kmeans.cpp

# include <cstdlib>
# include <iostream>
# include <fstream>
# include <iomanip>
# include <cmath>
# include <ctime>
# include <string>
# include "kmeans.hpp"

using namespace std;

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

char ch_cap(char ch)

//****************************************************************************80
//
//  Purpose:
//
//    CH_CAP capitalizes a single character.
//
//  Discussion:
//
//    This routine should be equivalent to the library "toupper" function.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    19 July 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH, the character to capitalize.
//
//    Output, char CH_CAP, the capitalized character.
//
{
	if (97 <= ch && ch <= 122) {
		ch = ch - 32;
	}

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

bool ch_eqi(char ch1, char ch2)

//****************************************************************************80
//
//  Purpose:
//
//    CH_EQI is true if two characters are equal, disregarding case.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH1, CH2, the characters to compare.
//
//    Output, bool CH_EQI, is true if the two characters are equal,
//    disregarding case.
//
{
	if (97 <= ch1 && ch1 <= 122) {
		ch1 = ch1 - 32;
	}
	if (97 <= ch2 && ch2 <= 122) {
		ch2 = ch2 - 32;
	}

	return (ch1 == ch2);
}
//****************************************************************************80

int ch_to_digit(char ch)

//****************************************************************************80
//
//  Purpose:
//
//    CH_TO_DIGIT returns the integer value of a base 10 digit.
//
//  Example:
//
//     CH  DIGIT
//    ---  -----
//    '0'    0
//    '1'    1
//    ...  ...
//    '9'    9
//    ' '    0
//    'X'   -1
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH, the decimal digit, '0' through '9' or blank are legal.
//
//    Output, int CH_TO_DIGIT, the corresponding integer value.  If the
//    character was 'illegal', then DIGIT is -1.
//
{
	int digit;

	if ('0' <= ch && ch <= '9') {
		digit = ch - '0';
	} else if (ch == ' ') {
		digit = 0;
	} else {
		digit = -1;
	}

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

double *cluster_energy_compute(int dim_num, int point_num, int cluster_num,
	double point[], int cluster[], double cluster_center[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//    CLUSTER_ENERGY_COMPUTE computes the energy of the clusters.
	//
	//  Discussion:
	//
	//    The cluster energy is defined as the sum of the distance
	//    squared from each point to its cluster center.  It is the goal
	//    of the H-means and K-means algorithms to find, for a fixed number
	//    of clusters, a clustering that minimizes this energy
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    07 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of data points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the data points.
	//
	//    Input, int CLUSTER[POINT_NUM], the cluster to which each
	//    data point belongs.  These values are 0-based.
	//
	//    Input, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], the 
	//    centers associated with the minimal energy clustering.
	//
	//    Output, double CLUSTER_ENERGY_COMPUTE[CLUSTER_NUM], the energy
	//    associated with each cluster.
	//
{
	double *cluster_energy;
	int i;
	int j;
	int k;
	double point_energy;

	cluster_energy = r8vec_zero_new(cluster_num);

	for (j = 0; j < point_num; j++) {
		k = cluster[i];
		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy
				+ pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + point_energy;
	}

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

double *cluster_initialize_1(int dim_num, int point_num, int cluster_num,
	double point[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//    CLUSTER_INITIALIZE_1 initializes the clusters to data points.
	//
	//  Discussion:
	//
	//    The cluster centers are simply chosen to be the first data points.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Output, double CLUSTER_INITIALIZE_1[DIM_NUM*CLUSTER_NUM],
	//    the coordinates of the cluster centers.
	//
{
	double *cluster_center;
	int i;
	int j;

	cluster_center = new double[dim_num*cluster_num];

	for (j = 0; j < cluster_num; j++) {
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + j*dim_num] = point[i + j*dim_num];
		}
	}

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

double *cluster_initialize_2(int dim_num, int point_num, int cluster_num,
	double point[], int *seed)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//    CLUSTER_INITIALIZE_2 initializes the cluster centers to random values.
	//
	//  Discussion:
	//
	//    In this case, the hyperbox containing the data is computed.
	//
	//    Then the cluster centers are chosen uniformly at random within
	//    this hyperbox.
	//
	//    Of course, if the data is not smoothly distributed throughout
	//    the box, many cluster centers will be isolated.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input/output, int *SEED, a seed for the random 
	//    number generator.
	//
	//    Output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], 
	//    the coordinates of the cluster centers.
	//
{
	double *cluster_center;
	int i;
	int j;
	double *r;
	double *r_max;
	double *r_min;

	cluster_center = new double[dim_num*cluster_num];

	r = new double[dim_num];
	r_min = new double[dim_num];
	r_max = new double[dim_num];

	j = 0;
	for (i = 0; i < dim_num; i++) {
		r_max[i] = point[i + j*dim_num];
		r_min[i] = point[i + j*dim_num];
	}

	for (j = 1; j < point_num; j++) {
		for (i = 0; i < dim_num; i++) {
			r_max[i] = r8_max(r_max[i], point[i + j*dim_num]);
			r_min[i] = r8_min(r_min[i], point[i + j*dim_num]);
		}
	}

	for (j = 0; j < cluster_num; j++) {
		r8vec_uniform_01(dim_num, seed, r);
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + j*dim_num] = (1.0 - r[i]) * r_min[i] + r[i] * r_max[i];
		}
	}
	delete[] r;
	delete[] r_max;
	delete[] r_min;

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

double *cluster_initialize_3(int dim_num, int point_num, int cluster_num,
	double point[], int *seed)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   CLUSTER_INITIALIZE_3 initializes the cluster centers to random values.
	//
	//  Discussion:
	//
	//    In this case, each point is randomly assigned to a cluster, and
	//    the cluster centers are then computed as the centroids of the points 
	//    in the cluster.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input/output, int SEED, a seed for the random 
	//    number generator.
	//
	//    Output, double CLUSTER_INITIALIZE_3[DIM_NUM*CLUSTER_NUM], 
	//    the coordinates of the cluster centers.
	//
{
	double *cluster_center;
	int *cluster_population;
	int i;
	int j;
	int k;
	//
	//  Assign one point to each cluster center.
	//
	cluster_center = new double[dim_num*cluster_num];

	for (k = 0; k < cluster_num; k++) {
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = point[i + k*dim_num];
		}
	}

	cluster_population = new int[cluster_num];

	for (k = 0; k < cluster_num; k++) {
		cluster_population[k] = 1;
	}
	//
	//  The rest of the points get assigned randomly.
	//
	for (j = cluster_num; j < point_num; j++) {
		k = i4_uniform(1, cluster_num, seed);
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ point[i + j*dim_num];
		}
		cluster_population[k] = cluster_population[k] + 1;
	}
	//
	//  Now average the points to get the centroid.
	//
	for (k = 0; k < cluster_num; k++) {
		if (cluster_population[k] != 0) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num] /
					(double)(cluster_population[k]);
			}
		}
	}

	delete[] cluster_population;

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

double *cluster_initialize_4(int dim_num, int point_num, int cluster_num,
	double point[], int *seed)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   CLUSTER_INITIALIZE_4 initializes the cluster centers to random values.
	//
	//  Discussion:
	//
	//    In this case, each data point is divided randomly among the
	//    the cluster centers.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input/output, int SEED, a seed for the random 
	//    number generator.
	//
	//    Output, double CLUSTER_INITIALIZE_4[DIM_NUM*CLUSTER_NUM], 
	//    the coordinates of the cluster centers.
	//
{
	double *cluster_center;
	double *cluster_factor;
	double *cluster_weight;
	double divisor;
	int i;
	int j;
	int k;

	cluster_center = r8vec_zero_new(dim_num * cluster_num);

	cluster_factor = new double[cluster_num];

	cluster_weight = r8vec_zero_new(cluster_num);

	for (j = 0; j < point_num; j++) {
		r8vec_uniform_01(cluster_num, seed, cluster_factor);

		divisor = r8vec_sum(cluster_num, cluster_factor);

		for (k = 0; k < cluster_num; k++) {
			cluster_factor[k] = cluster_factor[k] / divisor;
		}

		for (k = 0; k < cluster_num; k++) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
					+ cluster_factor[k] * point[i + j*dim_num];
			}
		}

		for (k = 0; k < cluster_num; k++) {
			cluster_weight[k] = cluster_weight[k] + cluster_factor[k];
		}
	}
	//
	//  Now normalize,  so that each cluster center is now a convex 
	//  combination of the points.
	//
	for (k = 0; k < cluster_num; k++) {
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				/ cluster_weight[k];
		}
	}

	delete[] cluster_factor;
	delete[] cluster_weight;

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

double *cluster_initialize_5(int dim_num, int point_num, int cluster_num,
	double point[], int *seed)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   CLUSTER_INITIALIZE_5 initializes the cluster centers to random values.
	//
	//  Discussion:
	//
	//    In this case, each cluster center is a random convex combination 
	//    of the data points.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input/output, int *SEED, a seed for the random 
	//    number generator.
	//
	//    Output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the coordinates of the cluster centers.
	//
{
	double *cluster_center;
	double column_sum;
	double *factor;
	int j;
	int k;
	//
	//  Get a PxC block of random factors.
	//
	factor = r8mat_uniform_01_new(point_num, cluster_num, seed);
	//
	//  Make each column of factors have unit sum.
	//
	for (k = 0; k < cluster_num; k++) {
		column_sum = 0.0;
		for (j = 0; j < point_num; j++) {
			column_sum = column_sum + factor[j + k*point_num];
		}
		for (j = 0; j < point_num; j++) {
			factor[j + k*point_num] = factor[j + k*point_num] / column_sum;
		}
	}
	//
	//  Set centers = points * factors.
	//
	cluster_center = r8mat_mm_new(dim_num, point_num, cluster_num, point,
		factor);

	delete[] factor;

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

void cluster_print_summary(int point_num, int cluster_num,
	int cluster_population[], double cluster_energy[], double cluster_variance[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   CLUSTER_PRINT_SUMMARY prints a summary of data about a clustering.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int CLUSTER_POPULATION[CLUSTER_NUM], the number of
	//    points assigned to each cluster.
	//
	//    Input, double CLUSTER_ENERGY[CLUSTER_NUM], the energy of 
	//    the clusters.
	//
	//    Input, double CLUSTER_VARIANCE[CLUSTER_NUM], the variance of 
	//    the clusters.
	//
{
	double ce;
	int cep;
	double ce_total;
	int cp;
	int cpp;
	double cv;
	int k;

	ce_total = r8vec_sum(cluster_num, cluster_energy);

	cout << "\n";
	cout << "  Clustering statistics:\n";
	cout << "\n";
	cout << "    Number of clusters is " << cluster_num << "\n";
	cout << "    Number of points is   " << point_num << "\n";
	cout << "    Total energy is       " << ce_total << "\n";
	cout << "\n";
	cout << "    Cluster   Population        Energy          Variance\n";
	cout << "    -------  -----------  -----------------  --------------\n";
	cout << "                  #    %     value        %\n";
	cout << "\n";

	for (k = 0; k < cluster_num; k++) {
		cp = cluster_population[k];
		cpp = (int)((double)(100 * cp) / (double)(point_num));
		ce = cluster_energy[k];
		cep = (int)((ce * 100.0) / ce_total);
		cv = cluster_variance[k];
		cout << "  " << setw(7) << k
			<< "  " << setw(8) << cp
			<< "  " << setw(3) << cpp
			<< "  " << setw(12) << ce
			<< "  " << setw(3) << cep
			<< "  " << setw(12) << cv << "\n";
	}

	cp = i4vec_sum(cluster_num, cluster_population);
	cpp = 100;
	ce = r8vec_sum(cluster_num, cluster_energy);
	cep = 100;
	cv = r8vec_i4vec_dot_product(cluster_num, cluster_variance,
		cluster_population) / cp;

	cout << "\n";
	cout << "  " << "  Total"
		<< "  " << setw(8) << cp
		<< "  " << setw(3) << cpp
		<< "  " << setw(12) << ce
		<< "  " << setw(3) << cep
		<< "  " << setw(12) << cv << "\n";

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

double *cluster_variance_compute(int dim_num, int point_num, int cluster_num,
	double point[], int cluster[], double cluster_center[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   CLUSTER_VARIANCE_COMPUTE computes the variance of the clusters.
	//
	//  Discussion:
	//
	//    The cluster variance (from the cluster center) is the average of the 
	//    sum of the squares of the distances of each point in the cluster to the 
	//    cluster center.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    06 October 2009
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of data points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the data points.
	//
	//    Input, int CLUSTER[POINT_NUM], the cluster to which each
	//    data point belongs.
	//
	//    Input, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], the 
	//    centers associated with the minimal energy clustering.
	//
	//    Output, double CLUSTER_VARIANCE_COMPUTE[CLUSTER_NUM], the variance
	//    associated with each cluster.
	//
{
	int *cluster_population;
	double *cluster_variance;
	int i;
	int j;
	int k;
	double point_variance;

	cluster_population = i4vec_zero_new(cluster_num);
	cluster_variance = r8vec_zero_new(cluster_num);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_variance = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_variance = point_variance +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_variance[k] = cluster_variance[k] + point_variance;
		cluster_population[k] = cluster_population[k] + 1;
	}

	for (k = 0; k < cluster_num; k++) {
		if (0 < cluster_population[k]) {
			cluster_variance[k] = cluster_variance[k] / cluster_population[k];
		}
	}

	delete[] cluster_population;

