Integer roots: Difference between revisions

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Task

Create a program that computes an approximation of the principal N^th root of X as the largest integer less than or equal to R for which R^N=X.

──where:

       N  is a positive integer. 
       X  is a non-negative integer. 
       R  (the root)   is a non-negative real number.

No arbitrary limits should be placed on the magnitudes of the numbers involved.

Example: With N=3 and X=8 you would calculate the number 2 because $2^{3}=8$

Example: With N=3 and X=9 you would again calculate the number 2 because 2 is the largest integer less than or equal to the root R.

Example: With N=2 and X=2×100^2,000 you would calculate a large integer consisting of the first 2,001 digits (in order) of the square root of two.

Arturo

Translation of: D

<lang arturo>iRoot: @(b n){

   if b<2 { return b }

   n1: n-1
   n2: n
   n3: n1
   c: 1
   d: (n3+b)/n2
   e: (n3*d + b/d^n1)/n2
   loop [and c!=d c!=e] {
       c: d
       d: e
       e: (n3*e + b/e^n1)/n2
   }
   if d<e { return d }
   return e

}

print "3rd root of 8 = " + [iRoot 8 3] print "3rd root of 9 = " + [iRoot 9 3] print "First 2001 digits of the square root of 2 = " + [iRoot 100^2000*2 2]</lang>

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First 2001 digits of the square root of 2 = 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

C

Translation of: C++

<lang C>#include <stdio.h>

include <math.h>

typedef unsigned long long ulong;

ulong root(ulong base, ulong n) {

   ulong n1, n2, n3, c, d, e;

   if (base < 2) return base;
   if (n == 0) return 1;

   n1 = n - 1;
   n2 = n;
   n3 = n1;
   c = 1;
   d = (n3 + base) / n2;
   e = (n3 * d + base / (ulong)powl(d, n1)) / n2;

   while (c != d && c != e) {
       c = d;
       d = e;
       e = (n3*e + base / (ulong)powl(e, n1)) / n2;
   }

   if (d < e) return d;
   return e;

}

int main() {

   ulong b = (ulong)2e18;

   printf("3rd root of 8 = %lld\n", root(8, 3));
   printf("3rd root of 9 = %lld\n", root(9, 3));
   printf("2nd root of %lld = %lld\n", b, root(b, 2));

   return 0;

}</lang>

Output:

3rd root of 8 = 2
3rd root of 9 = 2
2nd root of 2000000000000000000 = 1414213562

C#

Translation of: Java

<lang csharp>using System; using System.Numerics;

namespace IntegerRoots {

   class Program {
       static BigInteger IRoot(BigInteger @base, int n) {
           if (@base < 0 || n <= 0) {
               throw new ArgumentException();
           }

           int n1 = n - 1;
           BigInteger n2 = n;
           BigInteger n3 = n1;
           BigInteger c = 1;
           BigInteger d = (n3 + @base) / n2;
           BigInteger e = ((n3 * d) + (@base / BigInteger.Pow(d, n1))) / n2;
           while (c != d && c != e) {
               c = d;
               d = e;
               e = (n3 * e + @base / BigInteger.Pow(e, n1)) / n2;
           }
           if (d < e) {
               return d;
           }
           return e;
       }

       static void Main(string[] args) {
           Console.WriteLine("3rd integer root of 8 = {0}", IRoot(8, 3));
           Console.WriteLine("3rd integer root of 9 = {0}", IRoot(9, 3));

           BigInteger b = BigInteger.Pow(100, 2000) * 2;
           Console.WriteLine("First 2001 digits of the sqaure root of 2: {0}", IRoot(b, 2));
       }
   }

}</lang>

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the sqaure root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

C++

<lang cpp>#include <iostream>

include <math.h>

unsigned long long root(unsigned long long base, unsigned int n) { if (base < 2) return base; if (n == 0) return 1;

unsigned int n1 = n - 1; unsigned long long n2 = n; unsigned long long n3 = n1; unsigned long long c = 1; auto d = (n3 + base) / n2; auto e = (n3 * d + base / pow(d, n1)) / n2;

while (c != d && c != e) { c = d; d = e; e = (n3*e + base / pow(e, n1)) / n2; }

if (d < e) return d; return e; }

int main() { using namespace std;

cout << "3rd root of 8 = " << root(8, 3) << endl; cout << "3rd root of 9 = " << root(9, 3) << endl;

unsigned long long b = 2e18; cout << "2nd root of " << b << " = " << root(b, 2) << endl;

return 0; }</lang>

Output:

3rd root of 8 = 2
3rd root of 9 = 2
2nd root of 2000000000000000000 = 1414213562

D

Translation of: Kotlin

<lang D>import std.bigint; import std.stdio;

auto iRoot(BigInt b, int n) in {

   assert(b >=0 && n > 0);

} body {

   if (b < 2) return b;
   auto n1 = n - 1;
   auto n2 = BigInt(n);
   auto n3 = BigInt(n1);
   auto c = BigInt(1);
   auto d = (n3 + b) / n2;
   auto e = (n3 * d + b / d^^n1) / n2;
   while (c != d && c != e) {
       c = d;
       d = e;
       e = (n3 * e + b / e^^n1) / n2;
   }
   if (d < e) return d;
   return e;

