a double, not an integer
        tmp1 = (((tExp - 1#) * tmp2) + (tBase / (tmp2 ^ (tExp - 1#)))) / tExp
    LOOP WHILE (ABS(tmp1 - tmp2) > diffLimit)
    RootX = tmp1
END FUNCTION

Note that for the above to work in QBasic, the function definition needs to be changed like so:

FUNCTION RootX# (tBase AS DOUBLE, tExp AS DOUBLE, diffLimit AS DOUBLE)

The function is called like so:

PRINT "The "; e; "th root of "; b; " is "; RootX(b, e, .000001)

Sample output:

The  4th root of  16 is  2

For BASICs without the ^ operator, it would be trivial to write a function to reproduce it (as is done in the C example below).

See also the Liberty BASIC and PureBasic solutions.

BBC BASIC

Works with: BBC BASIC for Windows

      *FLOAT 64
      @% = &D0D
      PRINT "Cube root of 5 is "; FNroot(3, 5, 0)
      PRINT "125th root of 5643 is "; FNroot(125, 5643, 0)
      END
 
      DEF FNroot(n%, a, d)
      LOCAL x0, x1 : x0 = a / n% : REM Initial guess
      REPEAT
        x1 = ((n% - 1)*x0 + a/x0^(n%-1)) / n%
        SWAP x0, x1
      UNTIL ABS (x0 - x1) <= d
      = x0

Output:

Cube root of 5 is 1.709975946677
125th root of 5643 is 1.071549111198

C

Implemented without using math library, because if we were to use pow(), the whole exercise wouldn't make sense.

#include <stdio.h>
#include <float.h>
 
inline double abs_(double x) { return x >= 0 ? x : -x; }
double pow_(double x, int e)
{
	double ret = 1;
	for (ret = 1; e; x *= x, e >>= 1)
		if ((e & 1)) ret *= x;
	return ret;
}
 
double root(double a, int n)
{
	double d, x = 1;
	if (!a) return 0;
	if (n < 1 || (a < 0 && !(n&1))) return 0./0.; /* NaN */
 
	do {	d = (a / pow_(x, n - 1) - x) / n;
		x+= d;
	} while (abs_(d) >= abs_(x) * (DBL_EPSILON * 10));
 
	return x;
}
 
int main()
{
	double x = pow_(-3.14159, 15);
	printf("root(%g, 15) = %g\n", x, root(x, 15));
	return 0;
}

C#

Almost exactly how C works.

 
static void Main(string[] args)
{
	Console.WriteLine(NthRoot(81,2,.001));
        Console.WriteLine(NthRoot(1000,3,.001));
        Console.ReadLine();
}
 
public static double NthRoot(double A,int n,  double p)
{
	double _n= (double) n;
	double[] x = new double[2];		
	x[0] = A;
	x[1] = A/_n;
	while(Math.Abs(x[0] -x[1] ) > p)
	{
		x[1] = x[0];
		x[0] = (1/_n)*(((_n-1)*x[1]) + (A/Math.Pow(x[1],_n-1)));
 
	}
	return x[0];
}
 

C++

double NthRoot(double m_nValue, double index, double guess, double pc)
   {
       double result = guess;
       double result_next;
       do
       {
           result_next = (1.0/index)*((index-1.0)*result+(m_nValue)/(pow(result,(index-1.0))));
           result = result_next;
           pc--;
       }while(pc>1);
       return result;
   };
 

double NthRoot(double value, double degree)
{
    return pow(value, (double)(1 / degree));
};
 

CoffeeScript

 
nth_root = (A, n, precision=0.0000000000001) ->
  x = 1
  while true
    x_new = (1 / n) * ((n - 1) * x + A / Math.pow(x, n - 1))
    return x_new if Math.abs(x_new - x) < precision
    x = x_new
 
# tests
do -> 
  tests = [
    [8, 3]
    [16, 4]
    [32, 5]
    [343, 3]
    [1024, 10]
    [1000000000, 3]
    [1000000000, 9]
    [100, 2]
    [100, 3]
    [100, 5]
    [100, 10]
  ]
  for test in tests
    [x, n] = test
    root = nth_root x, n
    console.log "#{x} root #{n} = #{root} (root^#{n} = #{Math.pow root, n})"
 

output

 
> coffee nth_root.coffee 
8 root 3 = 2 (root^3 = 8)
16 root 4 = 2 (root^4 = 16)
32 root 5 = 2 (root^5 = 32)
343 root 3 = 7 (root^3 = 343)
1024 root 10 = 2 (root^10 = 1024)
1000000000 root 3 = 1000 (root^3 = 1000000000)
1000000000 root 9 = 10 (root^9 = 1000000000)
100 root 2 = 10 (root^2 = 100)
100 root 3 = 4.641588833612778 (root^3 = 99.99999999999997)
100 root 5 = 2.5118864315095806 (root^5 = 100.0000000000001)
100 root 10 = 1.5848931924611134 (root^10 = 99.99999999999993)
 

Common Lisp

This version does not check for cycles in x_i and x_i+1, but finishes when the difference between them drops below ε. The initial guess can be provided, but defaults to n-1.