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

int file_column_count(string filename)

//****************************************************************************80
//
//  Purpose:
//
//    FILE_COLUMN_COUNT counts the columns in the first line of a file.
//
//  Discussion:
//
//    The file is assumed to be a simple text file.
//
//    Most lines of the file are presumed to consist of COLUMN_NUM words,
//    separated by spaces.  There may also be some blank lines, and some
//    comment lines, which have a "#" in column 1.
//
//    The routine tries to find the first non-comment non-blank line and
//    counts the number of words in that line.
//
//    If all lines are blanks or comments, it goes back and tries to analyze
//    a comment line.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string FILENAME, the name of the file.
//
//    Output, int FILE_COLUMN_COUNT, the number of columns assumed
//    to be in the file.
//
{
	int column_num;
	ifstream input;
	bool got_one;
	string text;
	//
	//  Open the file.
	//
	input.open(filename.c_str());

	if (!input) {
		column_num = -1;
		cerr << "\n";
		cerr << "FILE_COLUMN_COUNT - Fatal error!\n";
		cerr << "  Could not open the file:\n";
		cerr << "  \"" << filename << "\"\n";
		exit(1);
	}
	//
	//  Read one line, but skip blank lines and comment lines.
	//
	got_one = false;

	for (; ; ) {
		getline(input, text);

		if (input.eof()) {
			break;
		}

		if (s_len_trim(text) <= 0) {
			continue;
		}

		if (text[0] == '#') {
			continue;
		}
		got_one = true;
		break;
	}

	if (!got_one) {
		input.close();

		input.open(filename.c_str());

		for (; ; ) {
			input >> text;

			if (input.eof()) {
				break;
			}

			if (s_len_trim(text) == 0) {
				continue;
			}
			got_one = true;
			break;
		}
	}

	input.close();

	if (!got_one) {
		cerr << "\n";
		cerr << "FILE_COLUMN_COUNT - Warning!\n";
		cerr << "  The file does not seem to contain any data.\n";
		return -1;
	}

	column_num = s_word_count(text);

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

int file_row_count(string input_filename)

//****************************************************************************80
//
//  Purpose:
//
//    FILE_ROW_COUNT counts the number of row records in a file.
//
//  Discussion:
//
//    It does not count lines that are blank, or that begin with a
//    comment symbol '#'.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int FILE_ROW_COUNT, the number of rows found.
//
{
	int bad_num;
	int comment_num;
	ifstream input;
	string line;
	int record_num;
	int row_num;

	row_num = 0;
	comment_num = 0;
	record_num = 0;
	bad_num = 0;

	input.open(input_filename.c_str());

	if (!input) {
		cerr << "\n";
		cerr << "FILE_ROW_COUNT - Fatal error!\n";
		cerr << "  Could not open the input file: \"" << input_filename << "\"\n";
		exit(1);
	}

	for (; ; ) {
		getline(input, line);

		if (input.eof()) {
			break;
		}

		record_num = record_num + 1;

		if (line[0] == '#') {
			comment_num = comment_num + 1;
			continue;
		}

		if (s_len_trim(line) == 0) {
			comment_num = comment_num + 1;
			continue;
		}

		row_num = row_num + 1;

	}

	input.close();

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

void hmeans_01(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], int cluster[], double cluster_center[],
	int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   HMEANS_01 applies the H-Means algorithm.
	//
	//  Discussion:
	//
	//    The data for the H-Means problem is a set of N points X in
	//    M-dimensions, and a desired number of clusters K.
	//
	//    The goal is to determine K points Z, called cluster centers, so that
	//    if we associate each point X with its nearest Z value, we minimize
	//    the standard deviation or cluster energy.  Writing CLUSTER(I) to
	//    indicate the index of the nearest cluster center to point X(I), the
	//    energy can be written as:
	//
	//      Energy = Sum ( 1 <= I <= N ) || X(I) - Z(CLUSTER(I)) ||^2
	//
	//    where
	//
	//      || X - Z ||^2 = Sum ( 1 <= J <= M ) ( X(J) - Z(J) )^2
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Reference:
	//
	//    Wendy Martinez, Angel Martinez,
	//    Computational Statistics Handbook with MATLAB,
	//    pages 373-376,
	//    Chapman and Hall / CRC, 2002.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of data points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int &IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the data points.
	//
	//    Input/output, int CLUSTER[POINT_NUM].  On input, the user 
	//    may specify an initial cluster for each point, or leave all entrie of
	//    CLUSTER set to 0.  On output, CLUSTER contains the index of the
	//    cluster to which each data point belongs.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], the 
	//    centers associated with the minimal energy clustering.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM],
	//    the populuation of each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the energy
	//    associated with each cluster.
	//
{
	double *centroid;
	bool debug = true;
	int i;
	int j;
	int k;
	int k2;
	int missed;
	double point_energy;
	double point_energy_min;
	int swap;
	//
	//  Data checks.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "HMEANS_01 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "HMEANS_01 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "HMEANS_01 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "HMEANS_01 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}
	//
	//  On input, legal entries in CLUSTER are preserved, but
	//  otherwise, each point is assigned to its nearest cluster.
	//
	for (j = 0; j < point_num; j++) {
		if (cluster[j] < 0 || cluster_num <= cluster[j]) {
			point_energy_min = r8_huge();

			for (k = 0; k < cluster_num; k++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
				}

				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k;
				}
			}
		}
	}
	it_num = 0;

	while (it_num < it_max) {
		it_num = it_num + 1;
		//
		//  #1:
		//  Assign each point to the cluster of its nearest center.
		//
		swap = 0;

		for (j = 0; j < point_num; j++) {
			point_energy_min = r8_huge();
			k = cluster[j];

			for (k2 = 0; k2 < cluster_num; k2++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k2*dim_num], 2);
				}

				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k2;
				}
			}

			if (k != cluster[j]) {
				swap = swap + 1;
			}
		}
		//
		//  Terminate if no points were swapped.
		//
		if (1 < it_num) {
			if (swap == 0) {
				break;
			}
		}
		//
		//  #2:
		//  Determine the total energy of the new clustering with current centroids.
		//
		r8vec_zero(cluster_num, cluster_energy);

		for (j = 0; j < point_num; j++) {
			k = cluster[j];

			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}
			cluster_energy[k] = cluster_energy[k] + point_energy;
		}
		//
		//  #3:
		//  Determine the centroids of the clusters.
		//
		centroid = r8vec_zero_new(dim_num * cluster_num);
		i4vec_zero(cluster_num, cluster_population);

		for (j = 0; j < point_num; j++) {
			k = cluster[j];
			cluster_population[k] = cluster_population[k] + 1;
			for (i = 0; i < dim_num; i++) {
				centroid[i + k*dim_num] = centroid[i + k*dim_num] + point[i + j*dim_num];
			}
		}
		//
		//  Now divide by the population to get the centroid.
		//  But if a center has no population, pick a point at random.
		//
		missed = 0;

		for (k = 0; k < cluster_num; k++) {
			if (cluster_population[k] != 0) {
				for (i = 0; i < dim_num; i++) {
					centroid[i + k*dim_num] = centroid[i + k*dim_num]
						/ (double)(cluster_population[k]);
				}
			} else {
				for (i = 0; i < dim_num; i++) {
					centroid[i + k*dim_num] = point[i + missed*dim_num];
				}
				missed = missed + 1;
			}
		}

		for (k = 0; k < cluster_num; k++) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = centroid[i + k*dim_num];
			}
		}

		delete[] centroid;
		//
		//  #4:
		//  Determine the total energy of the current clustering with new centroids.
		//
		r8vec_zero(cluster_num, cluster_energy);

		for (j = 0; j < point_num; j++) {
			k = cluster[j];

			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}
			cluster_energy[k] = cluster_energy[k] + point_energy;
		}
	}
	return;
}
//****************************************************************************80

void hmeans_02(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], int cluster[], double cluster_center[],
	int cluster_population[], double cluster_energy[], int *seed)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   HMEANS_02 applies the H-Means algorithm.
	//
	//  Discussion:
	//
	//    This is a simple routine to group a set of points into K clusters,
	//    each with a center point, in such a way that the total cluster 
	//    energy is minimized.  The total cluster energy is the sum of the
	//    squares of the distances of each point to the center of its cluster.
	//
	//    The algorithm begins with an initial estimate for the cluster centers:
	//
	//    1. The points are assigned to the nearest cluster centers.
	//
	//    2. The iteration exit ( 1 );s if the total energy has not changed 
	//        significantly, or we have reached the maximum number of iterations.
	//
	//    3. Each cluster center is replaced by the centroid of the points
	//       in the cluster.
	//
	//    4. Return to step 1.
	//
	//    The algorithm may fail to find the best solution.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input/output, int CLUSTER[POINT_NUM].  On input, the user 
	//    may specify an initial cluster for each point, or leave all entrie of
	//    CLUSTER set to 0.  On output, CLUSTER contains the index of the
	//    cluster to which each data point belongs.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the coordinates of the cluster centers.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number of
	//    points assigned to each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the energy of 
	//    the clusters.
	//
	//    Input/output, int *SEED, a seed for the random
	//    number generator.
	//
{
	bool debug = false;
	int i;
	int j;
	int k;
	int k2;
	double point_energy;
	double point_energy_min;
	int swap;
	//
	//  Data checks.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "HMEANS_02 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "HMEANS_02 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "HMEANS_02 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "HMEANS_02 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}
	//
	//  On input, legal entries in CLUSTER are preserved, but
	//  otherwise, each point is assigned to its nearest cluster.
	//
	for (j = 0; j < point_num; j++) {
		if (cluster[j] < 0 || cluster_num <= cluster[j]) {
			point_energy_min = r8_huge();
			for (k = 0; k < cluster_num; k++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
				}
				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k;
				}
			}
		}
	}

	it_num = 0;

	for (; ; ) {
		//
		//  Given centers, assign points to nearest center.
		//
		i4vec_zero(cluster_num, cluster_population);
		r8vec_zero(cluster_num, cluster_energy);

		swap = 0;

		for (j = 0; j < point_num; j++) {
			point_energy_min = r8_huge();
			k = cluster[j];

			for (k2 = 0; k2 < cluster_num; k2++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k2*dim_num], 2);
				}

				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k2;
				}
			}
			if (k != cluster[j]) {
				swap = swap + 1;
			}
			k = cluster[j];
			cluster_energy[k] = cluster_energy[k] + point_energy_min;
			cluster_population[k] = cluster_population[k] + 1;
		}

		if (debug) {
			cout << "  " << setw(3) << it_num
				<< "  " << setw(14) << r8vec_sum(cluster_num, cluster_energy) << "\n";
		}

		if (0 < it_num) {
			if (swap == 0) {
				break;
			}
		}

		if (it_max <= it_num) {
			break;
		}

		it_num = it_num + 1;
		//
		//  Given points in cluster, replace center by centroid.
		//
		r8vec_zero(dim_num * cluster_num, cluster_center);

		for (j = 0; j < point_num; j++) {
			k = cluster[j];
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
					+ point[i + j*dim_num];
			}
		}

		for (k = 0; k < cluster_num; k++) {
			if (cluster_population[k] != 0) {
				for (i = 0; i < dim_num; i++) {
					cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num] /
						(double)(cluster_population[k]);
				}
			} else {
				j = i4_uniform(0, point_num - 1, seed);
				for (i = 0; i < dim_num; i++) {
					cluster_center[i + k*dim_num] = point[i + j*dim_num];
				}
			}
		}
	}
	//
	//  Compute the energy based on the final value of the cluster centers.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + point_energy;
	}
	return;
}
//****************************************************************************80

