/* NUM2.c
 *
 * Copyright (C) 1993-2003 David Weenink
 *
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or (at
 * your option) any later version.
 *
 * This program is distributed in the hope that it will be useful, but
 * WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 */

/*
 djmw 20020819 GPL header
 djmw 20020819 Split nonGLP part off.
 djmw 20030108  Latest modification
*/

/* 20001109 djmw: changed stop criteria in NUMsvdcmp and NUMtqli
 djmw 20020819 Split into GPL and nonGPL part
 djmw 20021008: removed SVD_sort
 djmw 20030308 latest modification.
*/

#include "NUMclapack.h"
#ifndef _NUM_h_
	#include "NUM.h"
#endif
#include "NUM2.h"
#include "SVD.h"
#include "NUMmachar.h"
#include "melder.h"

#define my me ->

#define MAX(m,n) ((m) > (n) ? (m) : (n))
#define MIN(m,n) ((m) < (n) ? (m) : (n))
 
extern machar_Table NUMfpp;

int NUMdmatrix_hasInfinities (double **m, long rb, long re, long cb, long ce)
{
	long i, j; double min, max;
	min = max = m[rb][cb];
	for (i=rb; i <= re; i++)
	{
		for (j=cb; j <= ce; j++)
		{
			if (m[i][j] > max) max = m[i][j];
			else if (m[i][j] < min) min = m[i][j];
		}
	}
	if (! NUMfpp) NUMmachar ();
	return max >= NUMfpp -> rmax || min <= - NUMfpp -> rmax;
}

double *NUMstring_to_numbers (const char *s, long *numbers_found)
{
	char *dup = NULL, *token, *delimiter = " ,\t";
	long capacity = 100, n; double *numbers = NULL;
	
	*numbers_found = n = 0;
	
	if (((dup = Melder_strdup (s)) == NULL) ||
		((numbers = NUMdvector (1, capacity)) == NULL)) goto end;
	token = strtok (dup, delimiter);
	while (token)
	{
		double value = atof (token);
		if (n > capacity) 
		{
			long newsize = 2 * capacity; double *new;
			if (! (new = Melder_realloc (numbers, newsize))) goto end;
			numbers = new; capacity = newsize;
		}
		numbers[++n] = value;
		token = strtok (NULL, delimiter);
	}
end:
	*numbers_found = n;
	Melder_free (dup);
	if (Melder_hasError())
	{
		NUMdvector_free (numbers, 1);
		*numbers_found = 0;
	}
	return numbers;
}

int NUMstrings_equal (char **s1, char **s2, long lo, long hi)
{
	long i;
	for (i=lo; i <= hi; i++)
	{
		if (NUMstrcmp (s1[i], s2[i])) return 0;
	}
	return 1;
}

int NUMstrings_copyElements (char **from, char**to, long lo, long hi)
{
	long i;
	for (i=lo; i <= hi; i++)
	{
		if (from[i])
		{
			Melder_free (to[i]);
			if ((to[i] = Melder_strdup (from[i])) == NULL) return 0;
		}
	}
	return 1;
}

void NUMstrings_free (char **s, long lo, long hi)
{
	long i;
	if (s == NULL) return;
	for (i=lo; i <= hi; i++) Melder_free (s[i]);
	NUMpvector_free (s, lo);
}

char **NUMstrings_copy (char **from, long lo, long hi)
{
	char **to = (char**) NUMpvector (lo, hi);
	if (to == NULL || ! NUMstrings_copyElements (from, to, lo, hi))
		NUMstrings_free (to, lo, hi);
	return to;
}

static char *appendNumberToString (char *s, long number)
{
	char buf[12], *new;
	long ncharb, nchars = 0;
	
	ncharb = sprintf (buf, "%d", number);
	if (s != NULL) nchars = strlen (s);
	new = Melder_malloc (nchars + ncharb + 1);
	if (new == NULL) return NULL;
	if (nchars > 0) strncpy (new, s, nchars);
	strncpy (new + nchars, buf, ncharb + 1);
	return new;
} 

int NUMstrings_setSequentialNumbering (char **s, long lo, long hi, 
	char *pre, long number, long increment)
{
	long i;
	
	for (i = lo; i <= hi; i++, number += increment)
	{
		char *new = appendNumberToString (pre, number);
		if (new == NULL) return 0;
		Melder_free (s[i]);
		s[i] = new;
	}
	return 1;
}

#define HIGHBYTE(x) ((unsigned char) ((x) & 0xFF))
#define LOWBYTE(x)  ((unsigned char) ((x) >> 8 & 0xFF))

/* a+b=c in radix 256 */
void NUMstring_add (unsigned char *a, unsigned char *b, unsigned char *c, long n);
void NUMstring_add (unsigned char *a, unsigned char *b, unsigned char *c, long n)
{
	int j;
	unsigned short reg = 0;
	
	for (j = n; j > 1; j--)
	{
		reg = a[j] + b[j] + HIGHBYTE (reg);
		c[j+1] = LOWBYTE (reg);
	}
}

/*
int NUMstrings_setSequential (char **s, long lo, long hi, 
	char *pre, char *start, long increment, char *post)
{
	char *proc = "NUMstrings_setSequential";
	long d, i, iset = 0, istart, startlen, nsymbolsets = 3, plast;
	long startnum = 0, prelen, postlen, setsize, maxnum, maxchar = 100;
	int symbolSetSize[4] = {0, 10, 26, 26};
	char *p, *psymbols, *pt, *pinc;
	char *pres = pre == NULL ? "" : pre; 
	char *posts = post == NULL ? "" : post;
	char *number = NULL, *inc = NULL;
	char *symbolSet[] = {"", "0123456789", "abcdefghijklmnopqrstuvwxyz", 
		"ABCDEFGHIJKLMNOPQRSTUVWXYZ"};

	// First char of start determines the symbol set
		
	for (i = 1; i <= nsymbolsets; i++)
	{
		p = strchr (symbolSet[i], start[0]);
		if (p != NULL)
		{
			iset = i; istart = p - start;
			break;
		}
	}
	
	if (iset == 0) return Melder_error ("%s: Illegal first character in start.", proc);
	if (istart == 0) iset = 1; // empty start 
	
	if (((inc = Melder_malloc (maxchar)) == NULL) ||
		((startnum = Melder_malloc (maxchar)) == NULL)) goto end;
	
	// Translate the increment into the symbol system
	
	psymbols = symbolSet[iset];
	setsize = symbolSetSize[iset];
	
	pinc = inc;
	d = increment;
	while (d > 0)
	{
		long mod = d % setsize;
		d /= setsize;
		pt = strchr (psymbols, *pinc);
		pinc++;
	}
	// Get the 'number' of the start symbol in the corresponding
	// number system and check if all start's chars are in the set.
	