(defun nth-root (n a &optional (epsilon .0001) (guess (1- n)))
  (assert (and (> n 1) (> a 0)))
  (flet ((next (x)
           (/ (+ (* (1- n) x)
                 (/ a (expt x (1- n))))
              n)))
    (do* ((xi guess xi+1)
          (xi+1 (next xi) (next xi)))
         ((< (abs (- xi+1 xi)) epsilon) xi+1))))

nth-root may return rationals rather than floating point numbers, so easy checking for correctness may require coercion to floats. For instance,

(let* ((r (nth-root 3 10))
       (rf (coerce r 'float)))
  (print (* r r r ))
  (print (* rf rf rf)))

produces the following output.

1176549099958810982335712173626176/117654909634627320192156007194483 
10.0

D

import std.stdio, std.math;
 
real nthroot(in int n, in real A, in real p=0.001) pure nothrow {
    real[2] x = [A, A / n];
    while (abs(x[1] - x[0]) > p)
        x = [x[1], ((n - 1) * x[1] + A / (x[1] ^^ (n-1))) / n];
    return x[1];
}
 
void main() {
    writeln(nthroot(10, 7131.5 ^^ 10));
    writeln(nthroot(6, 64));
}

Output:

7131.5
2

Delphi

 
USES
   Math;
 
function NthRoot(A, Precision: Double; n: Integer): Double;
var
   x_p, X: Double;
begin
   x_p := Sqrt(A);
   while Abs(A - Power(x_p, n)) > Precision do
   begin
      x := (1/n) * (((n-1) * x_p) + (A/(Power(x_p, n - 1))));
      x_p := x;
   end;
   Result := x_p;
end;
 

E

Rather than choosing an arbitrary precision, this implementation continues until a cycle in the iterated result is found, thus producing an answer almost as precise as the number type.

(Disclaimer: This was not written by a numerics expert; there may be reasons this is a bad idea. Also, it might be that cycles are always of length 2, which would reduce the amount of calculation needed by 2/3.)

def nthroot(n, x) {
  require(n > 1 && x > 0)
  def np := n - 1
  def iter(g) { return (np*g + x/g**np) / n }
  var g1 := x
  var g2 := iter(g1)
  while (!(g1 <=> g2)) {
    g1 := iter(g1)
    g2 := iter(iter(g2))
  }
  return g1
}

Erlang

Done by finding the fixed point of a function, which aims to find a value of x for which f(x)=x:

fixed_point(F, Guess, Tolerance) ->
    fixed_point(F, Guess, Tolerance, F(Guess)).
fixed_point(_, Guess, Tolerance, Next) when abs(Guess - Next) < Tolerance ->
    Next;
fixed_point(F, _, Tolerance, Next) ->
    fixed_point(F, Next, Tolerance, F(Next)).

The nth root function algorithm defined on the wikipedia page linked above can advantage of this:

nth_root(N, X) -> nth_root(N, X, 1.0e-5).
nth_root(N, X, Precision) ->
    F = fun(Prev) -> ((N - 1) * Prev + X / math:pow(Prev, (N-1))) / N end,
    fixed_point(F, X, Precision).

Excel

This will work in any spreadsheet that uses Excel-compatible expressions -- i.e. KOffice's KCells (formerly KSpread), Calligra Tables, and OpenOffice.org Calc.

• Cell A1 is the base.
• Cell B1 is the exponent.
• Cell A2 is the first guess (any non-zero number will do).

In cell A3, enter this formula:

=((($B$1-1)*A2)+($A$1/(A2^($B$1-1))))/$B$1

Copy A3 down until you get 2 cells with the same value. (Once there are two visibly-identical cells, all cells below those two will also be identical.)

For example, here we calculate the cube root of 100:

	A	B
1	100	3
2	7	(first guess)
3	5.346938776
4	4.730544697
5	4.643251125
6	4.641589429
7	4.641588834
8	4.641588834

Alternately, Excel could use the BASIC example above as VBA code, deleting A2 and replacing A3's formula with something like this:

=RootX(A1,B1,.00000001)

F#

 
let nthroot n A =
    let rec f x =
        let m = n - 1.
        let x' = (m * x + A/x**m) / n
        match abs(x' - x) with
        | t when t < abs(x * 1e-9) -> x'
        | _ -> f x'
    f (A / double n)
 
[<EntryPoint>]
let main args =
    if args.Length <> 2 then
        eprintfn "usage: nthroot n A"
        exit 1
    let (b, n) = System.Double.TryParse(args.[0])
    let (b', A) = System.Double.TryParse(args.[1])
    if (not b) || (not b') then
        eprintfn "error: parameter must be a number"
        exit 1
    printf "%A" (nthroot n A)
    0
 

Compiled using fsc nthroot.fs example output:

nthroot 0.5 7
49.0

Forth

: th-root { F: a F: n -- a^1/n }
  a
  begin
    a fover n 1e f- f** f/
      fover n 1e f- f*
    f+ n f/
    fswap fover 1e-5 f~
  until ;
 