void hmeans_w_01(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], double weight[], int cluster[],
	double cluster_center[], int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   HMEANS_W_01 applies the weighted H-Means algorithm. 
	//
	//  Discussion:
	//
	//    The input data for the weight H-Means problem includes:
	//    * a set of N data points X in M dimensions, 
	//    * a set of N nonnegative weights W,
	//    * a desired number of clusters K.
	//    * an initial set of cluster centers Z,
	//    * an (optional) initial set of cluster assignments.
	//
	//    The goal is to determine K points Z, called cluster centers, and
	//    to assign each point X(I) to some cluster Z(J), so that we minimize
	//    the weighted standard deviation of the distance of each data point
	//    to the center of its cluster.  Writing J = CLUSTER(I) to
	//    indicate the index of the nearest cluster center Z(J) to the 
	//    point X(I), the quantity we are trying to minimize is the sum
	//    of the weighted cluster energies E(J), where:
	//
	//      E(J) = Sum ( 1 <= I <= N ) W(I) * || X(I) - Z(J) ||^2
	//
	//    Here, we assume that we are using the Euclidean norm, so that
	//    
	//      || X(I) - Z(J) ||^2 = Sum ( 1 <= K <= M )
	//        ( X(I)(K) - Z(J)(K) )^2
	//
	//    In this notation, X(I)(K) is the K-th spatial component of the
	//    I-th data point.
	//
	//    Note that this routine should give the same results as HMEANS_01
	//    in any case in which all the entries of the WEIGHT vector are equal.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    08 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Reference:
	//
	//    Wendy Martinez, Angel Martinez,
	//    Computational Statistics Handbook with MATLAB,
	//    pages 373-376,
	//    Chapman and Hall / CRC, 2002.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of data points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int &IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the data points.
	//
	//    Input, double WEIGHT[POINT_NUM], the weights
	//    assigned to the data points.  These must be nonnegative, and
	//    at least one must be strictly positive.
	//
	//    Input/output, int CLUSTER[POINT_NUM].  On input, the user 
	//    may specify an initial cluster for each point, or leave all entrie of
	//    CLUSTER set to 0.  On output, CLUSTER contains the index of the
	//    cluster to which each data point belongs.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], the
	//    centers associated with the minimal energy clustering.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the energy
	//    associated with each cluster.
	//
{
	double *centroid;
	double *cluster_weight;
	bool debug = true;
	double energy;
	int i;
	int j;
	int k;
	int k2;
	int missed;
	double point_energy;
	double point_energy_min;
	int swap;
	//
	//  Data checks.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "HMEANS_W_01 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "HMEANS_W_01 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "HMEANS_W_01 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "HMEANS_W_01 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}

	if (r8vec_any_negative(point_num, weight)) {
		cout << "\n";
		cout << "HMEANS_W_01 - Fatal error!\n";
		cout << "  Some weight entry is negative.\n";
		exit(1);
	}

	if (r8vec_all_nonpositive(point_num, weight)) {
		cout << "\n";
		cout << "HMEANS_W_01 - Fatal error!\n";
		cout << "  No weight entry is positive.\n";
		exit(1);
	}
	//
	//  On input, legal entries in CLUSTER are preserved, but
	//  otherwise, each point is assigned to its nearest cluster.
	//
	for (j = 0; j < point_num; j++) {
		if (cluster[j] < 0 || cluster_num <= cluster[j]) {
			point_energy_min = r8_huge();

			for (k = 0; k < cluster_num; k++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
				}

				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k;
				}
			}
		}
	}
	it_num = 0;

	while (it_num < it_max) {
		it_num = it_num + 1;
		//
		//  #1:
		//  Reassign points to clusters:
		//  Assign each point to the cluster whose center is nearest;
		//  Count the number of points whose cluster assignment is changed.
		//
		swap = 0;

		for (j = 0; j < point_num; j++) {
			point_energy_min = r8_huge();
			k = cluster[j];

			for (k2 = 0; k2 < cluster_num; k2++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k2*dim_num], 2);
				}

				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k2;
				}
			}

			if (k != cluster[j]) {
				swap = swap + 1;
			}
		}
		//
		//  If no point changed its cluster assignment, the algorithm can make no 
		//  more improvements, so terminate.
		//
		if (1 < it_num) {
			if (swap == 0) {
				break;
			}
		}
		//
		//  Determine the current energy.
		//
		energy = 0.0;
		for (j = 0; j < point_num; j++) {
			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k2*dim_num], 2);
			}
			energy = energy + weight[j] * point_energy;
		}
		cout << "  " << setw(4) << it_num
			<< "  " << setw(14) << energy << "\n";
		//
		//  #2:
		//  Determine the centroids of the clusters, and set the 
		//  cluster center to the cluster centroid.
		//
		centroid = r8vec_zero_new(dim_num * cluster_num);
		cluster_weight = r8vec_zero_new(cluster_num);

		for (j = 0; j < point_num; j++) {
			k = cluster[j];
			cluster_population[k] = cluster_population[k] + 1;
			cluster_weight[k] = cluster_weight[k] + weight[j];
			for (i = 0; i < dim_num; i++) {
				centroid[i + k*dim_num] = centroid[i + k*dim_num]
					+ weight[j] * point[i + j*dim_num];
			}
		}

		missed = 0;

		for (k = 0; k < cluster_num; k++) {
			if (cluster_weight[k] != 0.0) {
				for (i = 0; i < dim_num; i++) {
					centroid[i + k*dim_num] = centroid[i + k*dim_num] / cluster_weight[k];
				}
			} else {
				for (i = 0; i < dim_num; i++) {
					centroid[i + k*dim_num] = point[i + missed*dim_num];
				}
				missed = missed + 1;
			}
		}

		for (k = 0; k < cluster_num; k++) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = centroid[i + k*dim_num];
			}
		}

		delete[] centroid;
		delete[] cluster_weight;
	}
	//
	//  Compute the energy based on the final value of the cluster centers.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + weight[j] * point_energy;
	}
	return;
}
//****************************************************************************80

void hmeans_w_02(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], double weight[], int cluster[],
	double cluster_center[], int cluster_population[], double cluster_energy[],
	int *seed)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   HMEANS_W_02 applies the weighted H-Means algorithm.
	//
	//  Discussion:
	//
	//    The input data for the weight H-Means problem includes:
	//    * a set of N data points X in M dimensions, 
	//    * a set of N nonnegative weights W,
	//    * a desired number of clusters K.
	//    * an initial set of cluster centers Z,
	//    * an (optional) initial set of cluster assignments.
	//
	//    The goal is to determine K points Z, called cluster centers, and
	//    to assign each point X(I) to some cluster Z(J), so that we minimize
	//    the weighted standard deviation of the distance of each data point
	//    to the center of its cluster.  Writing J = CLUSTER(I) to
	//    indicate the index of the nearest cluster center Z(J) to the 
	//    point X(I), the quantity we are trying to minimize is the sum
	//    of the weighted cluster energies E(J), where:
	//
	//      E(J) = Sum ( 1 <= I <= N ) W(I) * || X(I) - Z(J) ||^2
	//
	//    Here, we assume that we are using the Euclidean norm, so that
	//    
	//      || X(I) - Z(J) ||^2 = Sum ( 1 <= K <= M )
	//        ( X(I)(K) - Z(J)(K) )^2
	//
	//    In this notation, X(I)(K) is the K-th spatial component of the
	//    I-th data point.
	//
	//    Note that this routine should give the same results as HMEANS_02
	//    in any case in which all the entries of the WEIGHT vector are equal.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    09 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int &IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input, double WEIGHT[POINT_NUM], the weights
	//    assigned to the data points.  These must be nonnegative, and
	//    at least one must be strictly positive.
	//
	//    Input/output, int CLUSTER[POINT_NUM].  On input, the user 
	//    may specify an initial cluster for each point, or leave all entrie of
	//    CLUSTER set to 0.  On output, CLUSTER contains the index of the
	//    cluster to which each data point belongs.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the coordinates of the cluster centers.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number of
	//    points assigned to each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the energy of 
	//    the clusters.
	//
	//    Input/output, int *SEED, a seed for the random 
	//    number generator.
	//
{
	double *cluster_weight;
	int i;
	int j;
	int k;
	int k2;
	double point_energy;
	double point_energy_min;
	int swap;
	//
	//  Data checks.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "HMEANS_W_02 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "HMEANS_W_02 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "HMEANS_W_02 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "HMEANS_W_02 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}

	if (r8vec_any_negative(point_num, weight)) {
		cout << "\n";
		cout << "HMEANS_W_02 - Fatal error!\n";
		cout << "  Some weight entry is negative.\n";
		exit(1);
	}

	if (r8vec_all_nonpositive(point_num, weight)) {
		cout << "\n";
		cout << "HMEANS_W_02 - Fatal error!\n";
		cout << "  No weight entry is positive.\n";
		exit(1);
	}
	//
	//  On input, legal entries in CLUSTER are preserved, but
	//  otherwise, each point is assigned to its nearest cluster.
	//
	for (j = 0; j < point_num; j++) {
		if (cluster[j] < 0 || cluster_num <= cluster[j]) {
			point_energy_min = r8_huge();
			for (k = 0; k < cluster_num; k++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
				}
				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k;
				}
			}
		}
	}
	it_num = 0;

	for (; ; ) {
		//
		//  Given centers, assign points to nearest center.
		//
		i4vec_zero(cluster_num, cluster_population);
		r8vec_zero(cluster_num, cluster_energy);
		cluster_weight = r8vec_zero_new(cluster_num);

		swap = 0;

		for (j = 0; j < point_num; j++) {
			point_energy_min = r8_huge();
			k = cluster[j];

			for (k2 = 0; k2 < cluster_num; k2++) {
				point_energy = 0.0;
				for (i = 0; i < dim_num; i++) {
					point_energy = point_energy +
						pow(point[i + j*dim_num] - cluster_center[i + k2*dim_num], 2);
				}

				if (point_energy < point_energy_min) {
					point_energy_min = point_energy;
					cluster[j] = k2;
				}
			}

			if (k != cluster[j]) {
				swap = swap + 1;
			}
			k = cluster[j];
			cluster_energy[k] = cluster_energy[k] + weight[j] * point_energy_min;
			cluster_population[k] = cluster_population[k] + 1;
			cluster_weight[k] = cluster_weight[k] + weight[j];
		}

		if (0 < it_num) {
			if (swap == 0) {
				break;
			}
		}

		if (it_max <= it_num) {
			break;
		}

		it_num = it_num + 1;
		//
		//  Given points in cluster, replace center by weighted centroid.
		//
		r8vec_zero(dim_num * cluster_num, cluster_center);

		for (j = 0; j < point_num; j++) {
			k = cluster[j];
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
					+ weight[j] * point[i + j*dim_num];
			}
		}

		for (k = 0; k < cluster_num; k++) {
			if (cluster_weight[k] != 0.0) {
				for (i = 0; i < dim_num; i++) {
					cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num] /
						cluster_weight[k];
				}
			} else {
				j = i4_uniform(0, point_num - 1, seed);
				for (i = 0; i < dim_num; i++) {
					cluster_center[i + k*dim_num] = point[i + j*dim_num];
				}
			}
		}
		delete[] cluster_weight;
	}
	//
	//  Compute the energy based on the final value of the cluster centers.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + weight[j] * point_energy;
	}

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

int i4_max(int i1, int i2)

//****************************************************************************80
//
//  Purpose:
//
//    I4_MAX returns the maximum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, are two integers to be compared.
//
//    Output, int I4_MAX, the larger of I1 and I2.
//
{
	int value;

	if (i2 < i1) {
		value = i1;
	} else {
		value = i2;
	}
	return value;
}
//****************************************************************************80

int i4_min(int i1, int i2)

//****************************************************************************80
//
//  Purpose:
//
//    I4_MIN returns the minimum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, two integers to be compared.
//
//    Output, int I4_MIN, the smaller of I1 and I2.
//
{
	int value;

	if (i1 < i2) {
		value = i1;
	} else {
		value = i2;
	}
	return value;
}
//****************************************************************************80

int i4_uniform(int a, int b, int *seed)

//****************************************************************************80
//
//  Purpose:
//
//    I4_UNIFORM returns a scaled pseudorandom I4.
//
//  Discussion:
//
//    The pseudorandom number should be uniformly distributed
//    between A and B.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    12 November 2006
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Springer Verlag, pages 201-202, 1983.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley Interscience, page 95, 1998.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, pages 362-376, 1986.
//
//    Peter Lewis, Allen Goodman, James Miller
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, pages 136-143, 1969.
//
//  Parameters:
//
//    Input, int A, B, the limits of the interval.
//
//    Input/output, int *SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, int I4_UNIFORM, a number between A and B.
//
{
	int k;
	float r;
	int value;

	if (*seed == 0) {
		cerr << "\n";
		cerr << "I4_UNIFORM - Fatal error!\n";
		cerr << "  Input value of SEED = 0.\n";
		exit(1);
	}

	k = *seed / 127773;

	*seed = 16807 * (*seed - k * 127773) - k * 2836;

	if (*seed < 0) {
		*seed = *seed + 2147483647;
	}

	r = (float)(*seed) * 4.656612875E-10;
	//
	//  Scale R to lie between A-0.5 and B+0.5.
	//
	r = (1.0 - r) * ((float)(i4_min(a, b)) - 0.5)
		+ r   * ((float)(i4_max(a, b)) + 0.5);
	//
	//  Use rounding to convert R to an integer between A and B.
	//
	value = r4_nint(r);

	value = i4_max(value, i4_min(a, b));
	value = i4_min(value, i4_max(a, b));

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

void i4mat_write(string output_filename, int m, int n, int table[])

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_WRITE writes an I4MAT file with no header.
//
//  Discussion:
//
//    An I4MAT is an array of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, int TABLE[M*N], the data.
//
{
	int i;
	int j;
	ofstream output;
	//
	//  Open the file.
	//
	output.open(output_filename.c_str());

	if (!output) {
		cerr << "\n";
		cerr << "I4MAT_WRITE - Fatal error!\n";
		cerr << "  Could not open the output file.\n";
		exit(1);
	}
	//
	//  Write the data.
	//
	for (j = 0; j < n; j++) {
		for (i = 0; i < m; i++) {
			output << "  " << setw(10) << table[i + j*m];
		}
		output << "\n";
	}
	//
	//  Close the file.
	//
	output.close();

