hammersley.cpp

# include <cmath>
# include <cstdlib>
# include <cstring>
# include <ctime>
# include <iomanip>
# include <iostream>

using namespace std;

# include "hammersley.hpp"

//****************************************************************************80

double *hammersley ( int i, int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    HAMMERSLEY computes an element of a Hammersley sequence.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    20 August 2016
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    John Hammersley,
//    Monte Carlo methods for solving multivariable problems,
//    Proceedings of the New York Academy of Science,
//    Volume 86, 1960, pages 844-874.
//
//  Parameters:
//
//    Input, int I, the index of the element of the sequence.
//    0 <= I.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the "base" for the first component.
//    1 <= N.
//
//    Output, double HAMMERSLEY[M], the element of the sequence with index I.
//
{
  int d;
  int i1;
  int j;
  double *prime_inv;
  double *r;
  int *t;

  t = new int[m];

  t[0] = 0;
  for ( j = 1; j < m; j++ )
  {
    t[j] = i;
  }
//
//  Carry out the computation.
//
  prime_inv = new double[m];

  prime_inv[0] = 1.0;
  for ( j = 1; j < m; j++ )
  {
    prime_inv[j] = 1.0 / ( double ) ( prime ( j ) );
  }

  r = new double[m];

  r[0] = ( double ) ( i % ( n + 1 ) ) / ( double ) ( n );
  for ( j = 1; j < m; j++ )
  {
    r[j] = 0.0;
  }

  while ( 0 < i4vec_sum ( m, t ) )
  {
    for ( j = 1; j < m; j++ )
    {
      d = ( t[j] % prime ( j ) );
      r[j] = r[j] + ( double ) ( d ) * prime_inv[j];
      prime_inv[j] = prime_inv[j] / ( double ) ( prime ( j ) );
      t[j] = ( t[j] / prime ( j ) );
    }
  }

  delete [] prime_inv;
  delete [] t;

  return r;
}
//****************************************************************************80

int hammersley_inverse ( double r[], int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    HAMMERSLEY_INVERSE inverts an element of the Hammersley sequence.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    20 August 2016
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double R[M], the I-th element of the Hammersley sequence.
//    0 <= R < 1.0
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the "base" for the first component.
//    1 <= N.
//
//    Output, int HAMMERSLEY_INVERSE, the index of the element of the sequence.
//
{
  int d;
  int i;
  int j;
  int p;
  double t;

  for ( j = 0; j < m; j++ )
  {
    if ( r[j] < 0.0 || 1.0 < r[j] )
    {
      cerr << "\n";
      cerr << "HAMMERSLEY_INVERSE - Fatal error!\n";
      cerr << "  0 <= R <= 1.0 is required.\n";
      exit ( 1 );
    }
  }

  if ( m < 1 )
  {
    cerr << "\n";
    cerr << "HAMMERSLEY_INVERSE - Fatal error!\n";
    cerr << "  1 <= M is required.\n";
    exit ( 1 );
  }

  if ( n < 1 )
  {
    cerr << "\n";
    cerr << "HAMMERSLEY_INVERSE - Fatal error!'\n";
    cerr << "  1 <= N is required.\n";
    exit ( 1 );
  }
//
//  Invert using the second component only, because working with base
//  2 is accurate.
//
  if ( 2 <= m )
  {
    i = 0;
    t = r[1];
    p = 1;

    while ( t != 0.0 )
    {
      t = t * 2.0;
      d = ( int ) ( t );
      i = i + d * p;
      p = p * 2;
      t = t - ( double ) ( d );
    }
  }
  else
  {
    i = ( int ) ( round ( ( double ) ( n ) * r[0] ) );
  }
 
  return i;
}
//****************************************************************************80