34e 5e th-root f.   \ 2.02439745849989
34e 5e 1/f f** f.   \ 2.02439745849989

Fortran

program NthRootTest
  implicit none
 
  print *, nthroot(10, 7131.5**10)
  print *, nthroot(5, 34.0)
 
contains
 
  function nthroot(n, A, p)
    real :: nthroot
    integer, intent(in)        :: n
    real, intent(in)           :: A
    real, intent(in), optional :: p
 
    real :: rp, x(2)
 
    if ( A < 0 ) then
       stop "A < 0"       ! we handle only real positive numbers
    elseif ( A == 0 ) then
       nthroot = 0
       return
    end if
 
    if ( present(p) ) then
       rp = p
    else
       rp = 0.001
    end if
 
    x(1) = A
    x(2) = A/n   ! starting "guessed" value...
 
    do while ( abs(x(2) - x(1)) > rp )
       x(1) = x(2)
       x(2) = ((n-1.0)*x(2) + A/(x(2) ** (n-1.0)))/real(n)
    end do
 
    nthroot = x(2)
 
  end function nthroot
 
end program NthRootTest

Go

func root(a float64, n int) float64 {
    n1 := n - 1
    n1f, rn := float64(n1), 1/float64(n)
    x, x0 := 1., 0.
    for {
        potx, t2 := 1/x, a
        for b := n1; b > 0; b >>= 1 {
            if b&1 == 1 {
                t2 *= potx
            }
            potx *= potx
        }
        x0, x = x, rn*(n1f*x+t2)
        if math.Abs(x-x0)*1e15 < x {
            break
        }
    }
    return x
}

Groovy

Solution:

import static Constants.tolerance
import static java.math.RoundingMode.HALF_UP
 
def root(double base, double n) {
    double xOld = 1
    double xNew = 0
    while (true) {
        xNew = ((n - 1) * xOld + base/(xOld)**(n - 1))/n
    if ((xNew - xOld).abs() < tolerance) { break }
        xOld = xNew
    }
    (xNew as BigDecimal).setScale(7, HALF_UP)
}
 

Test:

class Constants {
    static final tolerance = 0.00001
}
 
print '''
   Base   Power  Calc'd Root  Actual Root
-------  ------  -----------  -----------
'''
def testCases = [
    [b:32.0, n:5.0, r:2.0],
    [b:81.0, n:4.0, r:3.0],
    [b:Math.PI**2, n:4.0, r:Math.PI**(0.5)],
    [b:7.0, n:0.5, r:49.0],
]
 
testCases.each {
    def r = root(it.b, it.n)
    printf('%7.4f  %6.4f  %11.4f  %11.4f\n',
        it.b, it.n, r, it.r)
    assert (r - it.r).abs() <= tolerance
}

Output:

   Base   Power  Calc'd Root  Actual Root
-------  ------  -----------  -----------
32.0000  5.0000       2.0000       2.0000
81.0000  4.0000       3.0000       3.0000
 9.8696  4.0000       1.7725       1.7725
 7.0000  0.5000      49.0000      49.0000

Haskell

Function exits when there's no difference between two successive values.

n `nthRoot` x = fst $ until (uncurry(==)) (\(_,x0) -> (x0,((n-1)*x0+x/x0**(n-1))/n)) (x,x/n)

Use:

*Main> 2 `nthRoot` 2
1.414213562373095

*Main> 5 `nthRoot` 34
2.024397458499885

*Main> 10 `nthRoot` (734^10)
734.0

*Main> 0.5 `nthRoot` 7
49.0

HicEst

WRITE(Messagebox) NthRoot(5, 34)
WRITE(Messagebox) NthRoot(10, 7131.5^10)
 
FUNCTION NthRoot(n, A)
   REAL :: prec = 0.001
 
   IF( (n > 0) * (A > 0) ) THEN
       NthRoot = A / n
       DO i = 1, 1/prec
         x = ((n-1)*NthRoot + A/(NthRoot^(n-1))) / n
         IF( ABS(x - NthRoot) <= prec ) THEN
             RETURN
         ENDIF
         NthRoot = x
       ENDDO
   ENDIF
 
   WRITE(Messagebox, Name) 'Cannot solve problem for:', prec, n, A
END

Icon and Unicon

All Icon/Unicon reals are double precision.

procedure main()
   showroot(125,3)
   showroot(27,3)
   showroot(1024,10)
   showroot(39.0625,4)
   showroot(7131.5^10,10)
end
 
procedure showroot(a,n)
   printf("%i-th root of %i = %i\n",n,a,root(a,n))
end
 
procedure root(a,n,p) #: finds the n-th root of the number a to precision p
   if n < 0 | type(n) !== "integer" then runerr(101,n)     
   if a < 0 then runerr(205,a)   
   /p := 1e-14                  # precision
   xn := a / real(n)            # initial guess
   while abs(a - xn^n) > p do       
      xn := ((n - 1) * (xi := xn) + a / (xi ^ (n-1))) / real(n)
   return xn
end
 
link printf

Output:

3-th root of 125 = 5.0
3-th root of 27 = 3.0
10-th root of 1024 = 2.0
4-th root of 39.0625 = 2.5
10-th root of 3.402584077894253e+038 = 7131.5

J

Translation of: E

J has a built in Nth root primitive, %:. For example, 7131.5 = 10 %: 7131.5^10. Also, the exponentiation primitive supports exponents < 1, e.g. 7131.5 = (7131.5^10)^(1%10).

