Creating an Arbitrary-precision Nth Root of X Function in PHP

Submitted by Jay-Tanner on Monday, December 20, 2021 - 09:49.

Language

PHP

This math utility function computes the real Nth root of X to any specified number of decimals.

Above is the iteration equation evaluated by the function. It comes in handy for advanced scientific programming.

The archive contains a demo program.

The demo code below, should return:

5 root of 100 to 50 decimals =2.51188643150958011108503206779932739415851810078248

Here is a listing of the demo program and the function code:

<?php
 
/*
   This utility function computes the real Nth root of X
   to any specified number of decimals.
 
   It comes in handy for advanced science programming.
*/
 
 
/*
   ===================================
   This example demonstrates computing
   the 5th root of 100 to 50 decimals.
*/
 
// Set root value.
   $N = 5;
 
// Set test argument as a numeric string.
   $X = '100';
 
// Set number of decimals limit.
   $decimals = 50;
 
// Call the function to compute Nth root of X
// to the specified number of decimals.
   $NthRootX = BC_Nth_Root ($N, $X, $decimals);
 
// Print computed root to page.
   print
"<pre style='font-size:150%; font-weight:bold;'>
 
$N root of $X to $decimals decimals =
 
$NthRootX
</pre>";
 
 
 
/*
   =======================================================================
   This function computes the arbitrary-precision Nth root of a given
   numerical string X rounded to any specified number of decimals.
 
   ----------
   ARGUMENTS:
   N = Root to be iterated = Positive Integer: 1, 2, 3, 4, ..., etc.
   X = Numerical string for which we seek the Nth root.
 
   Decimals = Number of decimals at which to round off the result
              if the exact resolution is not possible.
 
   Based on the following Nth root of X iteration algorithm.
 
   ERRORS:
   An error message is returned if an argument is non-numeric.
 
   NO DEPENDENCIES
   =======================================================================
*/
 
   function BC_Nth_Root ($N, $X, $Decimals=16)
{
   $x = trim($X);
   $N = trim($N);
 
   if (!Is_Numeric($x) or !Is_Numeric($N))
  {
   return "ERROR: BC_Nth_Root()\n\(X) argument must be a numeric string.";
  }
 
   $N = floor(abs($N));
 
// Check for odd root value with a negative argument, in which
// case, the sign of the argument is preserved in the root.
   $sign = '';  if ($x < 0) {$x = Str_Replace('-', '', $x);  $sign = '-';}
 
// Generate 1st approximation = Native machine value.
   $a = SPrintF("%1.16f", pow($x, 1/$N));
   $n = floor(abs($N)) - 1;
   $d = floor(abs($Decimals));
   $D = 2*$d;
 
// Loop to iterate root value (limit = 50 iterations).
// This should never get anywhere near 50. This limit
// prevents a lockup if something goes wrong.
   for ($i=1;  $i <= 50;   $i++)
  {
// Compute next approximation (b) from current approximation (a).
   $b = bcDiv(bcAdd(bcMul($n,$a,$D),bcDiv($x,bcpow($a,$n,$D),$D),$D),$N,$D);
 
// If (a == b) to precision limit, then done.
// Otherwise, execute another loop cycle.
   if (bcComp($a,$b,$D) == 0) {break;} else {$a = $b;}
  }
// Done.  Return root value rounded to the specified number of decimals.
   $w = $sign.bcAdd($b, '0.'.Str_Repeat(0,$d).'5', $d);
 
   if ($w == bcAdd('0', $w))
      {
       $w = Rtrim(Rtrim($w, '0'), '.');
      }
 
   return $w;
 
} // end of  BC_Nth_Root (...)
 
?>
  

I hope this function I provided will help you. Thanks

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