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

void i4vec_negone(int n, int a[])

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_NEGONE sets an I4VEC to -1.
//
//  Discussion:
//
//    An I4VEC is a vector of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    12 October 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, int A[N], a vector of -1's.
//
{
	int i;

	for (i = 0; i < n; i++) {
		a[i] = -1;
	}
	return;
}
//****************************************************************************80

int *i4vec_negone_new(int n)

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_NEGONE_NEW creates an I4VEC and sets it to -1.
//
//  Discussion:
//
//    An I4VEC is a vector of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    12 October 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, int I4VEC_NEGONE_NEW[N], a vector of -1's.
//
{
	int *a;
	int i;

	a = new int[n];

	for (i = 0; i < n; i++) {
		a[i] = -1;
	}
	return a;
}
//****************************************************************************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

void i4vec_zero(int n, int a[])

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_ZERO zeroes an I4VEC.
//
//  Discussion:
//
//    An I4VEC is a vector of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 August 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, int A[N], a vector of zeroes.
//
{
	int i;

	for (i = 0; i < n; i++) {
		a[i] = 0;
	}
	return;
}
//****************************************************************************80

int *i4vec_zero_new(int n)

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_ZERO_NEW creates and zeroes an I4VEC.
//
//  Discussion:
//
//    An I4VEC is a vector of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    11 July 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, int I4VEC_ZERO_NEW[N], a vector of zeroes.
//
{
	int *a;
	int i;

	a = new int[n];

	for (i = 0; i < n; i++) {
		a[i] = 0;
	}
	return a;
}
//****************************************************************************80

void kmeans_01(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], int cluster[], double cluster_center[],
	int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_01 applies the K-Means algorithm.
	//
	//  Discussion:
	//
	//    Given a matrix of POINT_NUM observations on DIM_NUM variables, the
	//    observations are to be allocated to CLUSTER_NUM clusters in such 
	//    a way that the within-cluster sum of squares is minimized.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    09 October 2011
	//
	//  Author:
	//
	//    Original FORTRAN77 version by David Sparks.
	//    C++ version by John Burkardt.
	//
	//  Reference:
	//
	//    David Sparks,
	//    Algorithm AS 58: 
	//    Euclidean Cluster Analysis,
	//    Applied Statistics,
	//    Volume 22, Number 1, 1973, pages 126-130.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the points.
	//
	//    Output, int CLUSTER[POINT_NUM], indicates which cluster
	//    each point belongs to.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the cluster centers.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number 
	//    of points in each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the 
	//    cluster energies.
	//
{
	double dc;
	double de;
	double *f;
	int i;
	int il;
	int ir;
	int j;
	int j2;
	int k;
	double point_energy;
	double point_energy_min;
	int swap;

	it_num = 0;
	//
	//  Idiot checks.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "KMEANS_01 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "KMEANS_01 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "KMEANS_01 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}
	//
	//  For each observation, calculate the distance from each cluster
	//  center, and assign to the nearest.
	//
	for (j = 0; j < point_num; j++) {
		point_energy_min = r8_huge();
		cluster[j] = -1;

		for (k = 0; k < cluster_num; k++) {
			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}

			if (point_energy < point_energy_min) {
				point_energy_min = point_energy;
				cluster[j] = k;
			}
		}
	}
	//
	//  Determine the cluster population counts.
	//
	i4vec_zero(cluster_num, cluster_population);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_population[k] = cluster_population[k] + 1;
	}
	//
	//  Calculate the mean and sum of squares for each cluster.
	//
	r8vec_zero(dim_num * cluster_num, cluster_center);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ point[i + j*dim_num];
		}
	}

	for (k = 0; k < cluster_num; k++) {
		if (0 < cluster_population[k]) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num] /
					(double)cluster_population[k];
			}
		}
	}
	//
	//  Set the point energies.
	//
	f = r8vec_zero_new(point_num);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			f[j] = f[j] + pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
	}
	//
	//  Set the cluster energies.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_energy[k] = cluster_energy[k] + f[j];
	}
	//
	//  Adjust the point energies by a weight factor.
	//
	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		if (1 < cluster_population[k]) {
			f[j] = f[j] * (double)(cluster_population[k])
				/ (double)(cluster_population[k] - 1);
		}
	}
	//
	//  Examine each observation in turn to see if it should be
	//  reassigned to a different cluster.
	//
	it_num = 0;

	while (it_num < it_max) {
		it_num = it_num + 1;

		swap = 0;

		for (j = 0; j < point_num; j++) {
			il = cluster[j];
			ir = il;

			if (cluster_population[il] <= 1) {
				continue;
			}

			dc = f[j];

			for (k = 0; k < cluster_num; k++) {
				if (k != il) {
					de = 0.0;
					for (i = 0; i < dim_num; i++) {
						de = de + pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
					}
					de = de * (double)cluster_population[k]
						/ (double)(cluster_population[k] + 1);

					if (de < dc) {
						dc = de;
						ir = k;
					}
				}
			}
			//
			//  If the lowest value was obtained by staying in the current cluster,
			//  then cycle.
			//
			if (ir == il) {
				continue;
			}
			//
			//  Reassign the point from cluster IL to cluster IR.
			//
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + il*dim_num] = (cluster_center[i + il*dim_num]
					* (double)(cluster_population[il]) - point[i + j*dim_num])
					/ (double)(cluster_population[il] - 1);

				cluster_center[i + ir*dim_num] = (cluster_center[i + ir*dim_num]
					* (double)(cluster_population[ir]) + point[i + j*dim_num])
					/ (double)(cluster_population[ir] + 1);
			}
			cluster_energy[il] = cluster_energy[il] - f[j];
			cluster_energy[ir] = cluster_energy[ir] + dc;
			cluster_population[ir] = cluster_population[ir] + 1;
			cluster_population[il] = cluster_population[il] - 1;

			cluster[j] = ir;
			//
			//  Adjust the value of F for points in clusters IL and IR.
			//  
			for (j2 = 0; j2 < point_num; j2++) {
				k = cluster[j2];

				if (k == il || k == ir) {
					f[j2] = 0.0;
					for (i = 0; i < dim_num; i++) {
						f[j2] = f[j2] + pow(point[i + j2*dim_num] - cluster_center[i + k*dim_num], 2);
					}

					if (1 < cluster_population[k]) {
						f[j2] = f[j2] * (double)(cluster_population[k])
							/ ((double)(cluster_population[k] - 1));
					}
				}
			}
			swap = swap + 1;
		}
		//
		//  Exit if no reassignments were made during this iteration.
		//
		if (swap == 0) {
			break;
		}
	}
	//
	//  Compute the cluster energies.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + point_energy;
	}

	delete[] f;

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

void kmeans_02(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], int cluster[], double cluster_center[],
	int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_02 applies the K-Means algorithm.
	//
	//  Discussion:
	//
	//    The routine attempts to divide POINT_NUM points in 
	//    DIM_NUM-dimensional space into CLUSTER_NUM clusters so that the within 
	//    cluster sum of squares is minimized.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    13 October 2011
	//
	//  Author:
	//
	//    Original FORTRAN77 by John Hartigan, Manchek Wong.
	//    C++ version by John Burkardt.
	//
	//  Reference:
	//
	//    John Hartigan, Manchek Wong,
	//    Algorithm AS 136:
	//    A K-Means Clustering Algorithm,
	//    Applied Statistics,
	//    Volume 28, Number 1, 1979, pages 100-108.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int &IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Output, int CLUSTER[POINT_NUM], the cluster each 
	//    point belongs to.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the cluster centers.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number 
	//    of points in each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the 
	//    within-cluster sum of squares.
	//
{
	double *an1;
	double *an2;
	int *cluster2;
	double *d;
	double db;
	double dt[2];
	int i;
	int ifault;
	int il;
	int indx;
	int *itran;
	int j;
	int k;
	int *live;
	int *ncp;
	double point_energy;
	double temp;

	it_num = 0;

	if (cluster_num < 1) {
		cout << "\n";
		cout << "KMEANS_02 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (point_num <= cluster_num) {
		cout << "\n";
		cout << "KMEANS_02 - Fatal error!\n";
		cout << "  POINT_NUM <= CLUSTER_NUM.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "KMEANS_02 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "KMEANS_02 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}
	//
	//  For each point I, find its two closest centers, CLUSTER(I) and CLUSTER2(I).
	//  Assign it to CLUSTER(I).
	//
	cluster2 = new int[point_num];

	for (j = 0; j < point_num; j++) {
		cluster[j] = 0;
		cluster2[j] = 1;

		for (il = 0; il < 2; il++) {
			dt[il] = 0.0;
			for (i = 0; i < dim_num; i++) {
				dt[il] = dt[il] + pow(point[i + j*dim_num] - cluster_center[i + il*dim_num], 2);
			}
		}

		if (dt[1] < dt[0]) {
			cluster[j] = 1;
			cluster2[j] = 0;
			temp = dt[0];
			dt[0] = dt[1];
			dt[1] = temp;
		}

		for (k = 2; k < cluster_num; k++) {
			db = 0.0;
			for (i = 0; i < dim_num; i++) {
				db = db + pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}

			if (db < dt[0]) {
				dt[1] = dt[0];
				cluster2[j] = cluster[j];
				dt[0] = db;
				cluster[j] = k;
			} else if (db < dt[1]) {
				dt[1] = db;
				cluster2[j] = k;
			}
		}
	}
	//
	//  Update cluster centers to be the average of points contained
	//  within them.
	//
	i4vec_zero(cluster_num, cluster_population);
	r8vec_zero(dim_num * cluster_num, cluster_center);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_population[k] = cluster_population[k] + 1;
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ point[i + j*dim_num];
		}
	}
	//
	//  Check to see if there is any empty cluster.
	//
	an1 = new double[cluster_num];
	an2 = new double[cluster_num];
	itran = new int[cluster_num];
	ncp = new int[cluster_num];

	for (k = 0; k < cluster_num; k++) {
		if (cluster_population[k] == 0) {
			cout << "\n";
			cout << "KMEANS_02 - Fatal error!\n";
			cout << "  Cluster " << k << " has zero population.\n";
			exit(1);
		}

		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				/ (double)(cluster_population[k]);
		}
		//
		//  Initialize AN1, AN2, ITRAN and NCP
		//  AN1(K) = CLUSTER_POPULATION(K) / (CLUSTER_POPULATION(K) - 1)
		//  AN2(K) = CLUSTER_POPULATION(K) / (CLUSTER_POPULATION(K) + 1)
		//  ITRAN(K) = 1 if cluster K is updated in the quick-transfer stage,
		//           = 0 otherwise
		//  In the optimal-transfer stage, NCP(K) stores the step at which
		//  cluster K is last updated.
		//  In the quick-transfer stage, NCP(K) stores the step at which
		//  cluster K is last updated plus POINT_NUM.
		//
		an2[k] = (double)(cluster_population[k])
			/ (double)(cluster_population[k] + 1);

		if (1 < cluster_population[k]) {
			an1[k] = (double)(cluster_population[k])
				/ (double)(cluster_population[k] - 1);
		} else {
			an1[k] = r8_huge();
		}
		itran[k] = 1;
		ncp[k] = -1;
	}

	indx = 0;
	ifault = 2;
	it_num = 0;

	d = new double[point_num];
	live = new int[cluster_num];

	while (it_num < it_max) {
		it_num = it_num + 1;
		//
		//  In this stage, there is only one pass through the data.   Each
		//  point is re-allocated, if necessary, to the cluster that will
		//  induce the maximum reduction in within-cluster sum of squares.
		//
		kmeans_02_optra(dim_num, point_num, cluster_num, point,
			cluster_center, cluster, cluster2, cluster_population, an1, an2,
			ncp, d, itran, live, indx);
		//
		//  Stop if no transfer took place in the last POINT_NUM optimal transfer steps.
		//
		if (indx == point_num) {
			ifault = 0;
			break;
		}
		//
		//  Each point is tested in turn to see if it should be re-allocated
		//  to the cluster to which it is most likely to be transferred,
		//  CLUSTER2(I), from its present cluster, CLUSTER(I).   Loop through the
		//  data until no further change is to take place.
		//
		kmeans_02_qtran(dim_num, point_num, cluster_num, point,
			cluster_center, cluster, cluster2, cluster_population, an1, an2,
			ncp, d, itran, indx);
		//
		//  If there are only two clusters, there is no need to re-enter the
		//  optimal transfer stage.
		//
		if (cluster_num == 2) {
			ifault = 0;
			break;
		}
		//
		//  NCP has to be set to 0 before entering OPTRA.
		//
		i4vec_zero(cluster_num, ncp);
	}
	if (ifault == 2) {
		cout << "\n";
		cout << "KMEANS_02 - Warning!\n";
		cout << "  Maximum number of iterations reached\n";
		cout << "  without convergence.\n";
	}
	//
	//  Compute the within-cluster sum of squares for each cluster.
	//
	r8vec_zero(dim_num * cluster_num, cluster_center);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ point[i + j*dim_num];
		}
	}

	for (k = 0; k < cluster_num; k++) {
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				/ (double)cluster_population[k];
		}
	}
	//
	//  Compute the cluster energies.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + point_energy;
	}

	delete[] an1;
	delete[] an2;
	delete[] cluster2;
	delete[] d;
	delete[] itran;
	delete[] live;
	delete[] ncp;

