nth_root <- function ( a, n, tol = 1 / 1000 )

#*****************************************************************************80
#
## nth_root computes the n-th root of a number.
#
#  Licensing:
#
#    Copyright 2016 James P. Howard, II
#
#    The computer code and data files on this web page are distributed under
#    https://opensource.org/licenses/BSD-2-Clause, the BSD-2-Clause license.
#
#  Modified:
#
#    03 February 2020
#
#  Author:
#
#    Original R version by James Howard.
#    Modifications by John Burkardt.
#
#  Reference:
#
#    James Howard,
#    Computational Methods for Numerical Analysis with R,
#    CRC Press, 2017
#    ISBN13: 978-1-4987-2363-3.
#
#  Input:
#
#    real A: the number whose n-th root is desired.
#
#    integer N: the root desired.
#    2 <= N.
#
#    real TOL; a convergence tolerance, with a default of 1/1000.
#
#  Output:
#
#    real X: the value of the n-th root of A.
{
  x <- 1.0
  deltax <- tol * 10.0

  while ( abs ( deltax ) > tol )
  {
    deltax <- ( 1.0 / n ) * ( a / x ^ ( n - 1 ) - x )
    x <- x + deltax
  }

  return ( x )
}