	p = start;
	psymbols = symbolSet[iset];
	setsize = symbolSetSize[iset];
	while (*p)
	{
		pt = strchr (psymbols, *p);
		if (pt == NULL) return Melder_error ("%s: Illegal symbol in start.", proc); // mixed symbols
		startnum = startnum * setsize + (psymbols - pt);
		p++;
	}
	startlen = p - start; // length of start (== strlen(start))

	maxchar = MAX (100, startlen);	
	number = Melder_malloc (maxchar); // number in reverse
		
	prelen = strlen (pre);
	postlen = strlen (post);
	inclen =   
	for (i = lo; i <= hi; i++)
	{
		char *new;
		// add the strings
		for (j=1; j <= MAX (
		if (new == NULL) return 0;
		Melder_free (s[i]);
		s[i] = new;
	}
end:
	Melder_free (inc);
	Melder_free (startnum);
	return ! Melder_hasError ();
}
*/ 
char *strstr_regexp (const char *string, const char *search_regexp)
{
	char *charp = NULL, *compileMsg;
	regexp *compiled_regexp = CompileRE (search_regexp, &compileMsg, 0);
	
	if (compiled_regexp == NULL) return Melder_errorp (compileMsg);
	
	if (ExecRE(compiled_regexp, NULL, string, NULL, 0, '\0', '\0', NULL))
		charp = compiled_regexp -> startp[0];
	free (compiled_regexp);
	return charp;
}

char *str_replace_literal (char *string, const char *search, 
	const char *replace, long maximumNumberOfReplaces, long *nmatches)
{
	int len_replace, len_search, len_string, len_result;
	int i, nchar, result_nchar = 0;
	char *pos; 	/* current position / start of current match */
	char *posp; /* end of previous match */
	char *result;
	
	if (string == NULL || search == NULL || replace == NULL) return NULL;
	
	len_search = strlen (search);
	len_replace = strlen (replace);
	
	/*
		To allocate memory for 'result' only once, we have to know how many
		matches will occur.
	*/
	
	pos = string; *nmatches = 0;
	if (maximumNumberOfReplaces <= 0) maximumNumberOfReplaces = LONG_MAX;
	while ((pos = strstr (pos, search)) && *nmatches < maximumNumberOfReplaces)
	{
		pos += len_search; 
		(*nmatches)++;	
	}
	
	len_string = strlen (string);
	len_result = len_string + *nmatches * (len_replace - len_search);
	result = (char *) Melder_malloc (len_result + 1);
	result[len_result] = '\0';
	if (result == NULL) return NULL;
	
	pos = posp = string;
	for (i=1; i <= *nmatches; i++)
	{
		pos = strstr (pos, search);
		
		/* 
			Copy gap between end of previous match and start of current.
		*/
		
		nchar = (pos - posp);
		if (nchar > 0)
		{
			strncpy (result + result_nchar, posp, nchar);
			result_nchar += nchar;
		}
		
		/*
			Insert the replace string in result.
		*/
		
		strncpy (result + result_nchar, replace, len_replace);
		result_nchar += len_replace;
		
		/*
			Next search starts after the match.
		*/
		
		pos += len_search;
		posp = pos;
	}
	
	/*
		Copy gap between end of match and end of string.
	*/
	
	pos = string + len_string;
	nchar = pos - posp;
	if (nchar > 0) strncpy (result + result_nchar, posp, nchar);
	return result; 
}


char *str_replace_regexp (char *string, regexp *compiledSearchRE, 
	const char *replaceRE, long maximumNumberOfReplaces, long *nmatches)
{
	long i = maximumNumberOfReplaces > 0 ? 0 : -214748363;
	int buf_size; 					/* inclusive nul-byte */
	int buf_nchar = 0;				/* # characters in 'buf' */
	int string_length;				/* exclusive 0-byte*/
	int gap_copied = 0;
	int nchar, reverse = 0;
	char *pos; 	/* current position in 'string' / start of current match */
	char *posp; /* end of previous match */
	char *buf, *tbuf;

	*nmatches = 0;	
	if (string == NULL || compiledSearchRE == NULL || replaceRE == NULL)
		return NULL;
		
	string_length = strlen (string);
		  
	/*
		We do not know the size of the replaced string in advance,
		therefor, we allocate a replace buffer twice the size of the 
		original string. After all replaces have taken place we do a
		final realloc to the then exactly known size. 
		If during the replace, the size of the buffer happens to be too
		small (this is signalled when the last byte in the buffer equals 
		'\0' after the replace), we double its size and restart the replace.
		We cannot detect, however, whether the buffer was exactly filled up
		or it was too small. In the former case we are penalized by one 
		extra memory reallocation.
	*/
	  	
	buf_size = 2 * string_length + 1;	
	buf = (char *) Melder_malloc (buf_size);
	if (buf == NULL) return 0;
	
	/*
		The last byte of 'buf' is our buffer_full test.
		Make sure it is not '\0'.
	*/
	
	buf[buf_size - 1] = 'a';
	
	pos = posp = string;
	while (ExecRE(compiledSearchRE, NULL, pos, NULL, reverse, 
		pos == posp ? '\0' : pos[-1], '\0', NULL) && 
		i++ < maximumNumberOfReplaces)
	{
		/*
			Copy gap between the end of the previous match and the start
			of the current match.
			Check buffer overflow. 
		*/
		
		pos = compiledSearchRE -> startp[0];
		nchar = pos - posp;
		if (nchar > 0 && ! gap_copied)
		{
			if (buf_nchar + nchar > buf_size - 1)
			{
				buf_size *= 2;
				tbuf = (char *) Melder_realloc (buf, buf_size);
				if (tbuf == NULL) goto end;
				buf = tbuf;
				buf[buf_size - 1] = 'a';
			}
			strncpy (buf + buf_nchar, posp, nchar);
			buf_nchar += nchar;
		}
		
		gap_copied = 1;
		
		/*
			Do the substitution. We can only check afterwards for buffer
			overflow. SubstituteRE puts null byte at last replaced position.
		*/
		
		SubstituteRE (compiledSearchRE, replaceRE, buf + buf_nchar,
			buf_size - buf_nchar);
		
		/*
			Check buffer overflow; 
		*/
		
		if (buf[buf_size - 1] == '\0')
		{
			buf_size *= 2;
			tbuf = (char *) Melder_realloc (buf, buf_size);
			if (tbuf == NULL) goto end;
			buf = tbuf;
			buf[buf_size - 1] = 'a';
			/* redo */
			i--;
			continue;
		}
		
		/*
			Buffer is not full, get number of characters added;
		*/
		
		nchar = strlen (buf + buf_nchar);
		buf_nchar += nchar;
		
		/*
			Update next start position in search string.
		*/
		
		pos = posp = compiledSearchRE -> endp[0];
		gap_copied = 0;
		(*nmatches)++;
	}
	
	/*
		Were done, final allocation to exact size
	*/
	
	pos = string + string_length;
	nchar = pos - posp;
	buf_size = buf_nchar + nchar;
	tbuf = (char *) Melder_realloc (buf, buf_size + 1);
	if (tbuf == NULL) goto end;
	buf = tbuf;
	
	/*
		Copy the last part of 'string'.
	*/
	
	strncpy (buf + buf_nchar, posp, nchar);
	buf[buf_size] = '\0';
	
end:

	if (Melder_hasError ()) Melder_free (buf);
	return buf;
}

static char **strs_replace_literal (char **from, long lo, long hi, 
	const char *search, const char *replace, int maximumNumberOfReplaces,
	long *nmatches, long *nstringmatches)
{
	char **result;
	long i, nmatches_sub = 0;

	if (search == NULL || replace == NULL) return NULL;
	
	result = (char**) NUMpvector (lo, hi);
	if (result == NULL) goto end;
	