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

void kmeans_02_optra(int dim_num, int point_num, int cluster_num,
	double point[], double cluster_center[], int cluster[], int cluster2[],
	int cluster_population[], double an1[], double an2[], int ncp[],
	double d[], int itran[], int live[], int &indx)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_02_OPTRA carries out the optimal transfer stage.
	//
	//  Discussion:
	//
	//    Each point is re-allocated, if necessary, to the cluster that
	//    will induce a maximum reduction in the within-cluster sum of
	//    squares.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    09 October 2011
	//
	//  Author:
	//
	//    Original FORTRAN77 by John Hartigan, Manchek Wong.
	//    C++ version by John Burkardt.
	//
	//  Reference:
	//
	//    John Hartigan, Manchek Wong,
	//    Algorithm AS 136:
	//    A K-Means Clustering Algorithm,
	//    Applied Statistics,
	//    Volume 28, Number 1, 1979, pages 100-108.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates of 
	//    the points.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the cluster centers.
	//
	//    Input/output, int CLUSTER[POINT_NUM], the cluster 
	//    each point belongs to.
	//
	//    Input/output, int CLUSTER2[POINT_NUM], the cluster 
	//    to which each point is most likely to be transferred to.
	//
	//    Input/output, int CLUSTER_POPULATION[CLUSTER_NUM], 
	//    the number of points in each cluster.
	//
	//    Input/output, double AN1[CLUSTER_NUM], 
	//    CLUSTER_POPULATION(L) / (CLUSTER_POPULATION(L) - 1)
	//
	//    Input/output, double AN2[CLUSTER_NUM], 
	//    CLUSTER_POPULATION(L) / (CLUSTER_POPULATION(L) + 1)
	//
	//    Input/output, int NCP[CLUSTER_NUM], ?
	//
	//    Input/output, double D[POINT_NUM], ?
	//
	//    Input/output, int ITRAN[CLUSTER_NUM], 
	//    1 if cluster L is updated in the quick-transfer stage,
	//    0 otherwise.  Reset to zero on output.
	//
	//    Input/output, int LIVE[CLUSTER_NUM], ?
	//
	//    Input/output, int INDX, ?
	//
{
	double al1;
	double al2;
	double alt;
	double alw;
	double dc;
	int i;
	int j;
	int k;
	int l;
	int l1;
	int l2;
	int ll;
	double r2;
	double rr;
	//
	//  If cluster L is updated in the last quick-transfer stage, it
	//  belongs to the live set throughout this stage.   Otherwise, at
	//  each step, it is not in the live set if it has not been updated
	//  in the last POINT_NUM optimal transfer steps.
	//
	for (k = 0; k < cluster_num; k++) {
		if (itran[k] == 1) {
			live[k] = point_num + 1;
		}
	}

	for (j = 0; j < point_num; j++) {
		indx = indx + 1;
		l1 = cluster[j];
		l2 = cluster2[j];
		ll = l2;
		//
		//  If point J is the only member of cluster L1, no transfer.
		//
		if (1 < cluster_population[l1]) {
			//
			//  If L1 has been updated in this stage, re-compute D(I).
			//
			if (ncp[l1] != 0) {
				d[j] = 0.0;
				for (i = 0; i < dim_num; i++) {
					d[j] = d[j] + pow(point[i + j*dim_num] - cluster_center[i + l1*dim_num], 2);
				}
				d[j] = an1[l1] * d[j];
			}
			//
			//  Find the cluster with minimum R2.
			//
			r2 = 0.0;
			for (i = 0; i < dim_num; i++) {
				r2 = r2 + pow(point[i + j*dim_num] - cluster_center[i + l2*dim_num], 2);
			}
			r2 = an2[l2] * r2;

			for (l = 0; l < cluster_num; l++) {
				//
				//  If LIVE(L1) <= J, then L1 is not in the live set.   If this is
				//  true, we only need to consider clusters that are in the live set
				//  for possible transfer of point J.   
				//
				//  Otherwise, we need to consider all possible clusters.
				//
				if ((j < live[l1] || j < live[l]) && l != l1 && l != ll) {
					rr = r2 / an2[l];

					dc = 0.0;
					for (i = 0; i < dim_num; i++) {
						dc = dc + pow(point[i + j*dim_num] - cluster_center[i + l*dim_num], 2);
					}

					if (dc < rr) {
						r2 = dc * an2[l];
						l2 = l;
					}
				}
			}
			//
			//  If no transfer is necessary, L2 is the new CLUSTER2(J).
			// 
			if (d[j] <= r2) {
				cluster2[j] = l2;
			}
			//
			//  Update cluster centers, LIVE, NCP, AN1 and AN2 for clusters L1 and
			//  L2, and update CLUSTER(J) and CLUSTER2(J).
			//
			else {
				indx = 0;
				live[l1] = point_num + j;
				live[l2] = point_num + j;
				ncp[l1] = j;
				ncp[l2] = j;
				al1 = cluster_population[l1];
				alw = al1 - 1.0;
				al2 = cluster_population[l2];
				alt = al2 + 1.0;

				for (i = 0; i < dim_num; i++) {
					cluster_center[i + l1*dim_num] = (cluster_center[i + l1*dim_num] * al1
						- point[i + j*dim_num]) / alw;

					cluster_center[i + l2*dim_num] = (cluster_center[i + l2*dim_num] * al2
						+ point[i + j*dim_num]) / alt;
				}

				cluster_population[l1] = cluster_population[l1] - 1;
				cluster_population[l2] = cluster_population[l2] + 1;
				an2[l1] = alw / al1;

				if (1.0 < alw) {
					an1[l1] = alw / (alw - 1.0);
				} else {
					an1[l1] = r8_huge();
				}

				an1[l2] = alt / al2;
				an2[l2] = alt / (alt + 1.0);
				cluster[j] = l2;
				cluster2[j] = l1;
			}
		}

		if (indx == point_num) {
			return;
		}

	}
	//
	//  ITRAN(L) = 0 before entering QTRAN.
	//
	i4vec_zero(cluster_num, itran);
	//
	//  LIVE(L) has to be decreased by POINT_NUM before re-entering OPTRA.
	//
	for (k = 0; k < cluster_num; k++) {
		live[k] = live[k] - point_num;
	}

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

void kmeans_02_qtran(int dim_num, int point_num, int cluster_num,
	double point[], double cluster_center[], int cluster[], int cluster2[],
	int cluster_population[], double an1[], double an2[], int ncp[], double d[],
	int itran[], int &indx)

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_02_QTRAN carries out the quick transfer stage.
	//
	//  Discussion:
	//
	//    For each point I, CLUSTER(I) and CLUSTER2(I) are switched, if necessary, 
	//    to reduce within-cluster sum of squares.  The cluster centers are
	//    updated after each step.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    13 October 2011
	//
	//  Author:
	//
	//    Original FORTRAN77 by John Hartigan, Manchek Wong.
	//    C++ version by John Burkardt.
	//
	//  Reference:
	//
	//    John Hartigan, Manchek Wong,
	//    Algorithm AS 136:
	//    A K-Means Clustering Algorithm,
	//    Applied Statistics,
	//    Volume 28, Number 1, 1979, pages 100-108.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the coordinates 
	//    of the points.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the cluster centers.
	//
	//    Input/output, int CLUSTER[POINT_NUM], the cluster 
	//    each point belongs to.
	//
	//    Input/output, int CLUSTER2[POINT_NUM], the cluster to 
	//    which each point is most likely to be transferred to.
	//
	//    Input/output, int CLUSTER_POPULATION[CLUSTER_NUM], 
	//    the number of points in each cluster.
	//
	//    Input/output, double AN1[CLUSTER_NUM], 
	//    CLUSTER_POPULATION(L) / (CLUSTER_POPULATION(L) - 1).
	//
	//    Input/output, double AN2[CLUSTER_NUM], 
	//    CLUSTER_POPULATION(L) / (CLUSTER_POPULATION(L) + 1).
	//
	//    Input/output, int NCP[CLUSTER_NUM], ?
	//
	//    Input/output, double D[POINT_NUM], ?
	//
	//    Input/output, int ITRAN[CLUSTER_NUM], 
	//    1 if cluster L is updated in the quick-transfer stage,
	//    0 otherwise.
	//
	//    Input/output, int INDX, is set to 0 if any 
	//    updating occurs.
	//
{
	double al1;
	double al2;
	double alt;
	double alw;
	int count;
	double dd;
	int i;
	int j;
	int l1;
	int l2;
	double r2;
	int step;
	//
	//  In the optimal transfer stage, NCP(L) indicates the step at which
	//  cluster L is last updated.   In the quick transfer stage, NCP(L)
	//  is equal to the step at which cluster L is last updated plus POINT_NUM.
	//
	count = 0;
	step = 0;

	for (; ; ) {
		for (j = 0; j < point_num; j++) {
			count = count + 1;
			step = step + 1;
			l1 = cluster[j];
			l2 = cluster2[j];
			//
			//  If point I is the only member of cluster L1, no transfer.
			//
			if (1 < cluster_population[l1]) {
				//
				//  If NCP(L1) < STEP, no need to re-compute distance from point I to
				//  cluster L1.   Note that if cluster L1 is last updated exactly POINT_NUM
				//  steps ago, we still need to compute the distance from point I to
				//  cluster L1.
				//
				if (step <= ncp[l1]) {
					d[j] = 0.0;
					for (i = 0; i < dim_num; i++) {
						d[j] = d[j] + pow(point[i + j*dim_num] - cluster_center[i + l1*dim_num], 2);
					}
					d[j] = an1[l1] * d[j];
				}
				//
				//  If STEP >= both NCP(L1) and NCP(L2) there will be no transfer of
				//  point I at this step.
				//
				if (step < ncp[l1] || step < ncp[l2]) {
					r2 = d[j] / an2[l2];

					dd = 0.0;
					for (i = 0; i < dim_num; i++) {
						dd = dd + pow(point[i + j*dim_num] - cluster_center[i + l2*dim_num], 2);
					}
					//
					//  Update cluster centers, NCP, CLUSTER_POPULATION, ITRAN, AN1 and AN2 
					//  for clusters L1 and L2.   Also update CLUSTER(J) and CLUSTER2(J).   
					//
					//  Note that if any updating occurs in this stage, INDX is set back to 0.
					//
					if (dd < r2) {
						count = 0;
						indx = 0;
						itran[l1] = 1;
						itran[l2] = 1;
						ncp[l1] = step + point_num;
						ncp[l2] = step + point_num;
						al1 = cluster_population[l1];
						alw = al1 - 1.0;
						al2 = cluster_population[l2];
						alt = al2 + 1.0;

						for (i = 0; i < dim_num; i++) {
							cluster_center[i + l1*dim_num] = (cluster_center[i + l1*dim_num] * al1
								- point[i + j*dim_num]) / alw;

							cluster_center[i + l2*dim_num] = (cluster_center[i + l2*dim_num] * al2
								+ point[i + j*dim_num]) / alt;
						}

						cluster_population[l1] = cluster_population[l1] - 1;
						cluster_population[l2] = cluster_population[l2] + 1;
						an2[l1] = alw / al1;

						if (1.0 < alw) {
							an1[l1] = alw / (alw - 1.0);
						} else {
							an1[l1] = r8_huge();
						}
						an1[l2] = alt / al2;
						an2[l2] = alt / (alt + 1.0);
						cluster[j] = l2;
						cluster2[j] = l1;
					}
				}
			}
			//
			//  If no re-allocation took place in the last POINT_NUM steps, return.
			//
			if (count == point_num) {
				return;
			}
		}
	}
	return;
}
//****************************************************************************80

void kmeans_03(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], int cluster[], double cluster_center[],
	int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_03 applies the K-Means algorithm.
	//
	//  Discussion:
	//
	//    It is possible for a straightforward K-Means algorithm to
	//    halt at a non-optimal partition of the points.  This routine
	//    tries to improve the input partition if possible.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    10 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Reference:
	//
	//    Wendy Martinez, Angel Martinez,
	//    Computational Statistics Handbook with MATLAB,
	//    pages 373-376,
	//    Chapman and Hall / CRC, 2002.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of data points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the data points.
	//
	//    Output, int CLUSTER[POINT_NUM], the cluster to which
	//    each point belongs.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], the 
	//    centers associated with the clustering.  On output, these may 
	//    have been altered.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number
	//    of points in each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the energy of 
	//    the clusters.
	//
{
	int ci;
	int cj;
	bool debug = true;
	double *distsq;
	int i;
	int j;
	int k;
	double point_energy;
	double point_energy_min;
	int swap;
	//
	//  Check the input.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "KMEANS_03 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "KMEANS_03 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "KMEANS_03 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "KMEANS_03 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}
	//
	//  Assign each point to the nearest cluster center.
	//
	for (j = 0; j < point_num; j++) {
		point_energy_min = r8_huge();
		cluster[j] = -1;

