function r = i4_to_van_der_corput ( key, base )

%*****************************************************************************80
%
%% i4_to_van_der_corput() computes an element of a van der Corput sequence.
%
%  Discussion:
%
%    The van der Corput sequence is often used to generate a "subrandom"
%    sequence of points which have a better covering property
%    than pseudorandom points.
%
%    The van der Corput sequence generates a sequence of points in [0,1]
%    which (theoretically) never repeats.  Except for key = 0, the
%    elements of the van der Corput sequence are strictly between 0 and 1.
%
%    The van der Corput sequence writes an integer in a given base B,
%    and then its digits are "reflected" about the decimal point.
%    This maps the numbers from 1 to N into a set of numbers in [0,1],
%    which are especially nicely distributed if N is one less
%    than a power of the base.
%
%    Hammersley suggested generating a set of N nicely distributed
%    points in two dimensions by setting the first component of the
%    Ith point to I/N, and the second to the van der Corput 
%    value of I in base 2.  
%
%    Halton suggested that in many cases, you might not know the number 
%    of points you were generating, so Hammersley's formulation was
%    not ideal.  Instead, he suggested that to generated a nicely
%    distributed sequence of points in M dimensions, you simply
%    choose the first M primes, P(1:M), and then for the J-th component of
%    the I-th point in the sequence, you compute the van der Corput
%    value of I in base P(J).
%
%    Thus, to generate a Halton sequence in a 2 dimensional space,
%    it is typical practice to generate a pair of van der Corput sequences,
%    the first with prime base 2, the second with prime base 3.
%    Similarly, by using the first K primes, a suitable sequence
%    in K-dimensional space can be generated.
%
%    The generation is quite simple.  Given an integer key, the expansion
%    of key in base BASE is generated.  Then, essentially, the result R
%    is generated by writing a decimal point followed by the digits of
%    the expansion of key, in reverse order.  This decimal value is actually
%    still in base BASE, so it must be properly interpreted to generate
%    a usable value.
%
%  Example:
%
%    BASE = 2
%
%    key     key      van der Corput
%    decimal  binary    binary   decimal
%    -------  ------    ------   -------
%        0  =     0  =>  .0     = 0.0
%        1  =     1  =>  .1     = 0.5
%        2  =    10  =>  .01    = 0.25
%        3  =    11  =>  .11    = 0.75
%        4  =   100  =>  .001   = 0.125
%        5  =   101  =>  .101   = 0.625
%        6  =   110  =>  .011   = 0.375
%        7  =   111  =>  .111   = 0.875
%        8  =  1000  =>  .0001  = 0.0625
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    27 February 2003
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    J H Halton,
%    On the efficiency of certain quasi-random sequences of points
%    in evaluating multi-dimensional integrals,
%    Numerische Mathematik,
%    Volume 2, pages 84-90, 1960.
% 
%    J M Hammersley,
%    Monte Carlo methods for solving multivariable problems,
%    Proceedings of the New York Academy of Science,
%    Volume 86, pages 844-874, 1960.
%
%    J G van der Corput,
%    Verteilungsfunktionen I and II,
%    Nederl. Akad. Wetensch. Proc.,
%    Volume 38, 1935, pages 813-820, pages 1058-1066.
%
%  Input:
%
%    integer key, the key or index of the desired element.
%    key should be nonnegative.  Only the integer part of key is used.
%    key = 0 is allowed, and returns R = 0.
%
%    integer BASE, the van der Corput base, which is typically
%    a prime number.  Only the integer part of BASE is used.
%    BASE must be greater than 1.
%
%  Output:
%
%    real R, the key-th element of the van der Corput sequence
%    for base BASE.
%
  r = 0.0;
%
%  Ensure that BASE is an integer, and acceptable.
%
  base = floor ( base );

  if ( base <= 1 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4_TO_VAN_DER_CORPUT - Fatal error!\n' );
    fprintf ( 1, '  The input base BASE is <= 1!\n' );
    fprintf ( 1, '  BASE = %d\n', base );
    error ( 'I4_TO_VAN_DER_CORPUT - Fatal error!' );
  end
%
%  Ensure that key is an integer, and acceptable.
%
  key = floor ( key );

  if ( key < 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4_TO_VAN_DER_CORPUT - Fatal error!\n' );
    fprintf ( 1, '  The input base key is < 0!\n' );
    fprintf ( 1, '  key = %d\n', key );
    error ( 'I4_TO_VAN_DER_CORPUT - Fatal error!' );
  end
%
%  Carry out the computation.
%
  base_inv = 1.0 / base;

  while ( key ~= 0 )
    digit = mod ( key, base );
    r = r + digit * base_inv;
    base_inv = base_inv / base;
    key = floor ( key / base );
  end

  return
end