	*nmatches = 0; *nstringmatches = 0;
	for (i = lo; i <= hi; i++)
	{
		if (from[i] == NULL) continue;

		result[i] = str_replace_literal (from[i], search, replace, 
			maximumNumberOfReplaces, &nmatches_sub);
		if (result[i] == NULL) goto end;
		if (nmatches_sub > 0)
		{
			*nmatches += nmatches_sub;
			(*nstringmatches)++;
		}
	}
	
end:

	if (Melder_hasError ()) NUMstrings_free (result, lo, hi);
	return result;
}


static char **strs_replace_regexp (char **from, long lo, long hi, 
	const char *searchRE, const char *replaceRE, int maximumNumberOfReplaces,
	long *nmatches, long *nstringmatches)
{
	regexp *compiledRE;
	char *compileMsg, **result = NULL;
	long i, nmatches_sub = 0;

	if (searchRE == NULL || replaceRE == NULL) return NULL;
				
	compiledRE = CompileRE (searchRE, &compileMsg, 0);
	if (compiledRE == NULL) return Melder_errorp (compileMsg);

	result = (char **) NUMpvector (lo, hi);
	if (result == NULL) goto end;
	 
	*nmatches = 0; *nstringmatches = 0;
	for (i = lo; i <= hi; i++)
	{
		if (from[i] == NULL) continue;
		result[i] = str_replace_regexp (from[i], compiledRE, replaceRE, 
			maximumNumberOfReplaces, &nmatches_sub);
		if (result[i] == NULL) goto end;
		if (nmatches_sub > 0)
		{
			*nmatches += nmatches_sub;
			(*nstringmatches)++;
		}
	}
	
end:
	
	free (compiledRE);
	if (Melder_hasError ()) NUMstrings_free (result, lo, hi);
	return result;	
}

char **strs_replace (char **from, long lo, long hi, 
	const char *search, const char *replace, int maximumNumberOfReplaces,
	long *nmatches, long *nstringmatches, int use_regexp)
{
	if (use_regexp) return strs_replace_regexp (from, lo, hi, search,
		replace, maximumNumberOfReplaces, nmatches, nstringmatches);
	else return strs_replace_literal (from, lo, hi, search, replace,
		maximumNumberOfReplaces, nmatches, nstringmatches);
}

void NUMcentreRows_d (double **a, long rb, long re, long cb, long ce)
{
	long i, j;
	for (i=rb; i <= re; i++)
	{
		double rowmean = 0;
		for (j=cb; j <= ce; j++) rowmean += a[i][j];
		rowmean /= (ce - cb + 1);
		for (j=cb; j <= ce; j++) a[i][j] -= rowmean;
	}
}

void NUMcentreColumns_d (double **a, long rb, long re, long cb, long ce, 
	double *centres)
{
	long i, j;
	for (j=cb; j <= ce; j++)
	{
		double colmean = 0;
		for (i=rb; i <= re; i++) colmean += a[i][j];
		colmean /= (re - rb + 1);
		for (i=rb; i <= re; i++) a[i][j] -= colmean;
		if (centres != NULL) centres[j - cb + 1] = colmean;
	}
}

void NUMdoubleCentre_d (double **a, long rb, long re, long cb, long ce)
{
	NUMcentreRows_d (a, rb, re, cb, ce);
	NUMcentreColumns_d (a, rb, re, cb, ce, NULL);
}

void NUMnormalizeColumns_d (double **a, long nr, long nc, double norm)
{
	long i, j; double s;
	Melder_assert (norm > 0);
	for (j=1; j <= nc; j++)
	{
		for (s=0, i=1; i <= nr; i++) s += a[i][j] * a[i][j];
		if (s <= 0) continue;
		s = sqrt (norm / s);
		for (i=1; i <= nr; i++) a[i][j] *= s;
	}
}

void NUMnormalizeRows_d (double **a, long nr, long nc, double norm)
{
	long i, j; double s;
	Melder_assert (norm > 0);
	for (i=1; i <= nr; i++)
	{
		for (s=0, j=1; j <= nc; j++) s += a[i][j] * a[i][j];
		if (s <= 0) continue;
		s = sqrt (norm / s);
		for (j=1; j <= nc; j++) a[i][j] *= s;
	}
}

void NUMnormalize_d (double **a, long nr, long nc, double norm)
{
	double sq; long i, j;
	Melder_assert (norm > 0);
	for (sq=0, i=1; i <= nr; i++)
	{
		for (j=1; j <= nc; j++) sq += a[i][j] * a[i][j];
	}
	if (sq <= 0) return;
	norm = sqrt (norm / sq);
	for (i=1; i <= nr; i++)
	{
		for (j=1; j <= nc; j++) a[i][j] *= norm;
	}
}

void NUMstandardizeColumns (double **a, long rb, long re, long cb, long ce)
{
	long i, j, n = re - rb + 1;
	if (n < 2) return;
	for (j=cb; j <= ce; j++)
	{
		double ep = 0, s = 0, ave, sdev, var = 0;
		for (i=rb; i <= re; i++) s += a[i][j];
		ave = s / n;
		for (i=rb; i <= re; i++)
		{
			s = a[i][j] - ave;
			ep += s;
			var += s * s;
		}
		if (ave != 0) for (i=rb; i <= re; i++) a[i][j] -= ave;
		if (var > 0)
		{
			var = (var - ep * ep / n) / (n - 1);
			sdev = sqrt (var);
			for (i=rb; i <= re; i++) a[i][j] /= sdev;
		}
	}
}

void NUMstandardizeRows (double **a, long rb, long re, long cb, long ce)
{
	long i, j, n = ce - cb + 1;
	if (n < 2) return;
	for (i=rb; i <= re; i++)
	{
		double ep = 0, s = 0, ave, sdev, var = 0;
		for (j=cb; j <= ce; j++) s += a[i][j];
		ave = s / n;
		for (j=cb; j <= ce; j++)
		{
			s = a[i][j] - ave;
			ep += s;
			var += s * s;
		}
		if (ave != 0) for (j=cb; j <= ce; j++) a[i][j] -= ave;
		if (var > 0)
		{
			var = (var - ep * ep / n) / (n - 1);
			sdev = sqrt (var);
			for (j=cb; j <= ce; j++) a[i][j] /= sdev;
		}
	}
}

void NUMaverageColumns (double **a, long rb, long re, long cb, long ce)
{
	long i, j, n = re - rb + 1;
	if (n < 2) return;
	for (j = cb; j <= ce; j++)
	{
		double ave = 0;
		for (i = rb; i <= re; i++) ave += a[i][j];
		ave /= n;
		for (i = rb; i <= re; i++) a[i][j] = ave;
	}
	
}

void NUMcolumn_avevar (double **a, long nr, long nc, long icol, 
	double *average, double *variance)
{
	long i;
	double eps = 0, mean = 0, var = 0;
	
	Melder_assert (nr > 0 && nc > 0 && icol > 0 && icol <= nc);

	for (i = 1; i <= nr; i++)
	{
		mean += a[i][icol];
	}
	
	mean /= nr;
	
	if (average != NULL) *average = mean;
	
	if (variance == NULL) return;
	
	if (nr > 1)
	{
		for (i = 1; i <= nr; i++)
		{
			double s = a[i][icol] - mean;
			eps += s;
			var += s * s;
		}

		var = (var - eps * eps / nr) / (nr -1);
	}
	else
	{
		var = NUMundefined;
	}
	
	*variance = var;
}

void NUMcolumn2_avevar (double **a, long nr, long nc, long icol1, long icol2,
	double *average1, double *variance1, double *average2, double *variance2,
	double *covariance)
{
	long i, ndf = nr - 1;
	double eps1 = 0, eps2 = 0, mean1 = 0, mean2 = 0;
	double var1 = 0, var2 = 0, covar = 0;
	