double *hammersley_sequence ( int i1, int i2, int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    HAMMERSLEY_SEQUENCE computes elements I1 through I2 of a Hammersley sequence.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    20 August 2016
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    John Hammersley,
//    Monte Carlo methods for solving multivariable problems,
//    Proceedings of the New York Academy of Science,
//    Volume 86, 1960, pages 844-874.
//
//  Parameters:
//
//    Input, int I1, I2, the indices of the first and last 
//    elements of the sequence.  0 <= I1, I2.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the "base" for the first component.
//    1 <= N.
//
//    Output, double HAMMERSLEY_SEQUENCE[M*(abs(I1-I2)+1)], the elements of 
//    the sequence with indices I1 through I2.
//
{
  int d;
  int i;
  int i3;
  int j;
  int k;
  int l;
  double *prime_inv;
  double *r;
  int *t;

  if ( i1 <= i2 )
  {
    i3 = +1;
  }
  else
  {
    i3 = -1;
  }

  prime_inv = new double[m];
  prime_inv[0] = 1.0;

  l = abs ( i2 - i1 ) + 1;
  r = new double[m*l];

  for ( k = 0; k < l; k++ )
  {
    for ( j = 0; j < m; j++ )
    {
      r[j+k*m] = 0.0;
    }
  }

  i = i1;

  t = new int[m];
  for ( k = 0; k < l; k++ )
  {
    t[0] = 0;
    for ( j = 1; j < m; j++ )
    {
      t[j] = i;
    }
//
//  Carry out the computation.
//
    for ( j = 1; j < m; j++ )
    {
      prime_inv[j] = 1.0 / ( double ) ( prime ( j ) );
    }

    r[0+k*m] = ( double ) ( i % ( n + 1 ) ) / ( double ) ( n );

    while ( 0 < i4vec_sum ( m, t ) )
    {
      for ( j = 1; j < m; j++ )
      {
        d = ( t[j] % prime ( j ) );
        r[j+k*m] = r[j+k*m] + ( double ) ( d ) * prime_inv[j];
        prime_inv[j] = prime_inv[j] / ( double ) ( prime ( j ) );
        t[j] = ( t[j] / prime ( j ) );
      }
    }
    i = i + i3;
  }

  delete [] prime_inv;
  delete [] t;

  return r;
}
//****************************************************************************80

int i4vec_sum ( int n, int a[] )

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_SUM sums the entries of an I4VEC.
//
//  Discussion:
//
//    An I4VEC is a vector of I4's.
//
//  Example:
//
//    Input:
//
//      A = ( 1, 2, 3, 4 )
//
//    Output:
//
//      I4VEC_SUM = 10
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    26 May 1999
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, int A[N], the vector to be summed.
//
//    Output, int I4VEC_SUM, the sum of the entries of A.
//
{
  int i;
  int sum;

  sum = 0;
  for ( i = 0; i < n; i++ )
  {
    sum = sum + a[i];
  }

  return sum;
}
//****************************************************************************80

int prime ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    PRIME returns any of the first PRIME_MAX prime numbers.
//
//  Discussion:
//
//    PRIME_MAX is 1600, and the largest prime stored is 13499.
//
//    Thanks to Bart Vandewoestyne for pointing out a typo, 18 February 2005.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 February 2005
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz and Irene Stegun,
//    Handbook of Mathematical Functions,
//    US Department of Commerce, 1964, pages 870-873.
//
//    Daniel Zwillinger,
//    CRC Standard Mathematical Tables and Formulae,
//    30th Edition,
//    CRC Press, 1996, pages 95-98.
//
//  Parameters:
//
//    Input, int N, the index of the desired prime number.
//    In general, is should be true that 0 <= N <= PRIME_MAX.
//    N = -1 returns PRIME_MAX, the index of the largest prime available.
//    N = 0 is legal, returning PRIME = 1.
//
//    Output, int PRIME, the N-th prime.  If N is out of range, PRIME
//    is returned as -1.
//
{
# define PRIME_MAX 1600