		for (k = 0; k < cluster_num; k++) {
			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}

			if (point_energy < point_energy_min) {
				point_energy_min = point_energy;
				cluster[j] = k;
			}
		}
	}
	//
	//  Determine the cluster populations.
	//
	i4vec_zero(cluster_num, cluster_population);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_population[k] = cluster_population[k] + 1;
	}
	//
	//  Average the points in each cluster to get a new cluster center.
	//
	r8vec_zero(dim_num * cluster_num, cluster_center);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ point[i + j*dim_num];
		}
	}

	for (k = 0; k < cluster_num; k++) {
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				/ (double)(cluster_population[k]);
		}
	}
	//
	//  Carry out the iteration.
	//
	it_num = 0;
	distsq = new double[cluster_num];

	while (it_num < it_max) {
		it_num = it_num + 1;

		swap = 0;

		for (j = 0; j < point_num; j++) {
			ci = cluster[j];

			if (cluster_population[ci] <= 1) {
				continue;
			}

			for (cj = 0; cj < cluster_num; cj++) {
				if (cj == ci) {
					distsq[cj] = 0.0;
					for (i = 0; i < dim_num; i++) {
						distsq[cj] = distsq[cj]
							+ pow(point[i + j*dim_num] - cluster_center[i + cj*dim_num], 2);
					}
					distsq[cj] = distsq[cj] * (double)(cluster_population[cj])
						/ (double)(cluster_population[cj] - 1);
				} else if (cluster_population[cj] == 0) {
					for (i = 0; i < dim_num; i++) {
						cluster_center[i + cj*dim_num] = point[i + j*dim_num];
					}
					distsq[cj] = 0.0;
				} else {
					distsq[cj] = 0.0;
					for (i = 0; i < dim_num; i++) {
						distsq[cj] = distsq[cj]
							+ pow(point[i + j*dim_num] - cluster_center[i + cj*dim_num], 2);
					}
					distsq[cj] = distsq[cj] * (double)(cluster_population[cj])
						/ (double)(cluster_population[cj] + 1);
				}
			}
			//
			//  Find the index of the minimum value of DISTSQ.
			//
			k = r8vec_min_index(cluster_num, distsq);
			//
			//  If that is not the cluster to which point I now belongs, move it there.
			//
			if (k == ci) {
				continue;
			}

			cj = k;

			for (i = 0; i < dim_num; i++) {
				cluster_center[i + ci*dim_num] = ((double)(cluster_population[ci])
					* cluster_center[i + ci*dim_num] - point[i + j*dim_num])
					/ (double)(cluster_population[ci] - 1);

				cluster_center[i + cj*dim_num] = ((double)(cluster_population[cj])
					* cluster_center[i + cj*dim_num] + point[i + j*dim_num])
					/ (double)(cluster_population[cj] + 1);
			}
			cluster_population[ci] = cluster_population[ci] - 1;
			cluster_population[cj] = cluster_population[cj] + 1;

			cluster[j] = cj;
			swap = swap + 1;
		}
		//
		//  Exit if no reassignments were made during this iteration.
		//
		if (swap == 0) {
			break;
		}
	}
	//
	//  Compute the cluster energies.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + point_energy;
	}

	delete[] distsq;

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

void kmeans_w_01(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], double weight[], int cluster[],
	double cluster_center[], int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_W_01 applies the weighted K-Means algorithm.
	//
	//  Discussion:
	//
	//    The input data for the weight K-Means problem includes:
	//    * a set of N data points X in M dimensions, 
	//    * a set of N nonnegative weights W,
	//    * a desired number of clusters K.
	//    * an initial set of cluster centers Z,
	//    * an (optional) initial set of cluster assignments.
	//
	//    The goal is to determine K points Z, called cluster centers, and
	//    to assign each point X(I) to some cluster Z(J), so that we minimize
	//    the weighted standard deviation of the distance of each data point
	//    to the center of its cluster.  Writing J = CLUSTER(I) to
	//    indicate the index of the nearest cluster center Z(J) to the 
	//    point X(I), the quantity we are trying to minimize is the sum
	//    of the weighted cluster energies E(J), where:
	//
	//      E(J) = Sum ( 1 <= I <= N ) W(I) * || X(I) - Z(J) ||^2
	//
	//    Here, we assume that we are using the Euclidean norm, so that
	//    
	//      || X(I) - Z(J) ||^2 = Sum ( 1 <= K <= M )
	//         ( X(I)(K) - Z(J)(K) )^2
	//
	//    In this notation, X(I)(K) is the K-th spatial component of the
	//    I-th data point.
	//
	//    Note that this routine should give the same results as KMEANS_01
	//    in any case in which all the entries of the WEIGHT vector are equal.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    10 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Reference:
	//
	//    David Sparks,
	//    Algorithm AS 58: Euclidean Cluster Analysis,
	//    Applied Statistics,
	//    Volume 22, Number 1, 1973, pages 126-130.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int &IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the points.
	//
	//    Input, double WEIGHT[POINT_NUM], the weights.
	//
	//    Output, int CLUSTER[POINT_NUM], indicates which cluster
	//    each point belongs to.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM],
	//    the cluster centers.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number 
	//    of points in each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the 
	//    cluster energies.
	//
{
	double *cluster_weight;
	double dc;
	double de;
	double *f;
	int i;
	int il;
	int ir;
	int j;
	int k;
	double point_energy;
	double point_energy_min;
	int swap;

	it_num = 0;
	//
	//  Idiot checks.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "KMEANS_W_01 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "KMEANS_W_01 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "KMEANS_W_01 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "KMEANS_W_01 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}

	if (r8vec_any_negative(point_num, weight)) {
		cout << "\n";
		cout << "KMEANS_W_01 - Fatal error!\n";
		cout << "  Some weight entry is negative.\n";
		exit(1);
	}

	if (r8vec_all_nonpositive(point_num, weight)) {
		cout << "\n";
		cout << "KMEANS_W_01 - Fatal error!\n";
		cout << "  No weight entry is positive.\n";
		exit(1);
	}
	//
	//  Assign each point to the nearest cluster.
	//
	for (j = 0; j < point_num; j++) {
		point_energy_min = r8_huge();
		cluster[j] = -1;

		for (k = 0; k < cluster_num; k++) {
			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}

			if (point_energy < point_energy_min) {
				point_energy_min = point_energy;
				cluster[j] = k;
			}
		}
	}
	//
	//  Determine the cluster populations and weights.
	//
	i4vec_zero(cluster_num, cluster_population);
	cluster_weight = r8vec_zero_new(cluster_num);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_population[k] = cluster_population[k] + 1;
		cluster_weight[k] = cluster_weight[k] + weight[j];
	}
	//
	//  Calculate the mean and sum of squares for each cluster.
	//
	r8vec_zero(dim_num * cluster_num, cluster_center);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ weight[j] * point[i + j*dim_num];
		}
	}

	for (k = 0; k < cluster_num; k++) {
		if (0.0 < cluster_weight[k]) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
					/ cluster_weight[k];
			}
		}
	}
	//
	//  Set the point energies.
	//
	f = r8vec_zero_new(point_num);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			f[j] = f[j] + pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
	}
	//
	//  Set the cluster energies.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_energy[k] = cluster_energy[k] + weight[j] * f[j];
	}
	//
	//  Adjust the point energies by a weight factor.
	//
	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		if (weight[j] < cluster_weight[k]) {
			f[j] = f[j] * cluster_weight[k] / (cluster_weight[k] - weight[j]);
		}
	}
	//
	//  Examine each observation in turn to see if it should be
	//  reassigned to a different cluster.
	//
	it_num = 0;

	while (it_num < it_max) {
		it_num = it_num + 1;

		swap = 0;

		for (j = 0; j < point_num; j++) {
			il = cluster[j];
			ir = il;

			if (cluster_population[il] <= 1) {
				continue;
			}

			dc = f[j];

			for (k = 0; k < cluster_num; k++) {
				if (k != il) {
					de = 0.0;
					for (i = 0; i < dim_num; i++) {
						de = de + pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2)
							* cluster_weight[k] / (cluster_weight[k] + weight[j]);
					}

					if (de < dc) {
						dc = de;
						ir = k;
					}
				}
			}
			//
			//  If the lowest value was obtained by staying in the current cluster,
			//  then cycle.
			//
			if (ir == il) {
				continue;
			}
			//
			//  Reassign the point from cluster IL to cluster IR.
			//
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + il*dim_num] =
					(cluster_weight[il] * cluster_center[i + il*dim_num]
						- weight[j] * point[i + j*dim_num])
					/ (cluster_weight[il] - weight[j]);

				cluster_center[i + ir*dim_num] =
					(cluster_weight[ir] * cluster_center[i + ir*dim_num]
						+ weight[j] * point[i + j*dim_num])
					/ (cluster_weight[ir] + weight[j]);
			}
			cluster_weight[il] = cluster_weight[il] - weight[j];
			cluster_weight[ir] = cluster_weight[ir] + weight[j];

			cluster_energy[il] = cluster_energy[il] - f[j];
			cluster_energy[ir] = cluster_energy[ir] + dc;

			cluster_population[ir] = cluster_population[ir] + 1;
			cluster_population[il] = cluster_population[il] - 1;

			cluster[j] = ir;
			//
			//  Adjust the value of F for all points in clusters IL and IR.
			//
			for (j = 0; j < point_num; j++) {
				k = cluster[j];

				if (k == il || k == ir) {
					f[j] = 0.0;
					for (i = 0; i < dim_num; i++) {
						f[j] = f[j] + pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
					}
					if (weight[j] < cluster_weight[k]) {
						f[j] = f[j] * cluster_weight[k] / (cluster_weight[k] - weight[j]);
					}
				}
			}
			swap = swap + 1;
		}
		//
		//  Exit if no reassignments were made during this iteration.
		//
		if (swap == 0) {
			break;
		}
	}
	//
	//  Compute the energy based on the final value of the cluster centers.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + weight[j] * point_energy;
	}

	delete[] cluster_weight;
	delete[] f;

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

void kmeans_w_03(int dim_num, int point_num, int cluster_num, int it_max,
	int &it_num, double point[], double weight[], int cluster[],
	double cluster_center[], int cluster_population[], double cluster_energy[])

	//****************************************************************************80
	//
	//  Purpose:
	//
	//   KMEANS_W_03 applies the weighted K-Means algorithm.
	//
	//  Discussion:
	//
	//    The input data for the weight K-Means problem includes:
	//    * a set of N data points X in M dimensions, 
	//    * a set of N nonnegative weights W,
	//    * a desired number of clusters K.
	//    * an initial set of cluster centers Z,
	//    * an (optional) initial set of cluster assignments.
	//
	//    The goal is to determine K points Z, called cluster centers, and
	//    to assign each point X(I) to some cluster Z(J), so that we minimize
	//    the weighted standard deviation of the distance of each data point
	//    to the center of its cluster.  Writing J = CLUSTER(I) to
	//    indicate the index of the nearest cluster center Z(J) to the 
	//    point X(I), the quantity we are trying to minimize is the sum
	//    of the weighted cluster energies E(J), where:
	//
	//      E(J) = Sum ( 1 <= I <= N ) W(I) * || X(I) - Z(J) ||^2
	//
	//    Here, we assume that we are using the Euclidean norm, so that
	//    
	//      || X(I) - Z(J) ||^2 = Sum ( 1 <= K <= M )
	//        ( X(I)(K) - Z(J)(K) )^2
	//
	//    In this notation, X(I)(K) is the K-th spatial component of the
	//    I-th data point.
	//
	//    Note that this routine should give the same results as KMEANS_01
	//    in any case in which all the entries of the WEIGHT vector are equal.
	//
	//  Licensing:
	//
	//    This code is distributed under the GNU LGPL license. 
	//
	//  Modified:
	//
	//    10 October 2011
	//
	//  Author:
	//
	//    John Burkardt
	//
	//  Reference:
	//
	//    Wendy Martinez, Angel Martinez,
	//    Computational Statistics Handbook with MATLAB,
	//    pages 373-376,
	//    Chapman and Hall / CRC, 2002.
	//
	//  Parameters:
	//
	//    Input, int DIM_NUM, the number of spatial dimensions.
	//
	//    Input, int POINT_NUM, the number of data points.
	//
	//    Input, int CLUSTER_NUM, the number of clusters.
	//
	//    Input, int IT_MAX, the maximum number of iterations.
	//
	//    Output, int IT_NUM, the number of iterations taken.
	//
	//    Input, double POINT[DIM_NUM*POINT_NUM], the data points.
	//
	//    Input, double WEIGHT[POINT_NUM], the weights.
	//
	//    Input/output, int CLUSTER[POINT_NUM], the cluster 
	//    to which each point belongs.  On output, these may have been altered.
	//
	//    Input/output, double CLUSTER_CENTER[DIM_NUM*CLUSTER_NUM], the
	//    centers associated with the clustering.  On output, these may
	//    have been altered.
	//
	//    Output, int CLUSTER_POPULATION[CLUSTER_NUM], the number
	//    of points in each cluster.
	//
	//    Output, double CLUSTER_ENERGY[CLUSTER_NUM], the energy of
	//    the clusters.
	//
{
	int ci;
	int cj;
	double *cluster_weight;
	bool debug = true;
	double *distsq;
	int i;
	int j;
	int k;
	double point_energy;
	double point_energy_min;
	int swap;
	//
	//  Check the input.
	//
	if (cluster_num < 1) {
		cout << "\n";
		cout << "KMEANS_W_03 - Fatal error!\n";
		cout << "  CLUSTER_NUM < 1.\n";
		exit(1);
	}