	Melder_assert (icol1 > 0 && icol1 <= nc && icol2 > 0 && icol2 <= nc);

	for (i = 1; i <= nr; i++)
	{
		mean1 += a[i][icol1];
		mean2 += a[i][icol2];
	}
	
	mean1 /= nr;
	mean2 /= nr;
	
	if (average1 != NULL) *average1 = mean1;
	if (average2 != NULL) *average2 = mean2;

	if (variance1 == NULL && variance2 == NULL && covariance == NULL) return;
	
	if (nr > 1)
	{	
		for (i = 1; i <= nr; i++)
		{
			double s1 = a[i][icol1] - mean1;
			double s2 = a[i][icol2] - mean2;
			eps1 += s1;
			eps2 += s2;
			var1 += s1 * s1;
			var2 += s2 * s2;
			covar += s1 * s2;
		}

		var1 = (var1 - eps1 * eps1 / nr) / ndf;
		var2 = (var2 - eps2 * eps2 / nr) / ndf;
		covar /= ndf;
	}
	else
	{
		var1 = NUMundefined;
		var2 = NUMundefined;
		covar = NUMundefined;
	}
	
	if (variance1 != NULL) *variance1 = var1;
	if (variance2 != NULL) *variance2 = var2;
	if (covariance != NULL) *covariance = covar;
	if (icol1 == icol2) *covariance = *variance1;
}

#undef SQR
#define SQR(a) ((temp=(a)) == 0.0 ? 0.0 : temp*temp)

#define MACRO_pythag(TYPE)\
{\
	TYPE absa, absb, temp;\
	absa = fabs(a);\
	absb = fabs(b);\
	if (absa > absb)\
	{\
		return absa * sqrt (1.0 + SQR (absb / absa));\
	}\
	else\
	{\
		return (absb == 0.0 ? 0.0 : absb * sqrt (1.0 + SQR (absa / absb)));\
	}\
}

static double pythag_d (double a, double b)
MACRO_pythag(double)

static float pythag_f (float a, float b)
MACRO_pythag(float)

#undef MACRO_pythag
#undef SQR

void eigenSort_d (double d[], double **v, long n, int sort)
{
    long i, j, k;
	if (sort == 0) return;
    for (i=1; i < n; i++)
    {
        double temp = d[k=i];
        if (sort > 0)
		{
			for (j=i+1; j <= n; j++)
			{
				if (d[j] > temp) temp = d[k=j];
			}
		}
        else
		{
			for (j=i+1; j <= n; j++)
			{
				if (d[j] < temp) temp = d[k=j];
			}
		}
        if (k != i)
        {
            d[k] = d[i];
			d[i] = temp;
            if (v)
			{
				for (j=1; j <= n; j++)
				{
					temp = v[j][i];
					v[j][i] = v[j][k];
					v[j][k] = temp;
				}
			}
        }
    }
}

int NUMstrcmp (const char *s1, const char *s2)
{
	if (s1 == NULL || s1[0] == '\0')
	{
		if (s2 != NULL && s2[0] != '\0') return -1;
		else return 0;
	}
	else
	{
		if (s2 == NULL) return +1;
		else return strcmp (s1, s2);
	}
}

void NUMlocate_f (float *xx, long n, float x, long *index)
{
	long ju = n + 1, jm, jl = 0;
	int ascend = xx[n] >= xx[1];

	while (ju - jl > 1)
	{
		jm = (ju + jl) / 2;
		if (x >= xx[jm] == ascend) jl = jm; else ju = jm;
	}
	if (x == xx[1]) *index = 1;
	else if (x == xx[n]) *index = n - 1;
	else *index = jl;
}

void NUMlocate_d (double *xx, long n, double x, long *index)
{
	long ju = n + 1, jm, jl = 0;
	int ascend = xx[n] >= xx[1];

	while (ju - jl > 1)
	{
		jm = (ju + jl) / 2;
		if (x >= xx[jm] == ascend) jl = jm; else ju = jm;
	}
	if (x == xx[1]) *index = 1;
	else if (x == xx[n]) *index = n - 1;
	else *index = jl;
}

/*
	The following algorithm for monotone scaling is on the average
	3.5 times faster than
	Kruskal's algorithm for monotone regression (and much simpler).
	Regression is ascending
*/
void NUMmonotoneRegression_d (const double x[], long n, double xs[])
{
	double sum, xt;
	long i, j, nt;
	
	for (i = 1; i <= n; i++) xs[i] = x[i];
	for (i = 2; i <= n; i++)
	{
		if (xs[i] >= xs[i-1]) continue;
		sum = xs[i];
		nt = 1;
		for (j = 1; j <= i-1; j++)
		{
			sum += xs[i-j]; nt++;
			xt = sum / nt;
			if (j < i-1 && xt >= xs[i-j-1]) break;
		}
		for (j = i - nt + 1; j <= i; j++) xs[j] = xt;
	}
}

static int NUMdmonotoneRegressionKruskal (const double x[], long n, double xm[], int descending)
{
	long i, current = 1, nBlocks = n, *length, index = 1; int status = 0;
	double low = -1e38, high = 1e38;
	for (i=1; i <= n; i++) xm[i] = x[i];
	if ((length = NUMlvector (1, n)) == NULL) goto end;
	if (descending) { low = 1e38; high = -1e38; } 
	for (i=1; i <= n; i++) length[i] = 1;
	do
	{
		int up_satisfied, down_satisfied;
		index += current > 1 ? length[current-1] : 0;
		do
		{
			long l_current = length[current], l_join;
			long l_prev = current > 1 ? length[current-1] : 0;
			double x_current = xm[index], x_join;
			double x_prev = current > 1 ? xm[index-l_prev] : low;
			double x_next = current < nBlocks ? xm[index+l_current] : high;
			up_satisfied = descending ? x_current > x_next : x_current < x_next;
			if (! up_satisfied) /* join current and next block */
			{
				l_join = l_current + length[current+1];
				x_join = (x_current * l_current + x_next *
					length[current+1]) / l_join;
				for (i=1; i <= l_join; i++) xm[index + i - 1] = x_join;
				length[current] = l_join; nBlocks--;
				for (i=current+1; i <= nBlocks; i++) length[i] = length[i+1];
				x_current = x_join; l_current = l_join;
			}
			down_satisfied = descending ? x_current < x_prev :
				x_current > x_prev;
			if (! down_satisfied) /* join current and previous block */
			{
				l_join = l_current + l_prev;
				x_join = (x_current * l_current + x_prev * l_prev) / l_join;
				index -= l_prev;
				Melder_assert (index > 0);
				for (i=1; i <= l_join; i++) xm[index + i - 1] = x_join;
				current--; length[current] = l_join; nBlocks--;
				for (i=current+1; i <= nBlocks; i++) length[i] = length[i+1];
			}
		} while (! (up_satisfied && down_satisfied));
		current++;
	} while (current < nBlocks);
	status = 1;
end:
	NUMlvector_free (length, 1);
	return status;
}

void NUMfvector_moment2 (float v[], long lo, long hi, double *ave, double *var)
{
	double m, s, mp = v[lo], sp = 0;
	long k;
	
	for (k = lo+1; k <= hi; k++)
	{
		double xk = v[k];
		double tmp = xk - mp;
		m = mp + tmp / k;
		s = sp + tmp * (xk - m);
		mp = m; sp = s;
	}
	*ave = m;
	*var = s / (hi - lo);
}

float NUMfvector_getNorm1 (const float v[], long n)
{
	float norm = 0; long i;
	for (i=1; i <= n; i++) norm += fabs (v[i]);
	return norm;
}