  int npvec[PRIME_MAX] = {
        2,    3,    5,    7,   11,   13,   17,   19,   23,   29,
       31,   37,   41,   43,   47,   53,   59,   61,   67,   71,
       73,   79,   83,   89,   97,  101,  103,  107,  109,  113,
      127,  131,  137,  139,  149,  151,  157,  163,  167,  173,
      179,  181,  191,  193,  197,  199,  211,  223,  227,  229,
      233,  239,  241,  251,  257,  263,  269,  271,  277,  281,
      283,  293,  307,  311,  313,  317,  331,  337,  347,  349,
      353,  359,  367,  373,  379,  383,  389,  397,  401,  409,
      419,  421,  431,  433,  439,  443,  449,  457,  461,  463,
      467,  479,  487,  491,  499,  503,  509,  521,  523,  541,
      547,  557,  563,  569,  571,  577,  587,  593,  599,  601,
      607,  613,  617,  619,  631,  641,  643,  647,  653,  659,
      661,  673,  677,  683,  691,  701,  709,  719,  727,  733,
      739,  743,  751,  757,  761,  769,  773,  787,  797,  809,
      811,  821,  823,  827,  829,  839,  853,  857,  859,  863,
      877,  881,  883,  887,  907,  911,  919,  929,  937,  941,
      947,  953,  967,  971,  977,  983,  991,  997, 1009, 1013,
     1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,
     1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151,
     1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223,
     1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291,
     1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373,
     1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451,
     1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511,
     1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583,
     1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657,
     1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733,
     1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811,
     1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
     1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987,
     1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053,
     2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129,
     2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213,
     2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287,
     2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357,
     2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423,
     2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531,
     2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617,
     2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687,
     2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741,
     2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819,
     2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903,
     2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999,
     3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079,
     3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181,
     3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257,
     3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331,
     3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413,
     3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511,
     3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571,
     3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643,
     3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727,
     3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821,
     3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907,
     3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989,
     4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057,
     4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139,
     4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231,
     4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297,
     4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409,
     4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493,
     4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583,
     4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657,
     4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751,
     4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831,
     4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937,
     4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003,
     5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087,
     5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179,
     5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279,
     5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387,
     5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443,
     5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521,
     5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639,
     5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693,
     5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791,
     5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857,
     5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939,
     5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053,
     6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133,
     6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221,
     6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301,
     6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367,
     6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473,
     6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571,
     6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673,
     6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761,
     6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833,
     6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917,
     6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997,
     7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103,
     7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207,
     7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297,
     7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411,
     7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499,
     7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561,
     7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643,
     7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723,
     7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829,
     7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919,
     7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017,
     8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111,
     8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219,
     8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291,
     8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387,
     8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501,
     8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597,
     8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677,
     8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741,
     8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831,
     8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929,
     8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011,
     9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109,
     9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199,
     9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283,
     9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377,
     9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439,
     9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533,
     9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631,
     9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733,
     9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811,
     9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887,
     9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973,10007,
    10009,10037,10039,10061,10067,10069,10079,10091,10093,10099,
    10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,
    10181,10193,10211,10223,10243,10247,10253,10259,10267,10271,
    10273,10289,10301,10303,10313,10321,10331,10333,10337,10343,
    10357,10369,10391,10399,10427,10429,10433,10453,10457,10459,
    10463,10477,10487,10499,10501,10513,10529,10531,10559,10567,
    10589,10597,10601,10607,10613,10627,10631,10639,10651,10657,
    10663,10667,10687,10691,10709,10711,10723,10729,10733,10739,
    10753,10771,10781,10789,10799,10831,10837,10847,10853,10859,
    10861,10867,10883,10889,10891,10903,10909,10937,10939,10949,
    10957,10973,10979,10987,10993,11003,11027,11047,11057,11059,
    11069,11071,11083,11087,11093,11113,11117,11119,11131,11149,
    11159,11161,11171,11173,11177,11197,11213,11239,11243,11251,
    11257,11261,11273,11279,11287,11299,11311,11317,11321,11329,
    11351,11353,11369,11383,11393,11399,11411,11423,11437,11443,
    11447,11467,11471,11483,11489,11491,11497,11503,11519,11527,
    11549,11551,11579,11587,11593,11597,11617,11621,11633,11657,
    11677,11681,11689,11699,11701,11717,11719,11731,11743,11777,
    11779,11783,11789,11801,11807,11813,11821,11827,11831,11833,
    11839,11863,11867,11887,11897,11903,11909,11923,11927,11933,
    11939,11941,11953,11959,11969,11971,11981,11987,12007,12011,
    12037,12041,12043,12049,12071,12073,12097,12101,12107,12109,
    12113,12119,12143,12149,12157,12161,12163,12197,12203,12211,
    12227,12239,12241,12251,12253,12263,12269,12277,12281,12289,
    12301,12323,12329,12343,12347,12373,12377,12379,12391,12401,
    12409,12413,12421,12433,12437,12451,12457,12473,12479,12487,
    12491,12497,12503,12511,12517,12527,12539,12541,12547,12553,
    12569,12577,12583,12589,12601,12611,12613,12619,12637,12641,
    12647,12653,12659,12671,12689,12697,12703,12713,12721,12739,
    12743,12757,12763,12781,12791,12799,12809,12821,12823,12829,
    12841,12853,12889,12893,12899,12907,12911,12917,12919,12923,
    12941,12953,12959,12967,12973,12979,12983,13001,13003,13007,
    13009,13033,13037,13043,13049,13063,13093,13099,13103,13109,
    13121,13127,13147,13151,13159,13163,13171,13177,13183,13187,
    13217,13219,13229,13241,13249,13259,13267,13291,13297,13309,
    13313,13327,13331,13337,13339,13367,13381,13397,13399,13411,
    13417,13421,13441,13451,13457,13463,13469,13477,13487,13499 };