But, since the talk page discourages using built-in facilities, here is a reimplementation, using the E algorithm:

   '`N X NP' =.  (0 { [)`(1 { [)`(2 { [)
   iter      =.  N %~ (NP * ]) + X % ] ^ NP
   nth_root  =:  (, , _1+[) iter^:_ f. ]
   10 nth_root 7131.5^10
7131.5

Java

Translation of: Fortran

public static double nthroot(int n, double A) {
	return nthroot(n, A, .001);
}
public static double nthroot(int n, double A, double p) {
	if(A < 0) {
		System.err.println("A < 0");// we handle only real positive numbers
		return -1;
	} else if(A == 0) {
		return 0;
	}
	double x_prev = A;
	double x = A / n;  // starting "guessed" value...
	while(Math.abs(x - x_prev) > p) {
		x_prev = x;
		x = ((n - 1.0) * x + A / Math.pow(x, n - 1.0)) / n;
	}
	return x;
}

Translation of: E

public static double nthroot(int n, double x) {
  assert (n > 1 && x > 0);
  int np = n - 1;
  double g1 = x;
  double g2 = iter(g1, np, n, x);
  while (g1 != g2) {
    g1 = iter(g1, np, n, x);
    g2 = iter(iter(g2, np, n, x), np, n, x);
  }
  return g1;
}
 
private static double iter(double g, int np, int n, double x) {
  return (np * g + x / Math.pow(g, np)) / n;
}

JavaScript

Gives the n:nth root of num, with precision prec. (n defaults to 2 [e.g. sqrt], prec defaults to 12.)

function nthRoot(num, nArg, precArg) {
  var n = nArg || 2;
  var prec = precArg || 12;
 
  var x = 1; // Initial guess.
  for (var i=0; i<prec; i++) {
    x = 1/n * ((n-1)*x + (num / Math.pow(x, n-1)));
  }
 
  return x;
}

Liberty BASIC

 
print "First estimate is: ",        using( "#.###############",  NthRoot( 125, 5642, 0.001  ));
print "    ... and better is: ",    using( "#.###############",  NthRoot( 125, 5642, 0.00001))
print "125'th root of 5642 by LB's exponentiation operator is "; using( "#.###############", 5642^(1 /125))
 
print "27^(1 / 3)",                 using( "#.###############",  NthRoot(   3,   27, 0.00001))
print "2^(1 / 2)",                  using( "#.###############",  NthRoot(   2,    2, 0.00001))
print "1024^(1 /10)",               using( "#.###############",  NthRoot(  10, 1024, 0.00001))
 
wait
 
function NthRoot( n, A, p)
  x( 0) =A
  x( 1) =A /n
  while abs( x( 1) -x( 0)) >p
    x( 0) =x( 1)
    x( 1) =( ( n -1.0) *x( 1) +A /x( 1)^( n -1.0)) /n
  wend
  NthRoot =x( 1)
end function
 
end
 
 

First estimate is:          1.071559602191682    ... and better is:   1.071547591944771
125'th root of 5642 by LB's exponentiation operator is 1.071547591944767
27^(1 / 3)    3.000000000000002
2^(1 / 2)     1.414213562374690
1024^(1 /10)  2.000000000000000

Logo

to about :a :b
  output and [:a - :b < 1e-5] [:a - :b > -1e-5]
end
 
to root :n :a [:guess :a]
  localmake "next ((:n-1) * :guess + :a / power :guess (:n-1)) / n
  if about :guess :next [output :next]
  output (root :n :a :next)
end
 
show root 5 34   ; 2.02439745849989

Lua

 
function nth_root(num,root)
  return num^(1/root)
end
 

Mathematica

Root[A,n]

MATLAB

function answer = nthRoot(number,root)
 
    format long
 
    answer = number / root;
    guess = number;
 
    while not(guess == answer)
       guess = answer;
       answer = (1/root)*( ((root - 1)*guess) + ( number/(guess^(root - 1)) ) ); 
    end
 
end

Sample Output:

>> nthRoot(2,2)
 
ans =
 
   1.414213562373095

Maxima

nth_root(a, n) := block(
   [x, y, d, p: fpprec],
   fpprec: p + 10,
   x: bfloat(a),
   eps: 10.0b0^-p,
   y: do (
      d: bfloat((a / x^(n - 1) - x) / n),
      if abs(d) < eps * x then return(x),
      x: x + d
   ),
   fpprec: p,
   bfloat(y)
)$

Metafont

Metafont does not use IEEE floating point and we can't go beyond 0.0001 or it will loop forever.