float NUMfvector_getNorm2 (const float v[], long n)
{
	float norm = 0; long i;
	for (i=1; i <= n; i++) norm += v[i] * v[i];
	return sqrt (norm);
}

float NUMfvector_normalize1 (float v[], long n)
{
	float norm = 0; long i;
	for (i=1; i <= n; i++) norm += fabs (v[i]);
	for (i=1; i <= n; i++) v[i] /= norm;
	return norm;
}

float NUMfvector_normalize2 (float v[], long n)
{
	float norm = 0; long i;
	for (i=1; i <= n; i++) norm += v[i] * v[i];
	norm = sqrt (norm);
	for (i=1; i <= n; i++) v[i] /= norm;
	return norm;
}
double NUMdvector_getNorm1 (const double v[], long n)
{
	double norm = 0; long i;
	for (i=1; i <= n; i++) norm += fabs (v[i]);
	return norm;
}

double NUMdvector_getNorm2 (const double v[], long n)
{
	double norm = 0; long i;
	for (i=1; i <= n; i++) norm += v[i] * v[i];
	return sqrt (norm);
}

double NUMdvector_normalize1 (double v[], long n)
{
	double norm = 0; long i;
	for (i=1; i <= n; i++) norm += fabs (v[i]);
	for (i=1; i <= n; i++) v[i] /= norm;
	return norm;
}

double NUMdvector_normalize2 (double v[], long n)
{
	double norm = 0; long i;
	for (i=1; i <= n; i++) norm += v[i] * v[i];
	norm = sqrt (norm);
	for (i=1; i <= n; i++) v[i] /= norm;
	return norm;
}

#undef TINY

void NUMcholeskySolve (double **a, long n, double d[], double b[], double x[])
{
	long i, k;
	double sum;
	
	for (i=1; i <= n; i++) /* Solve L.y=b */
	{
		for (sum = b[i], k = i - 1; k >= 1; k--)
		{
			sum -= a[i][k] * x[k];
		}
		x[i] = sum / d[i];
	}
	for (i = n; i >= 1; i--) /* Solve L^T.x=y */
	{
		for (sum = x[i], k = i +1; k <= n; k++)
		{
			sum -= a[k][i] * x[k];
		}
		x[i] = sum / d[i];
	}
}

int NUMdeterminant_cholesky (double **a, long n, double *lnd)
{
	double *d;
	char uplo = 'U';
	long i, j, lda = n, info;

	/* Save the diagonal */
	
	if ((d = NUMdvector (1, n)) == NULL) return 0;
	for (i = 1; i <= n; i++) d[i] = a[i][i];

	/* Cholesky decomposition in lower, leave upper intact */
	
	(void) NUMlapack_dpotf2 (&uplo, &n, &a[1][1], &lda, &info);
	if (info != 0) goto end;

	/* Determinant from diagonal, restore diagonal */
		
	for (*lnd = 0, i = 1; i <= n; i++)
	{
		*lnd += log (a[i][i]);
		a[i][i] = d[i];
	}
	*lnd *= 2; /* because A = L . L' */
	
end:
	/* Restore lower from upper */
	
	for (i = 1; i < n; i++)
	{
		for (j = i + 1; j <= n; j++) a[j][i] = a[i][j];
	}
	
	NUMdvector_free (d, 1);
	return info == 0;
}
/*
int NUMdeterminant_cholesky2 (double **a, long n, double *lnd)
{
	double *d = NULL; 
	long i, j;
	
	if ((d = NUMdvector (1, n)) == NULL) return 0;
	
	if (NUMcholeskyDecomposition(a, n, d))
	{
		if (lnd != NULL)
		{
			for (*lnd = 0, i = 1; i <= n; i++) *lnd += log (d[i]);
			*lnd *= 2; 
		}
		
		for (i = 2; i <= n; i++)
		{
			for (j = 1; j < i; j++) a[i][j] = a[j][i];	
		}
	}
	NUMdvector_free (d, 1);
	return ! Melder_hasError ();
}
*/

int NUMinverse_cholesky (double **a, long n, double *lnd)
{
	char uplo = 'U', diag = 'N';
	long i, info;

	/* Cholesky decomposition in lower, leave upper intact
		Fortran storage -> use uplo='U' to get 'L'.
	*/

	(void) NUMlapack_dpotf2 (&uplo, &n, &a[1][1], &n, &info);
	if (info != 0) return 0;

	/* Determinant from diagonal, restore diagonal */

	for (*lnd = 0, i = 1; i <= n; i++)
	{
		*lnd += log (a[i][i]);
	}
	*lnd *= 2; /* because A = L . L' */

	/* Get the inverse */
	
	(void) NUMlapack_dtrtri (&uplo, &diag, &n, &a[1][1], &n, &info);

	return info == 0;
}

/*
int NUMinverse_cholesky2 (double **a, long n, double *lnd)
{
	double *d = NULL; long i, j, k;
	
	if ((d = NUMdvector (1, n)) == NULL) return 0;
	
	if (NUMcholeskyDecomposition (a, n, d))
	{
		if (lnd != NULL)
		{
			for (*lnd = 0, i = 1; i <= n; i++) *lnd += log (d[i]);
			*lnd *= 2;
		}
		for (i = 1; i <= n; i++)
		{
			a[i][i] = 1.0 / d[i];
			for (j = i + 1; j <= n; j++)
			{
				double sum = 0;
				for (k = i; k < j; k++) sum -= a[j][k] * a[k][i];
				a[j][i] = sum / d[j];
			}
		}
	}
	NUMdvector_free (d, 1);
	return ! Melder_hasError ();
}
*/


double NUMtrace (double **a, long n)
{
	long i; double trace = 0;
	for (i=1; i <= n; i++) trace += a[i][i];
	return trace;
}

double NUMtrace2 (double **a1, double **a2, long n)
{
	long i, k; double trace = 0;
	for (i=1; i <= n; i++) for (k=1; k <= n; k++) trace += a1[i][k] * a2[k][i];
	return trace;
}

int NUMeigensystem_d (double **a, long n, double **evec, double eval[],
	int sort)
{
	int status = 0, no_evecs = evec == NULL, no_evals = eval == NULL;
	double *e = NULL, **vectors = NULL;
	
	if (no_evecs && no_evals) return 1;
	
	if (((e = NUMdvector (1, n)) == NULL) ||
		(no_evals && ((eval = NUMdvector (1, n)) == NULL)) ||
		(no_evecs && ((evec = NUMdmatrix (1, n, 1, n)) == NULL))) goto end;
	