  if ( n == -1 )
  {
    return PRIME_MAX;
  }
  else if ( n == 0 )
  {
    return 1;
  }
  else if ( n <= PRIME_MAX )
  {
    return npvec[n-1];
  }
  else
  {
    cerr << "\n";
    cerr << "PRIME - Fatal error!\n";
    cerr << "  Unexpected input value of n = " << n << "\n";
    exit ( 1 );
  }

  return 0;
# undef PRIME_MAX
}
//****************************************************************************80

void r8mat_print ( int m, int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_PRINT prints an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//    Entry A(I,J) is stored as A[I+J*M]
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, double A[M*N], the M by N matrix.
//
//    Input, string TITLE, a title.
//
{
  r8mat_print_some ( m, n, a, 1, 1, m, n, title );

  return;
}
//****************************************************************************80

void r8mat_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi,
  int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_PRINT_SOME prints some of an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    26 June 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrix.
//    M must be positive.
//
//    Input, int N, the number of columns of the matrix.
//    N must be positive.
//
//    Input, double A[M*N], the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, designate the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title.
//
{
# define INCX 5

  int i;
  int i2hi;
  int i2lo;
  int j;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";

  if ( m <= 0 || n <= 0 )
  {
    cout << "\n";
    cout << "  (None)\n";
    return;
  }
//
//  Print the columns of the matrix, in strips of 5.
//
  for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX )
  {
    j2hi = j2lo + INCX - 1;
    if ( n < j2hi )
    {
      j2hi = n;
    }
    if ( jhi < j2hi )
    {
      j2hi = jhi;
    }
    cout << "\n";
//
//  For each column J in the current range...
//
//  Write the header.
//
    cout << "  Col:    ";
    for ( j = j2lo; j <= j2hi; j++ )
    {
      cout << setw(7) << j - 1 << "       ";
    }
    cout << "\n";
    cout << "  Row\n";
    cout << "\n";
//
//  Determine the range of the rows in this strip.
//
    if ( 1 < ilo )
    {
      i2lo = ilo;
    }
    else
    {
      i2lo = 1;
    }
    if ( ihi < m )
    {
      i2hi = ihi;
    }
    else
    {
      i2hi = m;
    }

    for ( i = i2lo; i <= i2hi; i++ )
    {
//
//  Print out (up to) 5 entries in row I, that lie in the current strip.
//
      cout << setw(5) << i - 1 << ": ";
      for ( j = j2lo; j <= j2hi; j++ )
      {
        cout << setw(12) << a[i-1+(j-1)*m] << "  ";
      }
      cout << "\n";
    }
  }

  return;
# undef INCX
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}