}

void main() {

   auto b = BigInt(8);
   writeln("3rd root of 8 = ", b.iRoot(3));
   b = BigInt(9);
   writeln("3rd root of 9 = ", b.iRoot(3));
   b = BigInt(100)^^2000*2;
   writeln("First 2001 digits of the square root of 2: ", b.iRoot(2));

}</lang>

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Elixir

Translation of: Ruby

<lang elixir>defmodule Integer_roots do

 def root(_, b) when b<2, do: b
 def root(a, b) do
   a1 = a - 1
   f = fn x -> (a1 * x + div(b, power(x, a1))) |> div(a) end
   c = 1
   d = f.(c)
   e = f.(d)
   until(c, d, e, f)
 end
 
 defp until(c, d, e, _) when c in [d, e], do: min(d, e)
 defp until(_, d, e, f), do: until(d, e, f.(e), f)
 
 defp power(_, 0), do: 1
 defp power(n, m), do: Enum.reduce(1..m, 1, fn _,acc -> acc*n end)
 
 def task do
   IO.puts root(3,8)
   IO.puts root(3,9)
   IO.puts "First 2,001 digits of the square root of two:"
   IO.puts root(2, 2 * power(100, 2000))
 end

end

Integer_roots.task</lang>

Output:

2
2
First 2,001 digits of the square root of two:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

F#

Translation of: C#

<lang fsharp>open System

let iroot (base_ : bigint) n =

   if base_ < bigint.Zero || n <= 0 then
       raise (ArgumentException "Bad parameter")

   let n1 = n - 1
   let n2 = bigint n
   let n3 = bigint n1
   let mutable c = bigint.One
   let mutable d = (n3 + base_) / n2
   let mutable e = ((n3 * d) + (base_ / bigint.Pow(d, n1))) / n2
   while c <> d && c <> e do
       c <- d
       d <- e
       e <- (n3 * e + base_ / bigint.Pow(e, n1)) / n2

   if d < e then
       d
   else
       e

[<EntryPoint>] let main _ =

   Console.WriteLine("3rd integer root of 8 = {0}", (iroot (bigint 8) 3))
   Console.WriteLine("3rd integer root of 9 = {0}", (iroot (bigint 9) 3))

   let b = bigint.Pow(bigint 100, 2000) * (bigint 2)
   Console.WriteLine("First 2001 digits of the sqaure root of 2: {0}", (iroot b 2))

   0 // return an integer exit code</lang>

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the sqaure root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Go

int

<lang go>package main

import "fmt"

func main() {

   fmt.Println(root(3, 8))
   fmt.Println(root(3, 9))
   fmt.Println(root(2, 2e18))

}

func root(N, X int) int {

   // adapted from https://en.wikipedia.org/wiki/Nth_root_algorithm
   for r := 1; ; {
       x := X
       for i := 1; i < N; i++ {
           x /= r
       }
       x -= r
       // A small complication here is that Go performs truncated integer
       // division but for negative values of x, Δr in the line below needs 
       // to be computed as the floor of x / N.  The following % test and
       // correction completes the floor division operation (for positive N.)
       Δr := x / N
       if x%N < 0 {
           Δr--
       }
       if Δr == 0 {
           return r
       }
       r += Δr
   }

}</lang>

Output:

2
2
1414213562

big.Int

<lang go>package main

import (

   "fmt"
   "math/big"

)

func main() {

   fmt.Println(root(3, "8"))
   fmt.Println(root(3, "9"))
   fmt.Println(root(2, "2000000000000000000"))
   fmt.Println(root(2, "200000000000000000000000000000000000000000000000000"))

}

var one = big.NewInt(1)

func root(N int, X string) *big.Int {

   var xx, x, Δr big.Int
   xx.SetString(X, 10)
   nn := big.NewInt(int64(N))
   for r := big.NewInt(1); ; {
       x.Set(&xx)
       for i := 1; i < N; i++ {
           x.Quo(&x, r)
       }
       // big.Quo performs Go-like truncated division and would allow direct
       // translation of the int-based solution, but package big also provides
       // Div which performs Euclidean rather than truncated division.
       // This gives the desired result for negative x so the int-based
       // correction is no longer needed and the code here can more directly
       // follow the Wikipedia article.
       Δr.Div(x.Sub(&x, r), nn)
       if len(Δr.Bits()) == 0 {
           return r
       }
       r.Add(r, &Δr)
   }

}</lang>

Output:

2
2
1414213562
14142135623730950488016887

Haskell

Translation of: Python

<lang haskell>root :: Integer -> Integer -> Integer root a b = findAns $ iterate (\x -> (a1 * x + b `div` (x ^ a1)) `div` a) 1

 where
   a1 = a - 1
   findAns (x:xs@(y:z:_))
     | x == y || x == z = min y z
     | otherwise = findAns xs

main :: IO () main = do

 print $ root 3 8
 print $ root 3 9
 print $ root 2 (2 * 100 ^ 2000) -- first 2001 digits of the square root of 2</lang>

Or equivalently, in terms of an applicative expression:

<lang haskell>integerRoot :: Integer -> Integer -> Integer integerRoot n x =

 go $ iterate ((`div` n) . ((+) . (pn *) <*> (x `div`) . (^ pn))) 1
 where
   pn = pred n
   go (x:xs@(y:z:_))
     | x == y || x == z = min y z
     | otherwise = go xs

main :: IO () main = mapM_ (print . uncurry integerRoot) [(3, 8), (3, 9), (2, 2 * 100 ^ 2000)]</lang>

Output:

2
2
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

J

<.@%: satisfies this task. Left argument is the task's N, right argument is the task's X:

Note: Depending on N, one must select the proper number of digits, that is, 2000, 2001, 2002, etc..., otherwise the result will be the digits of the nth root of 20, 2000, etc...
For example, If you use "3 <.@%: (2*10x^2*2000)" instead of "3 <.@%: (2*10x^2*2001)", you will get an output starting with "271441761659490657151808946...", which are the first digits of the cube root of 20, not 2.