vardef mnthroot(expr n, A) =
  x0 := A / n;
  m := n - 1;
  forever:
    x1 := (m*x0 + A/(x0 ** m)) / n;
    exitif abs(x1 - x0) < abs(x0 * 0.0001);
    x0 := x1;
  endfor;
  x1
enddef;
 
primarydef n nthroot A = mnthroot(n, A) enddef;
 
show 5 nthroot 34;  % 2.0244
show 0.5 nthroot 7; % 49.00528
 
bye

МК-61/52

1/x	<->	x^y	С/П

Instruction: number ^ degree В/О С/П

NetRexx

Translation of: REXX

 
/*NetRexx program to calculate the  Nth root of  X,  with  DIGS  accuracy. */
class nth_root
 
  method main(args=String[]) static
    if args.length < 2 then
      do
	say "at least 2 arguments expected"
	exit
      end
    x = args[0]
    root = args[1]
    if args.length > 2 then digs = args[2]
 
    if root=='' then root=2
    if digs = null, digs = '' then digs=20
    numeric digits digs
    say '     x	= ' x
    say '  root	= ' root
    say 'digits	= ' digs
    say 'answer	= ' root(x,root,digs)
 
  method root(x,r,digs) static --procedure; parse arg x,R 1 oldR  /*assign 2nd arg-->r and rOrig.  */
    /*this subroutine will use the   */
    /*digits from the calling prog.  */
    /*The default digits is  9.      */
    R = r
    oldR = r
    if r=0 then do
      say
      say '*** error! ***'
      say "a root of zero can't be specified."
      say
      return '[n/a]'
    end
 
    R=R.abs()                              /*use absolute value of root.    */
 
    if x<0 & (R//2==0) then do
      say
      say '*** error! ***'
      say "an even root can't be calculated for a" -
      'negative number,'
      say 'the result would be complex.'
      say
      return '[n/a]'
    end
 
    if x=0 | r=1 then return x/1           /*handle couple of special cases.*/
    Rm1=R-1                                /*just a fast version of  ROOT-1 */
    oldDigs=digs                           /*get the current number of digs.*/
    dm=oldDigs+5                           /*we need a little guard room.   */
    ax=x.abs()                             /*the absolute value of  X.      */
    g=(ax+1)/r**r                          /*take a good stab at 1st guess. */
 --   numeric fuzz 3                         /*fuzz digits for higher roots.  */
    d=5                                    /*start with only five digits.   */
    /*each calc doubles precision.   */
 
    loop forever
 
      d=d+d
      if d>dm then d = dm                        /*double the digits, but not>DM. */
      numeric digits d                     /*tell REXX to use   D   digits. */
      old=0                                /*assume some kind of old guess. */
 
      loop forever
	_=(Rm1*g**R+ax)/R/g**rm1           /*this is the nitty-gritty stuff.*/
	if _=g | _=old then leave          /*computed close to this before? */
	old=g                              /*now, keep calculation for OLD. */
	g=_                                /*set calculation to guesstimate.*/
      end
 
      if d==dm then leave                  /*found the root for DM digits ? */
    end
 
    _=g*x.sign()                           /*correct the sign (maybe).      */
    if oldR<0 then return _=1/_            /*root < 0 ?    Reciprocal it is.*/
    numeric digits oldDigs                 /*re-instate the original digits.*/
    return _/1                             /*normalize the number to digs.  */
 
 

OCaml

Translation of: C

let nthroot ~n ~a ?(tol=0.001) () =
   let nf = float n in let nf1 = nf -. 1.0 in
   let rec iter x =
      let x' = (nf1 *. x +. a /. (x ** nf1)) /. nf in
      if tol > abs_float (x -. x') then x' else iter x' in
   iter 1.0
;;
 
let () =
  Printf.printf "%g\n" (nthroot 10 (7131.5 ** 10.0) ());
  Printf.printf "%g\n" (nthroot 5 34.0 ());
;;

Octave

Octave has it's how nthroot function.

 
  r = A.^(1./n)
 

Here it is another implementation (after Tcl)

Translation of: Tcl

function r = m_nthroot(n, A)
  x0 = A / n;
  m  = n - 1;
  while(1)
    x1 = (m*x0 + A./ x0 .^ m) / n;
    if ( abs(x1-x0) < abs(x0 * 1e-9) )
      r = x1;
      return
    endif
    x0 = x1;
  endwhile
endfunction

Here is an more elegant way by computing the successive differences in an explicit way:

function r = m_nthroot(n, A)
  r = A / n;
  m = n - 1;
  do 
    d = (A ./ r .^ m - r) / n;
    r+= d;
  until (abs(d) < abs(r * 1e-9))
endfunction

Show its usage and the built-in nthroot function

m_nthroot(10, 7131.5 .^ 10)
nthroot(7131.5 .^ 10, 10)
m_nthroot(5, 34)
nthroot(34, 5)
m_nthroot(0.5, 7)
nthroot(7, .5)

Oz

declare
  fun {NthRoot NInt A}
     N = {Int.toFloat NInt}
 
     fun {Next X}
        ( (N-1.0)*X + A / {Pow X N-1.0} ) / N
     end
  in
     {Until Value.'==' Next A/N}
  end
 
  fun {Until P F X}
     case {F X}
     of NX andthen {P NX X} then X
     [] NX then {Until P F NX}
     end
  end
in
  {Show {NthRoot 2 2.0}}

PARI/GP

root(n,A)=A^(1/n);