	NUMdmatrix_copyElements (a, evec, 1, n, 1, n);
	NUMtred2_d (evec, n, eval, e);
	if (no_evecs) vectors = NULL; else vectors = evec;
	if (NUMtqli_d (eval, e, n, vectors)) status = 1;
	eigenSort_d (eval, vectors, n, sort);
end:
	NUMdvector_free (e, 1);
	if (no_evals) NUMdvector_free (eval, 1);
	if (no_evecs) NUMdmatrix_free (evec, 1, 1);
	return status;
}

int NUMdominantEigenvector_d (double **mns, long n, double *q,
	double *lambda, double tolerance)
{
	long k, l, iter = 0; double val, cval = 0, *z;
	
	if ((z = NUMdvector (1, n)) == NULL) return 0;
	
	for (k=1; k <= n; k++)
	{
		for (l=1; l <= n; l++) cval += q[k] * mns[k][l] * q[l];
	}
	if (cval == 0) goto end;
	do
	{
		double znorm2 = 0;
		for (l=1; l <= n; l++)
		{
			z[l] = 0;
			for (k=1; k <= n; k++) z[l] += mns[l][k] * q[k];
		}
		for (k=1; k <= n; k++) znorm2 += z[k] * z[k];
		znorm2 = sqrt (znorm2);
		for (k=1; k <= n; k++)	q[k] = z[k] / znorm2;
		val = cval; cval = 0;
		for (k=1; k <= n; k++)
		{
			for (l=1; l <= n; l++) cval += q[k] * mns[k][l] * q[l];
		}	
	} while (fabs(cval - val) > tolerance || ++iter > 30);
end:
	*lambda = cval;
	NUMdvector_free (z, 1);
	return 1;
}

int NUMprincipalComponents_d (double **a, long n, long nComponents, double **pc)
{
	long i, j, k; int status = 0; double **evec = NULL;
	if (((evec = NUMdmatrix (1, n, 1, n)) == NULL) ||
		! NUMeigensystem_d (a, n, evec, NULL, 1)) goto end;
    for (i=1; i <= n; i++) for (j=1; j<= nComponents; j++)
    {
    	double s = 0;
		for (k=1; k <= n; k++)
		{
			s += a[k][i] * evec[k][j]; /* times sqrt(eigenvalue) ?? */
		}
		pc[i][j] = s;
	}
    status = 1;
end:
	NUMdmatrix_free (evec, 1, 1);
	return status;
}

int NUMdmatrix_into_principalComponents (double **m, long nrows, long ncols,
	long numberOfComponents, double **pc)
{
	SVD svd = NULL; long i, j, k; double **mc = NULL;
	
	Melder_assert (numberOfComponents > 0 && numberOfComponents <= ncols);
	
	if ((mc = NUMdmatrix_copy (m, 1, nrows, 1, ncols)) == NULL) return 0;
	
	/*NUMcentreColumns_d (mc, nrows, ncols);*/
	if ((svd = SVD_create_d (mc, nrows, ncols)) == NULL) goto end;
	for (i = 1; i <= nrows; i++)
	{
		for (j = 1; j <= numberOfComponents; j++)
		{
			pc[i][j] = 0;
			for (k = 1; k <= ncols; k++)
			{
				pc[i][j] += svd -> v[k][j] * m[i][k]; 
			}
		}
	}
	
end:
	NUMdmatrix_free (mc, 1, 1);
	forget (svd);
	return ! Melder_hasError ();
}

int NUMpseudoInverse_d (double **y, long nr, long nc, double **yinv,
	double tolerance)
{
	double **yc = NULL, **v = NULL, *w = NULL, wmax = 0;
	int status = 0; long i, j, k;
	if (((yc = NUMdmatrix_copy ((double **)y, 1, nr, 1, nc)) == NULL) ||
		((w = NUMdvector (1, nc)) == NULL) ||
		((v = NUMdmatrix (1, nc, 1, nc)) == NULL) ||
		! NUMsvdcmp (yc, nr, nc, w, v)) goto end;
	for (i=1; i <= nc; i++) if (w[i] > wmax) wmax = w[i];
	for (i=1; i <= nc; i++) if (w[i] < wmax * tolerance) w[i] = 0;	
	for (i=1; i <= nc; i++) for (j=1; j <= nr; j++)
	{
		double s = 0;
		for (k=1; k <= nc; k++) if (w[k]) s += v[i][k] * yc[j][k] / w[k];
		yinv[i][j] = s;
	}
	status = 1;
end:
	NUMdmatrix_free (v, 1, 1);
	NUMdvector_free (w, 1);
	NUMdmatrix_free (yc, 1, 1);
	return status;
}

int NUMsolveEquation_d (double **a, long nr, long nc, double *b,
	double tolerance, double *result)
{
	int status = 0; long j; 
	double  **v = NULL, *w = NULL, *p = NULL, wmax = 0;
	double tol = tolerance > 0 ? tolerance : NUMeps * nr;
	
	if (nr == 0 || nc == 0) return 0;
	if (((v = NUMdmatrix (1, nc, 1, nc)) == NULL) ||
		((w = NUMdvector (1, nc)) == NULL) ||
		((p = NUMdvector (1, nc)) == NULL) ||
		! NUMsvdcmp (a, nr, nc, w, v)) goto end;
	for ( j=1; j <= nc; j++) if (w[j] > wmax) wmax = w[j];
	for ( j=1; j <= nc; j++) if (w[j] < tol * wmax) w[j] = 0.0;
	if ( ! NUMsvbksb (a, w, v, nr, nc, b, p)) goto end;
	for ( j=1; j <= nc; j++) result[j] = p[j];
	status = 1;
end:
	NUMdmatrix_free (v, 1, 1);
	NUMdvector_free (w, 1);
	NUMdvector_free (p, 1);
	return status;
}

int NUMsolveEquations_d (double **a, long nr, long nc, double **b,
	long ncb, double tolerance, double **x)
{
	int status = 0; long j, k; 
	double  **v = NULL, *w = NULL, *p = NULL, *bt = NULL, wmax = 0; 
	double tol = tolerance > 0 ? tolerance : NUMeps * nr;
	if (nr <= 0 || nc <= 0 || ncb <= 0) return 0;
	if (((v = NUMdmatrix (1, nc, 1, nc)) == NULL) ||
		((w = NUMdvector (1, nc)) == NULL) ||
		((p = NUMdvector (1, nc)) == NULL) ||
		((bt = NUMdvector (1, nr)) == NULL) ||
		! NUMsvdcmp (a, nr, nc, w, v)) goto end;
	for (j=1; j <= nc; j++) if (w[j] > wmax) wmax = w[j];
	for (j=1; j <= nc; j++) if (w[j] < tol * wmax) w[j] = 0.0;
	for (k=1; k <= ncb; k++)
	{
		for (j=1; j <= nr; j++) bt[j] = b[j][k];
		if (! NUMsvbksb (a, w, v, nr, nc, bt, p)) goto end;
		for (j=1; j <= nc; j++) x[j][k] = p[j];
	}
	status = 1;
end:
	NUMdmatrix_free (v, 1, 1); NUMdvector_free (w, 1);
	NUMdvector_free (p, 1); NUMdvector_free (bt, 1);
	return status;

}

void NUMsolveNonNegativeLeastSquaresRegression (double **m, long nr, long nc,
	double *d, double tol, long itermax, double *b)
{
	long i, j, l, iter; double difsq, difsqp = 0;
	
	for (iter=1; iter <= itermax; iter++)
	{
		/*
			Fix all weights except b[j]
		*/
		
		for (j=1; j <= nc; j++)
		{
			double mjr = 0, mjmj = 0;
			for (i=1; i <= nr; i++)
			{
				double ri = d[i], mij = m[i][j];
				for (l=1; l <= nc; l++)
				{
					if (l != j) ri -= b[l] * m[i][l];
				}
				mjr += mij * ri;
				mjmj += mij * mij;
			}
			b[j] = mjr / mjmj;
			if (b[j] < 0) b[j] = 0;
		}
		
		/*
			Calculate t(b) and compare with previous result.
		*/
		
		difsq = 0;
		for (i=1; i <= nr; i++)
		{
			double dmb = d[i];
			for (j=1; j <= nc; j++)
			{
				dmb -= m[i][j] * b[j];
			}
			difsq += dmb * dmb;
		}
		if (fabs (difsq - difsqp) / difsq < tol) break;
		difsqp = difsq;
	}
}

typedef struct rtdata { double *y, *delta; } *rtdata;