<lang J> 9!:37]0 4096 0 222 NB. set display truncation sufficiently high for our results

  2 <.@%: (2*10x^2*2000)

141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

  3 <.@%: (2*10x^2*2001)

125992104989487316476721060727822835057025146470150798008197511215529967651395948372939656243625509415431025603561566525939902404061373722845911030426935524696064261662500097747452656548030686718540551868924587251676419937370969509838278316139915512931369536618394746344857657030311909589598474110598116290705359081647801147352132548477129788024220858205325797252666220266900566560819947156281764050606648267735726704194862076214429656942050793191724414809204482328401274703219642820812019057141889964599983175038018886895942020559220211547299738488026073636974178877921579846750995396300782609596242034832386601398573634339097371265279959919699683779131681681544288502796515292781076797140020406056748039385612517183570069079849963419762914740448345402697154762285131780206438780476493225790528984670858052862581300054293885607206097472230406313572349364584065759169169167270601244028967000010690810353138529027004150842323362398893864967821941498380270729571768128790014457462271477023483571519055067220848184850092872392092826466067171742477537097370300127429180940544256965920750363575703751896037074739934610144901451576359604711119738452991329657262589048609788561801386773836157730098659836608059757560127871214868562426845564116515581793532280158962912994450040120842541416015752584162988142309735821530604057724253836453253356

  5 <.@%: (2*10x^2*2000)

114869835499703500679862694677792758944385088909779750551371111849360320625351305681147311301150847391457571782825280872990018972855371267615994917020637676959403854539263226492033301322122190625130645468320078386350285806907949085127708283982797043969640382563667945344431106523789654147255972578315704103326302050272017414235255993151553782375173884359786924137881735354092890268530342009402133755822717151679559278360263800840317501093689917495888199116488588871447782240220513546797235647742625493141141704109917646404017146978939243424915943739448283626010758721504375406023613552985026793701507511351368254645700768390780390334017990233124030682358360249760098999315658413563173197024899154512108923313999675829872581317721346549115423634135836394159076400636688679216398175376716152621781331348

  7 <.@%: (2*10x^2*2002)

1104089513673812337649505387623344721325326600780124165514532464142106322880380980716598289886302005146897159065579931253969214680430855796510648058388081961639198643922155838145512343974763395078906646859029211806139421440562835192195007740110439139292223389537903767320705032063903809884944457070845279252405827307254864679671836816589429995916822424590361601902611505690284386526869351720866524568004847701822070064334667580822044823960984514550922242408608825451442062850448298384317793721518676765230683406727811327252052334859250776811047221310365241746671294399050316</lang>

Java

Translation of: Kotlin

<lang Java>import java.math.BigInteger;

public class IntegerRoots {

   private static BigInteger iRoot(BigInteger base, int n) {
       if (base.compareTo(BigInteger.ZERO) < 0 || n <= 0) {
           throw new IllegalArgumentException();
       }

       int n1 = n - 1;
       BigInteger n2 = BigInteger.valueOf(n);
       BigInteger n3 = BigInteger.valueOf(n1);
       BigInteger c = BigInteger.ONE;
       BigInteger d = n3.add(base).divide(n2);
       BigInteger e = n3.multiply(d).add(base.divide(d.pow(n1))).divide(n2);
       while (!c.equals(d) && !c.equals(e)) {
           c = d;
           d = e;
           e = n3.multiply(e).add(base.divide(e.pow(n1))).divide(n2);
       }
       if (d.compareTo(e) < 0) {
           return d;
       }
       return e;
   }

   public static void main(String[] args) {
       BigInteger b = BigInteger.valueOf(8);
       System.out.print("3rd integer root of 8 = ");
       System.out.println(iRoot(b, 3));

       b = BigInteger.valueOf(9);
       System.out.print("3rd integer root of 9 = ");
       System.out.println(iRoot(b, 3));

       b = BigInteger.valueOf(100).pow(2000).multiply(BigInteger.valueOf(2));
       System.out.print("First 2001 digits of the square root of 2: ");
       System.out.println(iRoot(b, 2));
   }

}</lang>

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Julia

Works with: Julia version 1.3

Translation of: Python

<lang julia>function iroot(a, b)

   b < 2 && return b
   a1, c = a - 1, 1
   d = (a1 * c + b ÷ c^a1) ÷ a
   e = (a1 * d + b ÷ d^a1) ÷ a
   while d ≠ c ≠ e
       c, d, e = d, e, (a1 * e + b ÷ (e ^ a1)) ÷ a
   end

   min(d, e)

end

println("First 2,001 digits of the square root of two:\n", iroot(2, 2 * big(100) ^ 2000))</lang>

Output:

First 2,001 digits of the square root of two:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Kotlin