Pascal

See Delphi

Perl

Translation of: Tcl

use strict;
 
sub nthroot ($$)
{
    my ( $n, $A ) = @_;
 
    my $x0 = $A / $n;
    my $m = $n - 1.0;
    while(1) {
	my $x1 = ($m * $x0 + $A / ($x0 ** $m)) / $n;
	return $x1 if abs($x1 - $x0) < abs($x0 * 1e-9);
	$x0 = $x1;
    }
}

print nthroot(5, 34), "\n";
print nthroot(10, 7131.5 ** 10), "\n";
print nthroot(0.5, 7), "\n";

Perl 6

sub nth-root ($n, $A, $p=1e-9)
{
    my $x0 = $A / $n;
    loop {
        my $x1 = (($n-1) * $x0 + $A / ($x0 ** ($n-1))) / $n;
        return $x1 if abs($x1-$x0) < abs($x0 * $p);
        $x0 = $x1;
    }
}
 
say nth-root(3,8);

PHP

function nthroot($number, $root, $p = P)
{
    $x[0] = $number;
    $x[1] = $number/$root;
    while(abs($x[1]-$x[0]) > $p)
    {
        $x[0] = $x[1];
        $x[1] = (($root-1)*$x[1] + $number/pow($x[1], $root-1))/$root;
    }
    return $x[1];
}

PicoLisp

(load "@lib/math.l")
 
(de nthRoot (N A)
   (let (X1 A  X2 (*/ A N))
      (until (= X1 X2)
         (setq
            X1 X2
            X2 (*/
               (+
                  (* X1 (dec N))
                  (*/ A 1.0 (pow X1 (* (dec N) 1.0))) )
               N ) ) )
      X2 ) )
 
(prinl (format (nthroot 2  2.0) *Scl))
(prinl (format (nthroot 3 12.3) *Scl))
(prinl (format (nthroot 4 45.6) *Scl))

Output:

1.414214
2.308350
2.598611

PL/I

 
/* Finds the N-th root of the number A */
root: procedure (A, N) returns (float);
   declare A float, N fixed binary;
   declare (xi, xip1) float;
 
   xi = 1; /* An initial guess */
   do forever;
      xip1 = ((n-1)*xi + A/xi**(n-1) ) / n;
      if abs(xip1-xi) < 1e-5 then leave;
      xi = xip1;
   end;
   return (xi);
end root;
 

PureBasic

#Def_p=0.001
 
Procedure.d Nth_root(n.i, A.d, p.d=#Def_p)
  Protected Dim x.d(1) 
  x(0)=A: x(1)=A/n
  While Abs(x(1)-x(0))>p
    x(0)=x(1)
    x(1)=((n-1.0)*x(1)+A/Pow(x(1),n-1.0))/n
  Wend
  ProcedureReturn x(1)
EndProcedure
 
;//////////////////////////////
Debug "125'th root of 5642 is"
Debug Pow(5642,1/125)
Debug "First estimate is:"
Debug Nth_root(125,5642)
Debug "And better:"
Debug Nth_root(125,5642,0.00001)

Outputs

125'th root of 5642 is
1.0715475919447675
First estimate is:
1.0715596021916822
And better:
1.0715475919447714

Python

from decimal import Decimal, getcontext
 
def nthroot (n, A, precision):
    getcontext().prec = precision
 
    n = Decimal(n)
    x_0 = A / n #step 1: make a while guess.
    x_1 = 1     #need it to exist before step 2
    while True:
        #step 2:
        x_0, x_1 = x_1, (1 / n)*((n - 1)*x_0 + (A / (x_0 ** (n - 1))))
        if x_0 == x_1:
            return x_1

print nthroot(5, 34, 10)
print nthroot(10,42, 20)
print nthroot(2, 5, 400)

R

nthroot <- function(A, n, tol=sqrt(.Machine$double.eps))
{
   ifelse(A < 1, x0 <- A * n, x0 <- A / n)
   repeat
   {
      x1 <- ((n-1)*x0 + A / x0^(n-1))/n
      if(abs(x1 - x0) > tol) x0 <- x1 else break
   }
   x1
}
nthroot(7131.5^10, 10)   # 7131.5
nthroot(7, 0.5)          # 49

Racket

#lang racket
 
(define (nth-root number root (tolerance 0.001))
  (define (acceptable? next current)
    (< (abs (- next current)) tolerance))
 
  (define (improve current)
    (/ (+ (* (- root 1) current) (/ number (expt current (- root 1)))) root))
 
  (define (loop current)
    (define next-guess (improve current))
    (if (acceptable? next-guess current)
        next-guess
        (loop next-guess)))
  (loop 1.0))