/*
	f (lambda) = sum (y[i]^2 delta[i] / (delta[i]-lambda)^2, i=1..3)
	f'(lambda) = 2 * sum (y[i]^2 delta[i] / (delta[i]-lambda)^3, i=1..3)
*/

static void rtfuncd (void *data, double x, double *f, double *df)
{
	rtdata me = (rtdata) data; long i;
	*f = *df = 0;
	for (i=1; i <= 3; i++)
	{
		double t1 = (my delta[i] - x);
		double t2 = my y[i] / t1;
		double t3 = t2 * t2 * my delta[i];
		*f  += t3;
		*df += t3 * 2 / t1;
	} 	
}

int NUMsolveConstrainedLSQuadraticRegression (double **o, const double d[],
	long n, double *alpha, double *gamma)
{
	int status = 0; long i, j, k, l, n3 = 3, info;  char uplo = 'U';
	double **b = NULL, **ftinvp = NULL, **ftinv = NULL, **g = NULL, eps = 1e-5;
	double **p = NULL, *delta = NULL, *y = NULL, *diag = NULL;
	double t1, t2, t3, *chi = NULL, *otd = NULL, **ptfinv = NULL;
	double **ptfinvc = NULL, *w = NULL;
	if (((ftinv = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((b = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((g = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((p = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((delta = NUMdvector (1, n3)) == NULL) ||
		((ftinvp = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((ptfinv = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((otd = NUMdvector (1, n3)) == NULL) || 
		((ptfinvc = NUMdmatrix (1, n3, 1, n3)) == NULL) ||
		((y = NUMdvector (1, n3)) == NULL) ||
		((w = NUMdvector (1, n3)) == NULL) ||
		((chi = NUMdvector (1, n3)) == NULL) ||
		((diag = NUMdvector (1, n3)) == NULL)) goto end;

	/*
		Construct O'.O     [1..3][1..3].
	*/

	for (i=1; i <= n3; i++)
	{
		for (j=1; j <= n3; j++)
		{
			for (k=1; k <= n; k++)
			{
				ftinv[i][j] += o[k][i] * o[k][j];
			}
		}
	}
	
	/*
		Get lower triangular decomposition from O'.O and
		get F'^-1 from it (eq. (2)) (F^-1 not done ????)
	*/
	
	(void) NUMlapack_dpotf2 (&uplo, &n3, &ftinv[1][1], &n3, &info);
	if (info != 0) goto end;
	ftinv[1][2] = ftinv[1][3] = ftinv[2][3] = 0;
	
/*	
	if (! NUMcholeskyDecomposition (ftinv, n3, diag)) goto end;
	
	for (i=1; i <= 3; i++)
	{
		ftinv[i][i] = diag[i];
		for (j=i+1; j <= 3; j++)
		{
			ftinv[i][j] = 0;
		}
	}
*/	
	/*
		Construct G and its eigen-decomposition (eq. (4,5))
		Sort eigenvalues (& eigenvectors) ascending.
	*/
	
	b[3][1] = b[1][3] = -0.5; b[2][2] = 1;
	
	/*
		G = F^-1 B (F')^-1 (eq. 4)
	*/
	
	for (i=1; i <= 3; i++)
	{
		for (j=1; j <= 3; j++)
		{
			for (k=1; k <= 3; k++)
			{
				if (ftinv[k][i] != 0)
				{ 
					for (l=1; l <= 3; l++)
					{
						g[i][j] += ftinv[k][i] * b[k][l] * ftinv[l][j];
					}
				}
			}
		}
	}
	
	/*
		G's eigen-decomposition with eigenvalues sorted ascending. (eq. 5)
	*/
	
	if (! NUMeigensystem_d (g, 3, p, delta, -1)) goto end;
	
	/*
		Construct y = P'.F'.O'.d ==> Solve (F')^-1 . P .y = (O'.d)    (page 632)
		Get P'F^-1 from the transpose of (F')^-1 . P 
	*/
	
	for (i=1; i <= 3; i++)
	{
		for (j=1; j <= 3; j++)
		{
			if (ftinv[i][j] != 0)
			{
				for (k=1; k <= 3; k++)
				{
					ftinvp[i][k] += ftinv[i][j] * p[j][k];
				}
			}
		}
		for (k=1; k <= n; k++)
		{
			otd[i] += o[k][i] * d[k];
		}
	}
	
	for (i=1; i <= 3; i++)
	{
		for (j=1; j <= 3; j++)
		{
			ptfinvc[j][i] = ptfinv[j][i] = ftinvp[i][j];
		}
	}
	
	if (! NUMsolveEquation_d (ftinvp, 3, 3, otd, 1e-6, y)) goto end;
	
	/*
		The solution (3 cases) 
	*/
	
	if (fabs (y[1]) < eps)
	{
		/*
			Case 1: page 633
		*/
		
		t2 = y[2] / (delta[2] - delta[1]);
		t3 = y[3] / (delta[3] - delta[1]);
		/* +- */
		w[1] = sqrt (- delta[1] * (t2 * t2 * delta[2] + t3 * t3 * delta[3]));
		w[2] = t2 * delta[2];
		w[3] = t3 * delta[3];
		
		if (! NUMsolveEquation_d (ptfinv, 3, 3, w, 1e-6, chi)) goto end;
		
		w[1] = -w[1]; 
		if (fabs (chi[3] / chi[1]) < eps &&
			! NUMsolveEquation_d (ptfinvc, 3, 3, w, 1e-6, chi)) goto end;
	}
	else if (fabs (y[2]) < eps)
	{
		/*
			Case 2: page 633
		*/
		
		t1 = y[1] / (delta[1] - delta[2]);
		t3 = y[3] / (delta[3] - delta[2]);
		w[1] = t1 * delta[1];
		if((delta[2] < delta[3] &&
			(t2 = (t1 * t1 * delta[1] + t3 * t3 * delta[3])) < eps))
		{
			w[2] = sqrt (- delta[2] * t2); /* +- */
			w[3] = t3 * delta[3];
			if (! NUMsolveEquation_d (ptfinv, 3, 3, w, 1e-6, chi)) goto end;
			w[2] = -w[2]; 
			if (fabs (chi[3] / chi[1]) < eps &&
				! NUMsolveEquation_d (ptfinvc, 3, 3, w, 1e-6, chi)) goto end;
		}
		else if (((delta[2] < delta[3] + eps) || (delta[2] > delta[3] - eps))
			&& fabs (y[3]) < eps)
		{
			/*
				choose one value for w[2] from an infinite number
			*/
			
			w[2] = w[1];
			w[3] = sqrt (- t1 * t1 * delta[1] * delta[2] - w[2] * w[2]);
			if (! NUMsolveEquation_d (ptfinv, 3, 3, w, 1e-6, chi)) goto end;
		}
	}
	else 
	{
		/*
			Case 3: page 634 use Newton-Raphson root finder
		*/
		
		struct rtdata me; double xlambda, eps = (delta[2] - delta[1]) * 1e-6;
		me.y = y; me.delta = delta;
		
		if (! NUMrtsafe (rtfuncd, & me, delta[1] + eps, delta[2] - eps, 1e-6,
			& xlambda)) goto end;
		
		for (i=1; i <= 3; i++)
		{
			w[i] = y[i] / (1 - xlambda / delta[i]);
		}
		if (! NUMsolveEquation_d (ptfinv, 3, 3, w, 1e-6, chi)) goto end;
	}
	