Translation of: Python

<lang scala>// version 1.1.2

import java.math.BigInteger

val bigZero = BigInteger.ZERO val bigOne = BigInteger.ONE val bigTwo = BigInteger.valueOf(2L)

fun BigInteger.iRoot(n: Int): BigInteger {

   require(this >= bigZero && n > 0)
   if (this < bigTwo) return this
   val n1 = n - 1
   val n2 = BigInteger.valueOf(n.toLong())
   val n3 = BigInteger.valueOf(n1.toLong())
   var c = bigOne
   var d = (n3 + this) / n2
   var e = (n3 * d + this / d.pow(n1)) / n2
   while (c != d && c != e) {
       c = d
       d = e
       e = (n3 * e + this / e.pow(n1)) / n2
   }
   return if (d < e) d else e

}

fun main(args: Array<String>) {

   var b: BigInteger
   b = BigInteger.valueOf(8L)
   println("3rd integer root of 8 = ${b.iRoot(3)}\n")
   b = BigInteger.valueOf(9L)
   println("3rd integer root of 9 = ${b.iRoot(3)}\n")    
   b = BigInteger.valueOf(100L).pow(2000) * bigTwo
   println("First 2001 digits of the square root of 2:")
   println(b.iRoot(2))

}</lang>

Output:

3rd integer root of 8 = 2

3rd integer root of 9 = 2

First 2001 digits of the square root of 2:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Lua

Translation of: C

<lang lua>function root(base, n)

   if base < 2 then return base end
   if n == 0 then return 1 end

   local n1 = n - 1
   local n2 = n
   local n3 = n1
   local c = 1
   local d = math.floor((n3 + base) / n2)
   local e = math.floor((n3 * d + base / math.pow(d, n1)) / n2)

   while c ~= d and c ~= e do
       c = d
       d = e
       e = math.floor((n3 * e + base / math.pow(e, n1)) / n2)
   end

   if d < e then return d end
   return e

end

-- main local b = 2e18

print("3rd root of 8 = " .. root(8, 3)) print("3rd root of 9 = " .. root(9, 3)) print("2nd root of " .. b .. " = " .. root(b, 2))</lang>

Output:

3rd root of 8 = 2
3rd root of 9 = 2
2nd root of 2e+018 = 1414213562

Modula-2

<lang modula2>MODULE IntegerRoot; FROM FormatString IMPORT FormatString; FROM Terminal IMPORT WriteString,ReadChar;

PROCEDURE pow(b : LONGCARD; p : CARDINAL) : LONGCARD; VAR

   result : LONGCARD;

BEGIN

   result := 1;
   WHILE p > 0 DO
       IF p MOD 2 = 1 THEN
           DEC(p);
           result := result * b;
       END;
       p := p / 2;
       b := b * b
   END;
   RETURN result

END pow;

PROCEDURE root(base : LONGCARD; n : CARDINAL) : LONGCARD; VAR

   n1,n2,n3,c,d,e : LONGCARD;

BEGIN

   IF base < 2 THEN RETURN base END;
   IF n = 0 THEN RETURN 1 END;

   n1 := n - 1;
   n2 := n;
   n3 := n1;
   c := 1;
   d := (n3 + base) / n2;
   e := (n3 * d + base / pow(d, n1)) / n2;

   WHILE (c # d) AND (c # e) DO
       c := d;
       d := e;
       e := (n3 * e + base / pow(e, n1)) / n2
   END;

   IF d < e THEN RETURN d END;
   RETURN e

END root;

(* main *) VAR

   buf : ARRAY[0..63] OF CHAR;
   b : LONGCARD;

BEGIN

   FormatString("3rd root of 8 = %u\n", buf, root(8, 3));
   WriteString(buf);

   FormatString("3rd root of 9 = %u\n", buf, root(9, 3));
   WriteString(buf);

   b := 2000000000000000000;
   FormatString("2nd root of %u = %u\n", buf, b, root(b, 2));
   WriteString(buf);

   ReadChar

END IntegerRoot.</lang>

PARI/GP

<lang parigp>sqrtnint(8,3) sqrtnint(9,3) sqrtnint(2*100^2000,2)</lang>

Output:

%1 = 2
%2 = 2
%3 = 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Perl

Translation of: Ruby

<lang perl>use bigint;

sub integer_root {

   our($a,$b) = @_;
   our $a1 = $a - 1;
   my $c = 1;
   my $d = f($c);
   my $e = f($d);
   ($c, $d, $e) = ($d, $e, f($e)) until $c==$d || $c==$e;
   return $d < $e ? $d : $e;

   sub f { ($a1*$_[0]+$b/$_[0]**$a1)/$a }

}

print integer_root( 3, 8), "\n"; print integer_root( 3, 9), "\n"; print integer_root( 2, 2 * 100 ** 2000), "\n";</lang>

Output:

2
2
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Using a module

If using bigints, we can do this directly, which will be much faster than the method above: <lang perl>use bigint; print 8->babs->broot(3),"\n"; print 9->babs->broot(3),"\n"; print +(2*100**2000)->babs->broot(2),"\n";</lang>

The babs calls are only necessary if the input might be non-negative.

Even faster, using a module: <lang perl>use bigint; use ntheory "rootint"; print rootint(8,3),"\n"; print rootint(9,3),"\n"; print rootint(2*100**2000,2),"\n";</lang>

Both generate the same output as above.