REXX

/*REXX program calculates the  Nth root of   X,   with  DIGS  accuracy. */
parse arg x root digs .                /*get specified args from the CL.*/
if    x==''  then    x=2               /*Not specified? Then use default*/
if root==''  then root=2               /* "       "       "   "     "   */
if digs==''  then digs=65              /* "       "       "   "     "   */
numeric digits digs                    /*set the precision to   DIGS.   */
say '       x = '    x                 /*echo the value of  X.          */
say '    root = '    root              /*echo the value of  ROOT.       */
say '  digits = '    digs              /*echo the value of  DIGS.       */
say '  answer = '    root(x,root)      /*show the value of  ANSWER.     */
exit                                   /*stick a fork in it, we're done.*/
/*──────────────────────────────────ROOT subroutine─────────────────────*/
root: procedure; parse arg x 1 Ox,r . 1 Or  /*1st arg──►x&Ox, 2nd──►r&Or*/
if r==''  then r=2                     /*Was root specified?   Assume √.*/
if r=0    then return '[n/a]'          /*oops-ay!  Can't do zeroth root.*/
complex= x<0 & R//2==0                 /*will the result be complex?    */
oDigs=digits()                         /*get the current number of digs.*/
if x=0 | r=1  then return x/1          /*handle couple of special cases.*/
dm=oDigs+5                             /*we need a little guard room.   */
r=abs(r);   x=abs(x)                   /*the absolute values of R and X.*/
rm=r-1                                 /*just a fast version of  ROOT -1*/
numeric form                           /*take a good guess at the root─┐*/
parse value format(x,2,1,,0) 'E0' with ? 'E' _ .        /* ◄───────────┘*/
g=(?/r'E'_%r)+(x>1)                    /*kinda uses a crude "logrithm". */
numeric fuzz 3                         /*fuzz digits for higher roots.  */
d=5                                    /*start with only five digits.   */
     do until d==dm;   d=min(d+d,dm)   /*each interation doubles prec.  */
     numeric digits d                  /*tell REXX to use   D   digits. */
     old=-1                            /*assume some kind of old guess. */
            do until old=g;   old=g    /*where da rubber meets da road─┐*/
            g=(rm*g**r+x) / r / g**rm  /*nitty-gritty root computation◄┘*/
            end   /*until old=g*/      /*maybe until the cows come home.*/
     end          /*until d==dm*/      /*and wait for more cows to come.*/
 
if g=0        then return 0            /*in case the jillionth root = 0.*/
if Or<0       then g=1/g               /*root < 0 ?    Reciprocal it is!*/
if \complex   then g=g*sign(Ox)        /*adjust the sign  (maybe).      */
numeric digits oDigs                   /*reinstate the original digits. */
return g/1 || left('j',complex)        /*normalize # to digs, append j ?*/

output when using the defaults

       x =  2
    root =  2
  digits =  65
  answer =  1.414213562373095048801688724209698078569671875376948073176679738

output when using for input: 10 3

       x =  10
    root =  3
  digits =  65
  answer =  2.1544346900318837217592935665193504952593449421921085824892355063

output when using for input: 625 -4

       x =  625
    root =  -4
  digits =  65
  answer =  0.2

output when using for input: 100.666 47

       x =  100.666
    root =  47
  digits =  65
  answer =  1.1030990940616109102886569991014966919115206420386192403152621652

output when using for input: -256 8

       x =  -256
    root =  8
  digits =  65
  answer =  2j

output when using for input: 12345678900098765432100.00987654321000123456789e333 19

       x =  12345678900098765432100.00987654321000123456789e333
    root =  19
  digits =  65
  answer =  4886828567991886455.3257854108687610458584138783288904955196401434

Ruby

def nthroot(n, a, precision = 1e-5)
  x = Float(a)
  begin
    prev = x
    x = ((n - 1) * prev + a / (prev ** (n - 1))) / n
  end while (prev - x).abs > precision
  x 
end
 
p nthroot(5,34)  # => 2.02439745849989

Run BASIC

print "Root 125th Root of 5643 Precision .001   ";using( "#.###############",  NthRoot( 125, 5642, 0.001  ))
print "125th Root of 5643 Precision .001   ";using( "#.###############",  NthRoot( 125, 5642, 0.001  ))
print "125th Root of 5643 Precision .00001 ";using( "#.###############",  NthRoot( 125, 5642, 0.00001))
print "  3rd Root of   27 Precision .00001 ";using( "#.###############",  NthRoot(   3,   27, 0.00001))
print "  2nd Root of    2 Precision .00001 ";using( "#.###############",  NthRoot(   2,    2, 0.00001))
print " 10th Root of 1024 Precision .00001 ";using( "#.###############",  NthRoot(  10, 1024, 0.00001))
 
wait
 
function NthRoot( root, A, precision)
  x0 = A
  x1 = A /root
  while abs( x1 -x0) >precision
    x0 = x1
    x1 = x1 / 1.0                                ' force float
    x1 = (( root -1.0) *x1 +A /x1^( root -1.0)) /root
  wend
  NthRoot =x1
end function
 
end

125th Root of 5643 Precision .001   1.071559602456735
125th Root of 5643 Precision .00001 1.071547591944771
  3rd Root of   27 Precision .00001 3.000000000000001
  2nd Root of    2 Precision .00001 1.414213562374690
 10th Root of 1024 Precision .00001 2.000000000000000

Sather

Translation of: Octave

class MATH is
  nthroot(n:INT, a:FLT):FLT
    pre n > 0
  is  
    x0 ::= a / n.flt;
    m  ::= n - 1;
    loop
      x1 ::= (m.flt * x0 + a/(x0^(m.flt))) / n.flt;
      if (x1 - x0).abs < (x0 * 1.0e-9).abs then
        return x1;
      end;
      x0 := x1;
    end;
  end;
 
end;

class MAIN is
  main is
    a:FLT := 2.5 ^ 10.0;
    #OUT + MATH::nthroot(10, a) + "\n";
  end;
end;

