Integer square root algorithms

By Jul 14, 2016

Description:

Integer square root algorithms

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We will outline in this article methods to calculate the square root of a large integer.

But we are just going to calculate the integer square root of an integer corresponding to the largest integer whose square is less than or equal to the whole number.

Babylonian method (Heron's method)

This calculation method #[ is to choose ]# a #[ ]# relatively close to the square root value of an integer N and to calculate from it several consecutive values by applying the function   defined by:

as we seek the integer square root, we can only work with integers and apply the following function :

the brackets  denoting the integer part function.

The integer part of N/x can be obtained by a Euclidean division of N by x.

Digit-by-digit extraction method

This ancient method of extraction is described in Wikipedia. It has the advantage of giving the numbers one by one. But for large integers is not efficient because of divisions between integers that are becoming larger.

Program in C#

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Numerics;
 
 
namespace Sqrt
{
       class SquareRoot
       {
         const int a = 256;
         byte[] tableSqareRoot = new byte[]{
                       0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3,
                       3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5,
                       5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6,
                       6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
                       7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
                       8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9,
                       9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,
                       10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
                       10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11,
                       11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12,
                       12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12,
                       12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
                       13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
                       13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14,
                       14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14,
                       15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,
                       15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,
               15 };
 
               int BiggestDigit(BigInteger p, BigInteger c)
               {
                       BigInteger bigQuotient = c;
                       bigQuotient = bigQuotient >> 9; //divide by 512
                       bigQuotient = BigInteger.Divide(bigQuotient, p);
                       BigInteger up = bigQuotient + BigInteger.One;
                       BigInteger errd = up * up;
                       BigInteger errn = errd * up;
                       errd = errd + c;
                       BigInteger bigErr = BigInteger.Divide(errn, errd);
                       int err = (int)bigErr + 1;
                       int quotient = (int)bigQuotient;
                       int uu = (int)(quotient - err);
                       if (uu < 0) uu = 0;
                       BigInteger bigP = p << 9; //Multiply by 512
                       int e = a - 1;
                       BigInteger s = (bigP * uu) - c;
                       for (int i = uu; i < a; i++)
                       {
                               if ((s + i * i) > BigInteger.Zero)
                               {
                                       e = i - 1;
                                       break;
                               }
                               s = s + bigP;
                       }
                       return e;
               }
             
               public BigInteger Sqrt(BigInteger N)
               {
                       byte[] t = N.ToByteArray();
                       int n = t.Length - 1;
                       bool zero = true;
                       while (n >= 0 && zero)
                       {
                               if (t[n] != 0)
                               {
                                       zero = false;
                               }
                               else
                                       n--;
                       }
                       int pp = t[n];
                       if (n % 2 != 0)
                       {
                               n--;
                               pp = (pp << 8) + t[n];
                       }
                       n--;
                       BigInteger c = pp;
                       pp = (int)Math.Floor(Math.Sqrt(pp));
                       //It is not worth the upgrade.
                       BigInteger p = pp;
                       BigInteger y = p * p;
                       c = c - y;
                       int i = n;
                       while (i >= 0)
                       {
                               c = (c << 8) + t[i];
                               i--;
                               c = (c << 8) + t[i];
                               i--;
                               int x = BiggestDigit(p, c);
                               p = (p << 8);
                               y = (x + (p << 1)) * x;
                               p = p + x;
                               c = c - y;
                       }
                       return p;
               }
 
               BigInteger SqareRootN(BigInteger N)
               {
                       if (N < 256) return tableSqareRoot[(int)N];
                       int bitLength = Convert.ToInt32(Math.Ceiling(BigInteger.Log(N, 2)));
                       BigInteger X = N;
                       BigInteger Y = (N >> ((bitLength + 1) / 2));
                       while (Y < X)
                       {
                               X = Y;
                               Y = (BigInteger.Divide(N, Y) + Y) >> 1;
                       }
                       return X;
               }
 
               public void Test()
               {
 
                       string stringN = "30555555555555550";
                       BigInteger N = BigInteger.Parse(stringN);
                       BigInteger R;
                       BigInteger S;
                       var watch = System.Diagnostics.Stopwatch.StartNew();
                       for (int i = 0; i < 100000; i++)
                               R = SqareRootN(N);
                       watch.Stop();
                       var elapsedMs = watch.ElapsedMilliseconds;
                       Console.Out.WriteLine(elapsedMs);
                       Console.Out.WriteLine(R.ToString());
                       Console.Out.WriteLine();
                       watch = System.Diagnostics.Stopwatch.StartNew();
                       for (int i = 0; i < 100000; i++)
                               S = Sqrt(N);
                       watch.Stop();
                       elapsedMs = watch.ElapsedMilliseconds;
                       Console.Out.WriteLine(elapsedMs);
                       Console.Out.WriteLine(S.ToString());
                       Console.In.ReadLine();
               }
       }
}

CSharp

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