Phix

Library: Phix/mpfr

<lang Phix>include mpfr.e

function integer_root(integer n, object a) -- a must be integer or string

   mpz res = mpz_init()
   {} = mpz_root(res,mpz_init(a),n)
   return mpz_get_str(res)

end function

printf(1,"3rd root of 8 = %s\n", {integer_root(3,8)}) printf(1,"3rd root of 9 = %s\n", {integer_root(3,9)}) string s = integer_root(2,"2"&repeat('0',4000)) printf(1,"First digits of the square root of 2: %s\n", {shorten(s)}) s = integer_root(3,"2"&repeat('0',6000)) printf(1,"First digits of the cube root of 2: %s\n", {shorten(s)})</lang>

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First digits of the square root of 2: 14142135623730950488...47107578486024636008 (2,001 digits)
First digits of the  cube  root of 2: 12599210498948731647...22546828353183047061 (2,001 digits)

Python

<lang python>def root(a, b):

   if b < 2:
       return b
   a1 = a - 1
   c = 1
   d = (a1 * c + b // (c ** a1)) // a
   e = (a1 * d + b // (d ** a1)) // a
   while c not in (d, e):
       c, d, e = d, e, (a1 * e + b // (e ** a1)) // a
   return min(d, e)

print("First 2,001 digits of the square root of two:\n{}".format(

   root(2, 2 * 100 ** 2000)

))</lang>

Output:

First 2,001 digits of the square root of two:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Racket

See #Scheme, there’s very little can be done to improve it.

Raku

(formerly Perl 6)

Translation of: Python

<lang perl6>sub integer_root ( Int $p where * >= 2, Int $n --> Int ) {

   my Int $d = $p - 1;
   my $guess = 10**($n.chars div $p);
   my $iterator = { ( $d * $^x   +   $n div ($^x ** $d) ) div $p };
   my $endpoint = {      $^x      ** $p <= $n
                    and ($^x + 1) ** $p >  $n };
   min (+$guess, $iterator ... $endpoint)[*-1, *-2];

}

say integer_root( 2, 2 * 100 ** 2000 );</lang>

Output:

141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

REXX

No error checking is performed to ensure the root is a non-zero integer.

This version incorporates some optimization when computing square roots (because M is unity, there is no need to
multiply the guess [G] by unity, and no need to compute the guess to the 1^st power, bypassing some trivial arithmetic).

integer result only

<lang rexx>/*REXX program calculates the Nth root of a number to a specified number of decimal digs*/ parse arg num root digs . /*obtain the optional arguments from CL*/ if num== | num=="," then num= 2 /*Not specified? Then use the default.*/ if root== | root=="," then root= 2 /* " " " " " " */ if digs== | digs=="," then digs=2001 /* " " " " " " */ numeric digits digs /*utilize this number of decimal digits*/ say 'number=' num /*display the number that will be used.*/ say ' root=' root /* " " root " " " " */ say 'digits=' digs /* " dec. digits " " " " */ say /* " a blank line. */ say 'result:'; say rootI(num, root, digs) /* " what it is; display the root.*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ rootI: procedure; parse arg x,root,p /*obtain the numbers, Y is the root #.*/

      numeric digits p*root+length(x)           /*double the number of digits  + guard.*/
      if x<2  then return x                     /*B is one or zero?  Return that value.*/
      z=x*(10**root)**p                         /*calculate the number with appended 0s*/
      m=root - 1                                /*utilize a diminished (by one) power. */
      g=(1 + z) % root                          /*take a stab at the first root guess. */
      old=.                                     /* [↓]  When M=1, a fast path for sqrt.*/
      if m==1  then  do  until old==g;   old=g;     g=(g   + z %  g     )  % root;    end
               else  do  until old==g;   old=g;     g=(g*m + z % (g**m) )  % root;    end
      return left(g,p)                          /*return the  Nth root of Z to invoker.*/</lang>

output when the defaults are being used:

number= 2
  root= 2
digits= 2001

result:
14142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605714
70109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847
16038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016
20758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342194897278290641045072636881313739855256117322040245091227700226941127573627280495738108967504018369
86836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847
20896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558
69568685964595155501644724509836896036887323114389415576651040883914292338113206052433629485317049915771756228549741438999188021762430965206564211827316726257539594717255934637238632261482742622208671
15583959992652117625269891754098815934864008345708518147223181420407042650905653233339843645786579679651926729239987536661721598257886026336361782749599421940377775368142621773879919455139723127406689
83299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685
40575867999670121372239475821426306585132217408832382947287617393647467837431960001592188807347857617252211867490424977366929207311096369721608933708661156734585334833295254675851644710757848602463600
8