	if (dim_num < 1) {
		cout << "\n";
		cout << "KMEANS_W_03 - Fatal error!\n";
		cout << "  DIM_NUM < 1.\n";
		exit(1);
	}

	if (point_num < 1) {
		cout << "\n";
		cout << "KMEANS_W_03 - Fatal error!\n";
		cout << "  POINT_NUM < 1.\n";
		exit(1);
	}

	if (it_max < 0) {
		cout << "\n";
		cout << "KMEANS_W_03 - Fatal error!\n";
		cout << "  IT_MAX < 0.\n";
		exit(1);
	}

	if (r8vec_any_negative(point_num, weight)) {
		cout << "\n";
		cout << "KMEANS_W_03 - Fatal error!\n";
		cout << "  Some weight entry is negative.\n";
		exit(1);
	}

	if (r8vec_all_nonpositive(point_num, weight)) {
		cout << "\n";
		cout << "KMEANS_W_03 - Fatal error!\n";
		cout << "  No weight entry is positive.\n";
		exit(1);
	}
	//
	//  Assign each observation to the nearest cluster center.
	//
	for (j = 0; j < point_num; j++) {
		point_energy_min = r8_huge();
		cluster[j] = -1;

		for (k = 0; k < cluster_num; k++) {
			point_energy = 0.0;
			for (i = 0; i < dim_num; i++) {
				point_energy = point_energy +
					pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
			}

			if (point_energy < point_energy_min) {
				point_energy_min = point_energy;
				cluster[j] = k;
			}
		}
	}
	//
	//  Determine the cluster populations and weights.
	//
	i4vec_zero(cluster_num, cluster_population);
	cluster_weight = r8vec_zero_new(cluster_num);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		cluster_population[k] = cluster_population[k] + 1;
		cluster_weight[k] = cluster_weight[k] + weight[j];
	}
	//
	//  Average the points in each cluster to get a new cluster center.
	//
	r8vec_zero(dim_num * cluster_num, cluster_center);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];
		for (i = 0; i < dim_num; i++) {
			cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num]
				+ weight[j] * point[i + j*dim_num];
		}
	}

	for (k = 0; k < cluster_num; k++) {
		if (cluster_weight[k] != 0.0) {
			for (i = 0; i < dim_num; i++) {
				cluster_center[i + k*dim_num] = cluster_center[i + k*dim_num] /
					cluster_weight[k];
			}
		}
	}
	//
	//  Carry out the iteration.
	//
	it_num = 0;
	distsq = new double[cluster_num];

	while (it_num < it_max) {
		it_num = it_num + 1;

		swap = 0;

		for (j = 0; j < point_num; j++) {
			ci = cluster[j];

			if (cluster_population[ci] <= 1) {
				continue;
			}

			for (cj = 0; cj < cluster_num; cj++) {
				if (cj == ci) {
					distsq[cj] = 0.0;
					for (i = 0; i < dim_num; i++) {
						distsq[cj] = distsq[cj]
							+ pow(point[i + j*dim_num] - cluster_center[i + cj*dim_num], 2);
					}
					distsq[cj] = distsq[cj] * cluster_weight[cj]
						/ (cluster_weight[cj] - weight[j]);
				} else if (cluster_population[cj] == 0) {
					for (i = 0; i < dim_num; i++) {
						cluster_center[i + cj*dim_num] = point[i + j*dim_num];
					}
					distsq[cj] = 0.0;
				} else {
					distsq[cj] = 0.0;
					for (i = 0; i < dim_num; i++) {
						distsq[cj] = distsq[cj]
							+ pow(point[i + j*dim_num] - cluster_center[i + cj*dim_num], 2);
					}
					distsq[cj] = distsq[cj] * cluster_weight[cj]
						/ (cluster_weight[cj] + weight[j]);
				}
			}
			//
			//  Find the index of the minimum value of DISTSQ.
			//
			k = r8vec_min_index(cluster_num, distsq);
			//
			//  If that is not the cluster to which point I now belongs, move it there.
			//
			if (k == ci) {
				continue;
			}

			cj = k;

			for (i = 0; i < dim_num; i++) {
				cluster_center[i + ci*dim_num] =
					(cluster_weight[ci] * cluster_center[i + ci*dim_num]
						- weight[j] * point[i + j*dim_num])
					/ (cluster_weight[ci] - weight[j]);

				cluster_center[i + cj*dim_num] =
					(cluster_weight[cj] * cluster_center[i + cj*dim_num]
						+ weight[j] * point[i + j*dim_num])
					/ (cluster_weight[cj] + weight[j]);
			}
			cluster_population[ci] = cluster_population[ci] - 1;
			cluster_population[cj] = cluster_population[cj] + 1;

			cluster_weight[ci] = cluster_weight[ci] - weight[j];
			cluster_weight[cj] = cluster_weight[cj] + weight[j];

			cluster[j] = cj;

			swap = swap + 1;
		}
		//
		//  Exit if no reassignments were made during this iteration.
		//
		if (swap == 0) {
			break;
		}
	}
	//
	//  Compute the energy based on the final value of the cluster centers.
	//
	r8vec_zero(cluster_num, cluster_energy);

	for (j = 0; j < point_num; j++) {
		k = cluster[j];

		point_energy = 0.0;
		for (i = 0; i < dim_num; i++) {
			point_energy = point_energy +
				pow(point[i + j*dim_num] - cluster_center[i + k*dim_num], 2);
		}
		cluster_energy[k] = cluster_energy[k] + weight[j] * point_energy;
	}

	delete[] cluster_weight;
	delete[] distsq;

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

int r4_nint(float x)

//****************************************************************************80
//
//  Purpose:
//
//    R4_NINT returns the nearest integer to an R4.
//
//  Example:
//
//        X         R4_NINT
//
//      1.3         1
//      1.4         1
//      1.5         1 or 2
//      1.6         2
//      0.0         0
//     -0.7        -1
//     -1.1        -1
//     -1.6        -2
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    14 November 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, float X, the value.
//
//    Output, int R4_NINT, the nearest integer to X.
//
{
	int value;

	if (x < 0.0) {
		value = -(int)(-x + 0.5);
	} else {
		value = (int)(x + 0.5);
	}
	return value;
}
//****************************************************************************80

double r8_huge()

//****************************************************************************80
//
//  Purpose:
//
//    R8_HUGE returns a "huge" R8.
//
//  Discussion:
//
//    The value returned by this function is NOT required to be the
//    maximum representable R8.  This value varies from machine to machine,
//    from compiler to compiler, and may cause problems when being printed.
//    We simply want a "very large" but non-infinite number.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 October 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_HUGE, a "huge" R8 value.
//
{
	double value;

	value = 1.0E+30;

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

double r8_max(double x, double y)

//****************************************************************************80
//
//  Purpose:
//
//    R8_MAX returns the maximum of two R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MAX, the maximum of X and Y.
//
{
	double value;

	if (y < x) {
		value = x;
	} else {
		value = y;
	}
	return value;
}
//****************************************************************************80

double r8_min(double x, double y)

//****************************************************************************80
//
//  Purpose:
//
//    R8_MIN returns the minimum of two R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    31 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MIN, the minimum of X and Y.
//
{
	double value;

	if (y < x) {
		value = y;
	} else {
		value = x;
	}
	return value;
}
//****************************************************************************80

double r8_uniform_01(int *seed)

//****************************************************************************80
//
//  Purpose:
//
//    R8_UNIFORM_01 returns a unit pseudorandom R8.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//    If the initial seed is 12345, then the first three computations are
//
//      Input     Output      R8_UNIFORM_01
//      SEED      SEED
//
//         12345   207482415  0.096616
//     207482415  1790989824  0.833995
//    1790989824  2035175616  0.947702
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    11 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input/output, int *SEED, the "seed" value.  Normally, this
//    value should not be 0.  On output, SEED has been updated.
//
//    Output, double R8_UNIFORM_01, a new pseudorandom variate,
//    strictly between 0 and 1.
//
{
	int i4_huge = 2147483647;
	int k;
	double r;

	if (*seed == 0) {
		cerr << "\n";
		cerr << "R8_UNIFORM_01 - Fatal error!\n";
		cerr << "  Input value of SEED = 0.\n";
		exit(1);
	}

	k = *seed / 127773;

	*seed = 16807 * (*seed - k * 127773) - k * 2836;

	if (*seed < 0) {
		*seed = *seed + i4_huge;
	}
	r = (double)(*seed) * 4.656612875E-10;

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

double *r8mat_data_read(string input_filename, int m, int n)

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_DATA_READ reads the data from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//    The file is assumed to contain one record per line.
//
//    Records beginning with '#' are comments, and are ignored.
//    Blank lines are also ignored.
//
//    Each line that is not ignored is assumed to contain exactly (or at least)
//    M real numbers, representing the coordinates of a point.
//
//    There are assumed to be exactly (or at least) N such records.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Input, int M, the number of spatial dimensions.
//
//    Input, int N, the number of points.  The program
//    will stop reading data once N values have been read.
//
//    Output, double R8MAT_DATA_READ[M*N], the data.
//
{
	bool error;
	ifstream input;
	int i;
	int j;
	string line;
	double *table;
	double *x;

	input.open(input_filename.c_str());

	if (!input) {
		cerr << "\n";
		cerr << "R8MAT_DATA_READ - Fatal error!\n";
		cerr << "  Could not open the input file: \"" << input_filename << "\"\n";
		exit(1);
	}

	table = new double[m*n];

	x = new double[m];

	j = 0;

	while (j < n) {
		getline(input, line);

		if (input.eof()) {
			break;
		}

		if (line[0] == '#' || s_len_trim(line) == 0) {
			continue;
		}

		error = s_to_r8vec(line, m, x);

		if (error) {
			continue;
		}

		for (i = 0; i < m; i++) {
			table[i + j*m] = x[i];
		}
		j = j + 1;

	}

	input.close();

	delete[] x;

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

void r8mat_header_read(string input_filename, int *m, int *n)

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_HEADER_READ reads the header from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int *M, the number of spatial dimensions.
//
//    Output, int *N, the number of points.
//
{
	*m = file_column_count(input_filename);

	if (*m <= 0) {
		cerr << "\n";
		cerr << "R8MAT_HEADER_READ - Fatal error!\n";
		cerr << "  FILE_COLUMN_COUNT failed.\n";
		exit(1);
	}

	*n = file_row_count(input_filename);

	if (*n <= 0) {
		cerr << "\n";
		cerr << "R8MAT_HEADER_READ - Fatal error!\n";
		cerr << "  FILE_ROW_COUNT failed.\n";
		exit(1);
	}

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

double *r8mat_mm_new(int n1, int n2, int n3, double a[], double b[])

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_MM_NEW multiplies two matrices.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//    For this routine, the result is returned as the function value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, N3, the order of the matrices.
//
//    Input, double A[N1*N2], double B[N2*N3], the matrices to multiply.
//
//    Output, double R8MAT_MM[N1*N3], the product matrix C = A * B.
//
{
	double *c;
	int i;
	int j;
	int k;

	c = new double[n1*n3];

	for (i = 0; i < n1; i++) {
		for (j = 0; j < n3; j++) {
			c[i + j*n1] = 0.0;
			for (k = 0; k < n2; k++) {
				c[i + j*n1] = c[i + j*n1] + a[i + k*n1] * b[k + j*n2];
			}
		}
	}

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

void r8mat_uniform_01(int m, int n, int *seed, double r[])

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input/output, int *SEED, the "seed" value.  Normally, this
//    value should not be 0.  On output, SEED has
//    been updated.
//
//    Output, double R[M*N], a matrix of pseudorandom values.
//
{
	int i;
	int i4_huge = 2147483647;
	int j;
	int k;

	if (*seed == 0) {
		cerr << "\n";
		cerr << "R8MAT_UNIFORM_01 - Fatal error!\n";
		cerr << "  Input value of SEED = 0.\n";
		exit(1);
	}

	for (j = 0; j < n; j++) {
		for (i = 0; i < m; i++) {
			k = *seed / 127773;

			*seed = 16807 * (*seed - k * 127773) - k * 2836;

			if (*seed < 0) {
				*seed = *seed + i4_huge;
			}

			r[i + j*m] = (double)(*seed) * 4.656612875E-10;
		}
	}
	return;
}
//****************************************************************************80

double *r8mat_uniform_01_new(int m, int n, int *seed)