Scala

 
object NthRoot {
 
  def main(args: Array[String]) {
    println(nthroot(3, 32))
  }
 
  def nthroot1(n: Int, a: Double): Double = {
    def loop(x0: Double) : Double = { 
      val x1 = (1.0d/n * ((n - 1) * x0 + a/math.pow(x0, n-1)))
      if (x0 <= x1) x0
      else loop(x1)
    } 
 
    return loop(a/2)
  }
}
 

Scheme

(define (root number degree tolerance)
  (define (good-enough? next guess)
    (< (abs (- next guess)) tolerance))
  (define (improve guess)
    (/ (+ (* (- degree 1) guess) (/ number (expt guess (- degree 1)))) degree))
  (define (*root guess)
    (let ((next (improve guess)))
      (if (good-enough? next guess)
          guess
          (*root next))))
  (*root 1.0))
 
(display (root (expt 2 10) 10 0.1))
(newline)
(display (root (expt 2 10) 10 0.01))
(newline)
(display (root (expt 2 10) 10 0.001))
(newline)

Output:

2.04732932236839
2.00463204835482
2.00004786858167

Seed7

The nth root of the number 'a' can be computed with the exponentiation operator: 'a ** (1 / n)'. An alternate function which uses Newton's method is:

const func float: nthRoot (in integer: n, in float: a) is func
  result
    var float: x1 is 0.0;
  local
    var float: x0 is 0.0;
  begin 
    x0 := a;
    x1 := a / flt(n);
    while abs(x1 - x0) >= abs(x0 * 1.0E-9) do
      x0 := x1;
      x1 := (flt(pred(n)) * x0 + a / x0 ** pred(n)) / flt(n);
    end while;
  end func;

Original source: [1]

Smalltalk

Works with: GNU Smalltalk

Translation of: Tcl

Number extend [
    nthRoot: n [
	|x0 m x1|
	x0 := (self / n) asFloatD.
	m := n - 1.
	[true] whileTrue: [
	    x1 := ( (m * x0) + (self/(x0 raisedTo: m))) / n.
	    ((x1 - x0) abs) < ((x0 * 1e-9) abs)
		ifTrue: [ ^ x1 ].
	    x0 := x1
	]
    ]
].

(34 nthRoot: 5) displayNl.
((7131.5 raisedTo: 10) nthRoot: 10) displayNl.
(7 nthRoot: 0.5) displayNl.

Tcl

The easiest way is to just use the pow function (or exponentiation operator) like this:

proc nthroot {n A} {
    expr {pow($A, 1.0/$n)}
}

However that's hardly tackling the problem itself. So here's how to do it using Newton-Raphson and a self-tuning termination test.

Works with: Tcl version 8.5

proc nthroot {n A} {
    set x0 [expr {$A / double($n)}]
    set m [expr {$n - 1.0}]
    while 1 {
        set x1 [expr {($m*$x0 + $A/$x0**$m) / $n}]
        if {abs($x1 - $x0) < abs($x0 * 1e-9)} {
            return $x1
        }
        set x0 $x1
    }
}

Demo:

puts [nthroot 2 2]
puts [nthroot 5 34]
puts [nthroot 5 [expr {34**5}]]
puts [nthroot 10 [expr 7131.5**10]]
puts [nthroot 0.5 7]; # Squaring!

Output:

1.414213562373095
2.0243974584998847
34.0
7131.5
49.0

Ursala

The nthroot function defined below takes a natural number n to the function that returns the n-th root of its floating point argument. Error is on the order of machine precision because the stopping criterion is either a fixed point or a repeating cycle.

#import nat
#import flo
 
nthroot =
 
-+
   ("n","n-1"). "A". ("x". div\"n" plus/times("n-1","x") div("A",pow("x","n-1")))^== 1.,
   float^~/~& predecessor+-

This implementation is unnecessary in practice due to the availability of the library function pow, which performs exponentiation and allows fractional exponents. Here is a test program.

#cast %eL
 
examples =
 
<
   nthroot2 2.,
   nthroot5 34.,
   nthroot5 pow(34.,5.),
   nthroot10 pow(7131.5,10.)>

output:

<
   1.414214e+00,
   2.024397e+00,
   3.400000e+01,
   7.131500e+03>

XPL0

include c:\cxpl\stdlib;
 
func real NRoot(A, N);  \Return the Nth root of A
real A, N;
real X, X0, Y;
int  I;
[X:= 1.0;               \initial guess
repeat  X0:= X;
        Y:= 1.0;
        for I:= 1 to fix(N)-1 do Y:= Y*X0;
        X:= ((N-1.0)*X0 + A/Y) / N;
until   abs(X-X0) < 1.0E-15;            \(until X=X0 doesn't always work)
return X;
];
 
[Format(5, 15);
RlOut(0, NRoot( 2., 2.));  CrLf(0);
RlOut(0, Power( 2., 0.5)); CrLf(0);     \for comparison
RlOut(0, NRoot(27., 3.));  CrLf(0);
RlOut(0, NRoot(1024.,10.)); CrLf(0);
]

Output:

    1.414213562373100
    1.414213562373100
    3.000000000000000
    2.000000000000000