true results

Negative and complex roots are supported. The expressed root may have a decimal point. <lang rexx>/*REXX program calculates the Nth root of a number to a specified number of decimal digs*/ parse arg num root digs . /*obtain the optional arguments from CL*/ if num== | num=="," then num= 2 /*Not specified? Then use the default.*/ if root== | root=="," then root= 2 /* " " " " " " */ if digs== | digs=="," then digs=2001 /* " " " " " " */ numeric digits digs /*utilize this number of decimal digits*/ say 'number=' num /*display the number that will be used.*/ say ' root=' root /* " " root " " " " */ say 'digits=' digs /* " dec. digits " " " " */ say /* " a blank line. */ say 'result:'; say iRoot(num, root) /* " what it is; display the root.*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ iRoot: procedure; parse arg x 1 ox, y 1 oy /*obtain the numbers, Y is the root #.*/ i=; x=abs(x); y=abs(y) /*use the absolute values of X and Y. */ if ox<0 & oy//2==0 then do; i='i'; ox=x; end /*if the results will be imaginary ··· */ od=digits() /*the current number of decimal digits.*/ a=od+9 /*bump the decimal digits by nine. */ numeric form /*number will be in exponential form.*/ parse value format(x,2,1,,0) 'E0' with ? 'E' _ . /*obtain exponent so we can do division*/ g=(?/y'E'_ % y) + (x>1) /*this is a best first guess of a root.*/ m=y-1 /*define a (fast) variable for later. */ d=5 /*start with only five decimal digits. */

            do  until d==a                      /*keep computing 'til we're at max digs*/
            d=min(d+d,a);           dm=d-2      /*bump number of (growing) decimal digs*/
            numeric digits d                    /*increase the number of decimal digits*/
            o=0                                 /*set the old value to zero (1st time).*/
                do  until o=g;      o=g         /*keep computing as long as  G changes.*/
                g=format((m*g**y+x)/y/g**m,,dm) /*compute the  Yth  root of  X.        */
                end   /*until o=g*/
            end       /*until d==a*/

_=g*sign(ox) /*change the sign of the result, maybe.*/ numeric digits od /*set numeric digits to the original.*/ if oy<0 then return (1/_)i /*Is the root negative? Use reciprocal*/

             return (_/1)i                      /*return the  Yth root of X to invoker.*/</lang>

output when the defaults are being used:

number= 2
  root= 2
digits= 2001

result:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571
47010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884
71603868999706990048150305440277903164542478230684929369186215805784631115966687130130156185689872372352885092648612494977154218334204285686060146824720771435854874155657069677653720226485447015858801
62075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836
98683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884
72089694633862891562882765952635140542267653239694617511291602408715510135150455381287560052631468017127402653969470240300517495318862925631385188163478001569369176881852378684052287837629389214300655
86956868596459515550164472450983689603688732311438941557665104088391429233811320605243362948531704991577175622854974143899918802176243096520656421182731672625753959471725593463723863226148274262220867
11558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668
98329989895386728822856378697749662519966583525776198939322845344735694794962952168891485492538904755828834526096524096542889394538646625744927556381964410316979833061852019379384940057156333720548068
54057586799967012137223947582142630658513221740883238294728761739364746783743196000159218880734785761725221186749042497736692920731109636972160893370866115673458533483329525467585164471075784860246360
08

output when using the input of: -81

number= -81
  root= 2
digits= 2001

result:
9i

output when using the input of: 4 -2

number= 4
  root= -2
digits= 2001

result:
0.5

Ring

Project : Integer roots

see root(3, 8) see root(3, 9) see root(4, 167)

func root(n, x)

      for nr = floor(sqrt(x)) to 1 step -1
           if pow(nr, n) <= x
              see nr + nl
              exit
           ok
      next

</lang> Output:

2
2
3

Ruby

Translation of: Python, zkl

<lang ruby>def root(a,b)

 return b if b<2
 a1, c = a-1, 1
 f = -> x {(a1*x+b/(x**a1))/a}  # a lambda with argument x
 d = f[c]
 e = f[d]
 c, d, e = d, e, f[e] until [d,e].include?(c)
 [d,e].min

end

puts "First 2,001 digits of the square root of two:" puts root(2, 2*100**2000) </lang>

Output:

First 2,001 digits of the square root of two:
14142135623730950488016887242096(...)46758516447107578486024636008

Scala

Functional solution, tail recursive, no immutables

<lang Scala>import scala.annotation.tailrec

object IntegerRoots extends App {

 println("3rd integer root of 8 = " + iRoot(8, 3))

 println("3rd integer root of 9 = " + iRoot(9, 3))

 val result = iRoot(BigInt(100).pow(2000) * BigInt(2), 2)
 println(s"All ${result.toString.length} digits of the square root of 2: \n$result")

 private def iRoot(base: BigInt, degree: Int): BigInt = {
   require(base >= 0 && degree > 0,
     "Base has to be non-negative while the degree must be positive.")

   val (n1, n2) = (degree - 1, BigInt(degree))
   val d = (n1 + base) / n2

   @tailrec
   def loop(c: BigInt, d: BigInt, e: BigInt): BigInt = {
     if (c == d || c == e) if (d < e) d else e
     else loop(d, e, (n1 * e + (base / e.pow(n1))) / n2)
   }

   loop(1, (n1 + base) / n2, (n1 * d + (base / d.pow(n1))) / n2)
 }

}</lang>

Output:

See it running in your browser by ScalaFiddle (JavaScript, non JVM) or by Scastie (JVM).