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_UNIFORM_01_NEW returns a unit pseudorandom R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8's,  stored as a vector
//    in column-major order.
//
//    This routine implements the recursion
//
//      seed = 16807 * seed mod ( 2^31 - 1 )
//      unif = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Springer Verlag, pages 201-202, 1983.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, pages 362-376, 1986.
//
//    Philip Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, pages 136-143, 1969.
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input/output, int *SEED, the "seed" value.  Normally, this
//    value should not be 0, otherwise the output value of SEED
//    will still be 0, and R8_UNIFORM will be 0.  On output, SEED has
//    been updated.
//
//    Output, double R8MAT_UNIFORM_01_NEW[M*N], a matrix of pseudorandom values.
//
{
	int i;
	int j;
	int k;
	double *r;

	r = new double[m*n];

	for (j = 0; j < n; j++) {
		for (i = 0; i < m; i++) {
			k = *seed / 127773;

			*seed = 16807 * (*seed - k * 127773) - k * 2836;

			if (*seed < 0) {
				*seed = *seed + 2147483647;
			}
			r[i + j*m] = (double)(*seed) * 4.656612875E-10;
		}
	}

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

void r8mat_write(string output_filename, int m, int n, double table[])

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_WRITE writes an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    29 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, double TABLE[M*N], the data.
//
{
	int i;
	int j;
	ofstream output;
	//
	//  Open the file.
	//
	output.open(output_filename.c_str());

	if (!output) {
		cerr << "\n";
		cerr << "R8MAT_WRITE - Fatal error!\n";
		cerr << "  Could not open the output file.\n";
		exit(1);
	}
	//
	//  Write the data.
	//
	for (j = 0; j < n; j++) {
		for (i = 0; i < m; i++) {
			output << "  " << setw(24) << setprecision(16) << table[i + j*m];
		}
		output << "\n";
	}
	//
	//  Close the file.
	//
	output.close();

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

bool r8vec_all_nonpositive(int n, double a[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_ALL_NONPOSITIVE: ( all ( A <= 0 ) ) for R8VEC's.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 October 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries.
//
//    Input, double A[N], the vector to check.
//
//    Output, bool R8VEC_ALL_NONPOSITIVE is TRUE if all entries
//    of A are less than or equal to zero.
//
{
	int i;
	bool value;

	for (i = 0; i < n; i++) {
		if (0.0 < a[i]) {
			value = false;
			return value;
		}
	}
	value = true;

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

bool r8vec_any_negative(int n, double a[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_ANY_NEGATIVE: ( any ( A < 0 ) ) for R8VEC's.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    09 October 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries.
//
//    Input, double A[N], the vector to check.
//
//    Output, bool R8VEC_ANY_NEGATIVE is TRUE if any entry
//    of A is less than zero.
//
{
	int i;
	bool value;

	for (i = 0; i < n; i++) {
		if (a[i] < 0.0) {
			value = true;
			return value;
		}
	}
	value = false;

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

double r8vec_i4vec_dot_product(int n, double r8vec[], int i4vec[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_I4VEC_DOT_PRODUCT computes the dot product of an R8VEC and an I4VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//    An I4VEC is a vector of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    30 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, double R8VEC[N], the first vector.
//
//    Input, int I4VEC[N], the second vector.
//
//    Output, double R8VEC_I4VEC_DOT_PRODUCT, the dot product of the vectors.
//
{
	int i;
	double value;

	value = 0.0;
	for (i = 0; i < n; i++) {
		value = value + r8vec[i] * (double)(i4vec[i]);
	}
	return value;
}
//****************************************************************************80

int r8vec_min_index(int n, double a[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_MIN_INDEX returns the index of the minimum value in an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 August 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the array.
//
//    Input, double A[N], the array.
//
//    Output, int R8VEC_MIN_INDEX, the index of the smallest entry.
//
{
	int i;
	int min_index;

	if (n <= 0) {
		min_index = -1;
	} else {
		min_index = 0;

		for (i = 1; i < n; i++) {
			if (a[i] < a[min_index]) {
				min_index = i;
			}
		}
	}

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

double r8vec_sum(int n, double a[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_SUM returns the sum of an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    15 October 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double A[N], the vector.
//
//    Output, double R8VEC_SUM, the sum of the vector.
//
{
	int i;
	double value;

	value = 0.0;
	for (i = 0; i < n; i++) {
		value = value + a[i];
	}
	return value;
}
//****************************************************************************80

void r8vec_uniform_01(int n, int *seed, double r[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    19 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input/output, int *SEED, a seed for the random number generator.
//
//    Output, double R[N], the vector of pseudorandom values.
//
{
	int i;
	int i4_huge = 2147483647;
	int k;

	if (*seed == 0) {
		cerr << "\n";
		cerr << "R8VEC_UNIFORM_01 - Fatal error!\n";
		cerr << "  Input value of SEED = 0.\n";
		exit(1);
	}

	for (i = 0; i < n; i++) {
		k = *seed / 127773;

		*seed = 16807 * (*seed - k * 127773) - k * 2836;

		if (*seed < 0) {
			*seed = *seed + i4_huge;
		}

		r[i] = (double)(*seed) * 4.656612875E-10;
	}

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

void r8vec_zero(int n, double a[])

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_ZERO zeroes an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 July 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, double A[N], a vector of zeroes.
//
{
	int i;

	for (i = 0; i < n; i++) {
		a[i] = 0.0;
	}
	return;
}
//****************************************************************************80

double *r8vec_zero_new(int n)

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_ZERO_NEW creates and zeroes an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    10 July 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, double R8VEC_ZERO_NEW[N], a vector of zeroes.
//
{
	double *a;
	int i;

	a = new double[n];

	for (i = 0; i < n; i++) {
		a[i] = 0.0;
	}
	return a;
}
//****************************************************************************80

int s_len_trim(string s)

//****************************************************************************80
//
//  Purpose:
//
//    S_LEN_TRIM returns the length of a string to the last nonblank.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, a string.
//
//    Output, int S_LEN_TRIM, the length of the string to the last nonblank.
//    If S_LEN_TRIM is 0, then the string is entirely blank.
//
{
	int n;

	n = s.length();

	while (0 < n) {
		if (s[n - 1] != ' ') {
			return n;
		}
		n = n - 1;
	}

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

double s_to_r8(string s, int *lchar, bool *error)

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_R8 reads an R8 from a string.
//
//  Discussion:
//
//    This routine will read as many characters as possible until it reaches
//    the end of the string, or encounters a character which cannot be
//    part of the real number.
//
//    Legal input is:
//
//       1 blanks,
//       2 '+' or '-' sign,
//       2.5 spaces
//       3 integer part,
//       4 decimal point,
//       5 fraction part,
//       6 'E' or 'e' or 'D' or 'd', exponent marker,
//       7 exponent sign,
//       8 exponent integer part,
//       9 exponent decimal point,
//      10 exponent fraction part,
//      11 blanks,
//      12 final comma or semicolon.
//
//    with most quantities optional.
//
//  Example:
//
//    S                 R
//
//    '1'               1.0
//    '     1   '       1.0
//    '1A'              1.0
//    '12,34,56'        12.0
//    '  34 7'          34.0
//    '-1E2ABCD'        -100.0
//    '-1X2ABCD'        -1.0
//    ' 2E-1'           0.2
//    '23.45'           23.45
//    '-4.2E+2'         -420.0
//    '17d2'            1700.0
//    '-14e-2'         -0.14
//    'e2'              100.0
//    '-12.73e-9.23'   -12.73 * 10.0**(-9.23)
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string containing the
//    data to be read.  Reading will begin at position 1 and
//    terminate at the end of the string, or when no more
//    characters can be read to form a legal real.  Blanks,
//    commas, or other nonnumeric data will, in particular,
//    cause the conversion to halt.
//
//    Output, int *LCHAR, the number of characters read from
//    the string to form the number, including any terminating
//    characters such as a trailing comma or blanks.
//
//    Output, bool *ERROR, is true if an error occurred.
//
//    Output, double S_TO_R8, the real value that was read from the string.
//
{
	char c;
	int ihave;
	int isgn;
	int iterm;
	int jbot;
	int jsgn;
	int jtop;
	int nchar;
	int ndig;
	double r;
	double rbot;
	double rexp;
	double rtop;
	char TAB = 9;

	nchar = s_len_trim(s);
	*error = false;
	r = 0.0;
	*lchar = -1;
	isgn = 1;
	rtop = 0.0;
	rbot = 1.0;
	jsgn = 1;
	jtop = 0;
	jbot = 1;
	ihave = 1;
	iterm = 0;

	for (; ; ) {
		c = s[*lchar + 1];
		*lchar = *lchar + 1;
		//
		//  Blank or TAB character.
		//
		if (c == ' ' || c == TAB) {
			if (ihave == 2) {
			} else if (ihave == 6 || ihave == 7) {
				iterm = 1;
			} else if (1 < ihave) {
				ihave = 11;
			}
		}
		//
		//  Comma.
		//
		else if (c == ',' || c == ';') {
			if (ihave != 1) {
				iterm = 1;
				ihave = 12;
				*lchar = *lchar + 1;
			}
		}
		//
		//  Minus sign.
		//
		else if (c == '-') {
			if (ihave == 1) {
				ihave = 2;
				isgn = -1;
			} else if (ihave == 6) {
				ihave = 7;
				jsgn = -1;
			} else {
				iterm = 1;
			}
		}
		//
		//  Plus sign.
		//
		else if (c == '+') {
			if (ihave == 1) {
				ihave = 2;
			} else if (ihave == 6) {
				ihave = 7;
			} else {
				iterm = 1;
			}
		}
		//
		//  Decimal point.
		//
		else if (c == '.') {
			if (ihave < 4) {
				ihave = 4;
			} else if (6 <= ihave && ihave <= 8) {
				ihave = 9;
			} else {
				iterm = 1;
			}
		}
		//
		//  Exponent marker.
		//
		else if (ch_eqi(c, 'E') || ch_eqi(c, 'D')) {
			if (ihave < 6) {
				ihave = 6;
			} else {
				iterm = 1;
			}
		}
		//
		//  Digit.
		//
		else if (ihave < 11 && '0' <= c && c <= '9') {
			if (ihave <= 2) {
				ihave = 3;
			} else if (ihave == 4) {
				ihave = 5;
			} else if (ihave == 6 || ihave == 7) {
				ihave = 8;
			} else if (ihave == 9) {
				ihave = 10;
			}

			ndig = ch_to_digit(c);

			if (ihave == 3) {
				rtop = 10.0 * rtop + (double)ndig;
			} else if (ihave == 5) {
				rtop = 10.0 * rtop + (double)ndig;
				rbot = 10.0 * rbot;
			} else if (ihave == 8) {
				jtop = 10 * jtop + ndig;
			} else if (ihave == 10) {
				jtop = 10 * jtop + ndig;
				jbot = 10 * jbot;
			}

		}
		//
		//  Anything else is regarded as a terminator.
		//
		else {
			iterm = 1;
		}
		//
		//  If we haven't seen a terminator, and we haven't examined the
		//  entire string, go get the next character.
		//
		if (iterm == 1 || nchar <= *lchar + 1) {
			break;
		}

	}
	//
	//  If we haven't seen a terminator, and we have examined the
	//  entire string, then we're done, and LCHAR is equal to NCHAR.
	//
	if (iterm != 1 && (*lchar) + 1 == nchar) {
		*lchar = nchar;
	}
	//
	//  Number seems to have terminated.  Have we got a legal number?
	//  Not if we terminated in states 1, 2, 6 or 7!
	//
	if (ihave == 1 || ihave == 2 || ihave == 6 || ihave == 7) {
		*error = true;
		return r;
	}
	//
	//  Number seems OK.  Form it.
	//
	if (jtop == 0) {
		rexp = 1.0;
	} else {
		if (jbot == 1) {
			rexp = pow(10.0, jsgn * jtop);
		} else {
			rexp = jsgn * jtop;
			rexp = rexp / jbot;
			rexp = pow(10.0, rexp);
		}

	}

	r = isgn * rexp * rtop / rbot;

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

bool s_to_r8vec(string s, int n, double rvec[])

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_R8VEC reads an R8VEC from a string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be read.
//
//    Input, int N, the number of values expected.
//
//    Output, double RVEC[N], the values read from the string.
//
//    Output, bool S_TO_R8VEC, is true if an error occurred.
//
{
	int begin;
	bool error;
	int i;
	int lchar;
	int length;

	begin = 0;
	length = s.length();
	error = 0;

	for (i = 0; i < n; i++) {
		rvec[i] = s_to_r8(s.substr(begin, length), &lchar, &error);

		if (error) {
			return error;
		}
		begin = begin + lchar;
		length = length - lchar;
	}

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

int s_word_count(string s)

//****************************************************************************80
//
//  Purpose:
//
//    S_WORD_COUNT counts the number of "words" in a string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be examined.
//
//    Output, int S_WORD_COUNT, the number of "words" in the string.
//    Words are presumed to be separated by one or more blanks.
//
{
	bool blank;
	int char_count;
	int i;
	int word_count;

	word_count = 0;
	blank = true;

	char_count = s.length();

	for (i = 0; i < char_count; i++) {
		if (isspace(s[i])) {
			blank = true;
		} else if (blank) {
			word_count = word_count + 1;
			blank = false;
		}
	}

	return word_count;
}
//****************************************************************************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
}