Scheme

Translation of: Python

<lang scheme>(define (root a b)

 (define // quotient)
 (define (y a a1 b c d e)
   (if (or (= c d) (= c e))
     (min d e)
     (y a a1 b d e (// (+ (* a1 e) (// b (expt e a1))) a))))
 (if (< b 2)
   b
   (let* ((a1 (- a 1))
          (c 1)
          (d (// (+ (* a1 c) (// b (expt c a1))) a))
          (e (// (+ (* a1 d) (// b (expt d a1))) a)))
     (y a a1 b c d e))))

(display "First 2,001 digits of the cube root of two:\n") (display (root 3 (* 2 (expt 1000 2000))))</lang>

Output:

First 2,001 digits of the cube root of two:
125992104989487316476721060727822835057025146470150798008197511215529967651395948372939656243625509415431025603561566525939902404061373722845911030426935524696064261662500097747452656548030686718540551868924587251676419937370969509838278316139915512931369536618394746344857657030311909589598474110598116290705359081647801147352132548477129788024220858205325797252666220266900566560819947156281764050606648267735726704194862076214429656942050793191724414809204482328401274703219642820812019057141889964599983175038018886895942020559220211547299738488026073636974178877921579846750995396300782609596242034832386601398573634339097371265279959919699683779131681681544288502796515292781076797140020406056748039385612517183570069079849963419762914740448345402697154762285131780206438780476493225790528984670858052862581300054293885607206097472230406313572349364584065759169169167270601244028967000010690810353138529027004150842323362398893864967821941498380270729571768128790014457462271477023483571519055067220848184850092872392092826466067171742477537097370300127429180940544256965920750363575703751896037074739934610144901451576359604711119738452991329657262589048609788561801386773836157730098659836608059757560127871214868562426845564116515581793532280158962912994450040120842541416015752584162988142309735821530604057724253836453253356595511725228557956227724036656284687590154306675351908548451181817520429124123378096317252135754114181146612736604578303605744026513096070968164006888185657231009008428452608641405950336900307918699355691335183428569382625543135589735445023330285314932245513412195545782119650083395771426685063328419619686512109255789558850899686190154670043896878665545309854505763765036008943306510356935777537249548436821370317162162183495809356208726009626785183418345652239744540004476021778894208183802786665306532663261864116007400747475473558527701689502063754132232329694243701742343491617690600723853902227681129777413872079823430391031628546452083111122546828353183047061

Sidef

Translation of: Ruby

<lang ruby>func root(a, b) {

   b < 2 && return(b)
   var (a1, c) = (a-1, 1)
   var f = {|x| (a1*x + b//(x**a1)) // a }
   var d = f(c)
   var e = f(d)
   while (c !~ [d, e]) {
       (c, d, e) = (d, e, f(e))
   }
   [d, e].min

}

say "First 2,001 digits of the square root of two:" say root(2, 2 * 100**2000)</lang>

Output:

First 2,001 digits of the square root of two:
14142135623730950488016887242096980[...]32952546758516447107578486024636008

Tcl

Tcl is not made for number crunching. The execution is quite slow compared to compiled languages.

On the other hand, everything is very straightforward, no libraries necessary. <lang tcl> proc root {this n} {

 if {$this < 2} {return $this}
 set n1 [expr $n - 1]
 set n2 $n
 set n3 $n1
 set c 1
 set d [expr ($n3 + $this) / $n2]
 set e [expr ($n3 * $d + $this / ($d ** $n1)) / $n2]
 while {$c != $d && $c != $e} {
   set c $d
   set d $e
   set e [expr ($n3 * $e + $this / ($e ** $n1)) / $n2]
 }
 return [expr min($d, $e)]

}

puts [root 8 3] puts [root 9 3] puts [root [expr 2* (100**2000)] 2] </lang>

Output:

2
2
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Visual Basic .NET

Library: System.Numerics

From the method described on the Wikipedia page. Included is an Integer Square Root function to compare results to the Integer Nth Square root function. One must choose the exponents carefully, otherwise one will obtain the digits of the nth root of 20, 200, 2000, etc..., instead of 2. For example, 4008 was chosen because it works for both n = 2 and n = 3, whereas 4004 was chosen for n = 7<lang vbnet>Imports System Imports System.Numerics Imports Microsoft.VisualBasic.Strings

Public Module Module1

   Public Function IntSqRoot(v As BigInteger) As BigInteger
       Dim digs As Integer = Math.Max(0, v.ToString().Length / 2 - 1)
       IntSqRoot = BigInteger.Parse("3" & StrDup(digs, "0"))
       Dim term As BigInteger
       Do
           term = v / IntSqRoot
           If Math.Abs(CDbl(term - IntSqRoot)) < 2 Then Exit Do
           IntSqRoot = (IntSqRoot + term) / 2
       Loop Until False
   End Function

   Public Function IntNthRoot(n As Integer, v As BigInteger) As BigInteger
       Dim digs As Integer = Math.Max(0, v.ToString().Length / n - 1)
       IntNthRoot = BigInteger.Parse(If(digs > 1, 3, 2).ToString() & StrDup(digs, "0"))
       Dim va As BigInteger, dr As BigInteger
       Do
           va = v : For i As Integer = 2 To n : va /= IntNthRoot : Next
           va -= IntNthRoot
           dr = va / n : If dr = 0 Then Exit Do
           IntNthRoot += dr
       Loop Until False
   End Function

   Public Sub Main()
       Dim b As BigInteger = BigInteger.Parse("2" & StrDup(4008, "0"))
       Console.WriteLine("Integer Cube Root of 8:")
       Console.WriteLine(IntNthRoot(3, 8).ToString()) ' given example
       Console.WriteLine("Integer Cube Root of 9:")
       Console.WriteLine(IntNthRoot(3, 9).ToString()) ' given example
       Console.WriteLine("Integer Square Root of 2, (actually 2 * 10 ^ 4008, square root method):")
       Console.WriteLine(IntSqRoot(b).ToString()) ' reality check
       Console.WriteLine("Integer Square Root of 2, (actually 2 * 10 ^ 4008, nth root method):")
       Console.WriteLine(IntNthRoot(2, b).ToString()) ' given example
       Console.WriteLine("Integer Cube Root of 2, (actually 2 * 10 ^ 4008):")
       Console.WriteLine(IntNthRoot(3, b).ToString()) ' bonus example
       b /= 10000
       Console.WriteLine("Integer 7th Root of 2, (actually 2 * 10 ^ 4004):")
       Console.WriteLine(IntNthRoot(7, b).ToString()) ' bonus example
       If Diagnostics.Debugger.IsAttached Then Console.Read()
   End Sub

End Module </lang>

Output:

Integer Cube Root of 8:
2
Integer Cube Root of 9:
2
Integer Square Root of 2, (actually 2 * 10 ^ 4008, square root method):
1414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237235288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836986836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847208969463386289156288276595263514054226765323969461751129160240871551013515045538128756005263146801712740265396947024030051749531886292563138518816347800156936917688185237868405228783762938921430065586956868596459515550164472450983689603688732311438941557665104088391429233811320605243362948531704991577175622854974143899918802176243096520656421182731672625753959471725593463723863226148274262220867115583959992652117625269891754098815934864008345708518147223181420407042650905653233339843645786579679651926729239987536661721598257886026336361782749599421940377775368142621773879919455139723127406689832998989538672882285637869774966251996658352577619893932284534473569479496295216889148549253890475582883452609652409654288939453864662574492755638196441031697983306185201937938494005715633372054806854057586799967012137223947582142630658513221740883238294728761739364746783743196000159218880734785761725221186749042497736692920731109636972160893370866115673458533483329525467585164471075784860246360083444
Integer Square Root of 2, (actually 2 * 10 ^ 4008, nth root method):
1414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237235288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836986836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847208969463386289156288276595263514054226765323969461751129160240871551013515045538128756005263146801712740265396947024030051749531886292563138518816347800156936917688185237868405228783762938921430065586956868596459515550164472450983689603688732311438941557665104088391429233811320605243362948531704991577175622854974143899918802176243096520656421182731672625753959471725593463723863226148274262220867115583959992652117625269891754098815934864008345708518147223181420407042650905653233339843645786579679651926729239987536661721598257886026336361782749599421940377775368142621773879919455139723127406689832998989538672882285637869774966251996658352577619893932284534473569479496295216889148549253890475582883452609652409654288939453864662574492755638196441031697983306185201937938494005715633372054806854057586799967012137223947582142630658513221740883238294728761739364746783743196000159218880734785761725221186749042497736692920731109636972160893370866115673458533483329525467585164471075784860246360083445
Integer Cube Root of 2, (actually 2 * 10 ^ 4008):
12599210498948731647672106072782283505702514647015079800819751121552996765139594837293965624362550941543102560356156652593990240406137372284591103042693552469606426166250009774745265654803068671854055186892458725167641993737096950983827831613991551293136953661839474634485765703031190958959847411059811629070535908164780114735213254847712978802422085820532579725266622026690056656081994715628176405060664826773572670419486207621442965694205079319172441480920448232840127470321964282081201905714188996459998317503801888689594202055922021154729973848802607363697417887792157984675099539630078260959624203483238660139857363433909737126527995991969968377913168168154428850279651529278107679714002040605674803938561251718357006907984996341976291474044834540269715476228513178020643878047649322579052898467085805286258130005429388560720609747223040631357234936458406575916916916727060124402896700001069081035313852902700415084232336239889386496782194149838027072957176812879001445746227147702348357151905506722084818485009287239209282646606717174247753709737030012742918094054425696592075036357570375189603707473993461014490145157635960471111973845299132965726258904860978856180138677383615773009865983660805975756012787121486856242684556411651558179353228015896291299445004012084254141601575258416298814230973582153060405772425383645325335660
Integer 7th Root of 2, (actually 2 * 10 ^ 4004):
110408951367381233764950538762334472132532660078012416551453246414210632288038098071659828988630200514689715906557993125396921468043085579651064805838808196163919864392215583814551234397476339507890664685902921180613942144056283519219500774011043913929222338953790376732070503206390380988494445707084527925240582730725486467967183681658942999591682242459036160190261150569028438652686935172086652456800484770182207006433466758082204482396098451455092224240860882545144206285044829838431779372151867676523068340672781132725205233485925077681104722131036524174667129439905032

zkl

Translation of: Python

Uses GNU GMP library <lang zkl>var [const] BN=Import("zklBigNum"); fcn root(n,r){

  f:='wrap(z){ (n/z.pow(r-1) + z*(r-1))/r or 1 };  //--> v or 1
  c,d,e:=1,f(c),f(d);
  while(c!=d and c!=e){ c,d,e=d,e,f(e) }
  if(d<e) d else e

}</lang> <lang zkl>a:=BN(100).pow(2000)*2; println("Does GMP agree: ",root(a,3)==a.root(3));</lang>

Output:

Does GMP agree: True

@@ Line 696: / Line 696: @@
 =={{header|Phix}}==
-{{libheader|mpfr}}
+{{libheader|Phix/mpfr}}
 <lang Phix>include mpfr.e