	*alpha = chi[1]; *gamma = chi[3];
	
end:
	NUMdmatrix_free (ftinv, 1, 1);
	NUMdmatrix_free (b, 1, 1);
	NUMdmatrix_free (g, 1, 1);
	NUMdmatrix_free (p, 1, 1);
	NUMdvector_free (delta, 1);
	NUMdmatrix_free (ftinvp, 1, 1);
	NUMdmatrix_free (ptfinv, 1, 1);
	NUMdvector_free (otd, 1);
	NUMdmatrix_free (ptfinvc, 1, 1);
	NUMdvector_free (y, 1);
	NUMdvector_free (w, 1);
	NUMdvector_free (chi, 1);
	NUMdvector_free (diag, 1);
	return status;
}

/*
	f (b) = delta - b / (2 alpha) - sum (x[i]^2 / (c[i] - b)^2, i=1..n)
	f'(b) = - 1 / (2 alpha) + 2 * sum (x[i]^2 / (c[i] - b)^3, i=1..n)
*/
typedef struct rtdata2 { long m; double delta, alpha, *x, *c; } *rtdata2;

static void rtfuncd2 (void *data, double b, double *f, double *df)
{
	rtdata2 me = (rtdata2) data; long i;
	*df = - 0.5 / my alpha;
	*f = my delta + *df * b;
	for (i=1; i <= my m; i++)
	{
		double c1 = (my c[i] - b);
		double c2 = my x[i] / c1;
		double c2sq = c2 * c2;
		*f -= c2sq; *df += 2 * c2sq / c1;
	}
}

int NUMsolveWeaklyConstrainedLinearRegression (double **f, long n, long m,
	double phi[], double alpha, double delta, double t[])
{
	struct rtdata2 me; int status = 0;
	long i, j, k, q = 1, r = m, *indx = NULL;
	double **fc = NULL, **ut = NULL, **u = NULL;
	double *sqrtc = NULL, *c = NULL, *x = NULL;
	double xqsq = 0, tol = 1e-6,  xCx = 0, bmin, b0, eps;
	
	if (((u = NUMdmatrix (1, m, 1, m)) == NULL) ||
		((ut = NUMdmatrix (1, m, 1, m)) == NULL) ||
		((fc = NUMdmatrix_copy (f, 1, n, 1, m)) == NULL) ||
		((c = NUMdvector (1, m)) == NULL) ||
		((sqrtc = NUMdvector (1, m)) == NULL) ||
		((x = NUMdvector (1, n)) == NULL) ||
		((indx = NUMlvector (1, m)) == NULL)) goto end;
		
	for (j=1; j <= m; j++) t[j] = 0;
	
	if (alpha == 0) /* standard least squares */
	{
		if (! NUMsolveEquation_d (f, n, m, phi, tol, t)) goto end;
		status = 1; goto end;
	}
	
	/*
		Step 1: Compute U and C from the eigendecomposition F'F = UCU'
		Evaluate q, the multiplicity of the smallest eigenvalue in C
	*/
	
	if (! NUMsvdcmp (f, n, m, sqrtc, ut)) goto end; /* destroys f! */
	
	NUMindexx_d (sqrtc, m, indx);
	
	for (j=m; j > 0; j--)
	{
		double tmp = sqrtc[ indx[j] ];
		c[m-j+1] = tmp * tmp;
		for (k=1; k <= m; k++)
		{
			u[m-j+1][k] = ut[ indx[j] ][k];
		}
	}
	
	q = 1; while (q < m && (c[m-q] - c[m]) < tol) q++;
	
	/*
		step 2: x = U'F'phi
	*/
	
	for (i=1; i <= m; i++)
	{
		for (j=1; j <= m; j++)
		{
			for (k=1; k <= n; k++)
			{
				x[i] += u[j][i] * fc[k][j] * phi[k];
			}
		}
	}
	
	/*
		step 3:
	*/
	
	me.m = m;
	me.delta = delta;
	me.alpha = alpha;
	me.x = x;
	me.c = c;
	for (j=m-q+1; j <= m; j++)
	{
		xqsq += x[j] * x[j];
	}
	if (xqsq < tol) /* xqsq == 0 */
	{
		double fm, df;
		r = m - q;
		me.m = r;
		rtfuncd2 (& me, c[m], &fm, &df);
		if (fm >= 0) /* step 3.b1 */
		{
			x[r+1] = sqrt (fm);
			for (j=1; j <= r; j++)
			{
				x[j] /= c[j] - c[m];
			} 
			for (j=1; j <= r+1; j++)
			{
				for (k=1; k <= r+1; k++)
				{
					t[j] += u[j][k] * x[k];
				}
			}
			status = 1; goto end;
		}
		/* else continue with r = m - q */
	}
	
	/*
		step 3a & 3b2, determine interval lower bound for Newton-Raphson 
		root finder
	*/
	
	for (j=1; j <= r; j++)
	{
		xCx += x[j] * x[j] / c[j];
	}
	bmin = delta > 0 ? - xCx / delta : -2 * sqrt (alpha * xCx);
	eps = (c[m] - bmin) * tol;
	
	/*
		find the root of d(psi(b)/db in interval (bmin, c[m])
	*/
	if (! NUMrtsafe (rtfuncd2, & me, bmin + eps, c[m] - eps, tol, & b0))
		goto end;

	for (j=1; j <= r; j++)
	{
		for (k=1; k <= r; k++)
		{
			t[j] += u[j][k] * x[k] / (c[k] - b0);
		}
	}
	status = 1;
end:
	NUMlvector_free (indx, 1); NUMdvector_free (x, 1);
	NUMdvector_free (sqrtc, 1); NUMdvector_free (c, 1);
	NUMdmatrix_free (fc, 1, 1); NUMdmatrix_free (ut, 1, 1);
	NUMdmatrix_free (u, 1, 1);
	return status;
}

int NUMmspline (double knot[], long nKnots, long order, long i,
	double x, double *y)
{
	long ito = i + order - 1, j, k, nSplines = nKnots - order;
	double *m; 
		
	*y = 0;
	
	/*
		Find the interval where x is located. 
		M-splines of order k have degree k-1. 
		M-splines are zero outside interval [ knot[i], knot[i+order] ).
		First and last 'order' knots are equal, i.e.,
		knot[1] = ... = knot[order] && knot[nKnots-order+1] = ... knot[nKnots].
	*/
	
	Melder_assert (nSplines > 0);
	if (i > nSplines || order < 1) return 0;
	for (j=order; j <= nKnots-order+1; j++)
	{
		if (x < knot[j]) break;
	}
	if (j < i || j > i + order || j == order ||
		j > nKnots - order + 1) return 1;
	
	if (! (m = NUMdvector (i, ito))) return 0;
	
	/*
		Calculate M[i](x|1,t) according to eq.2.
	*/
	
	for (j=i; j <= ito; j++)
	{
		if (x >= knot[j] && x < knot[j+1]) m[j] = 1 / (knot[j+1] - knot[j]);
	}
	
	/*
		Iterate to get M[i](x|k,t)
	*/
	
	for (k=2; k <= order; k++)
	{
		for (j=i; j <= i + order - k; j++)
		{
			double kj = knot[j], kjpk = knot[j+k];
			if (kjpk > kj)
			{
				m[j] = k * ((x - kj) * m[j] + (kjpk - x) * m[j+1]) /
					((k - 1) * (kjpk - kj));
			}
		}
	}
	
	*y = m[i];
	NUMdvector_free (m, i);
	return 1;
}

int NUMispline (double aknot[], long nKnots, long order, long i,
	double x, double *y)
{
	long j, m, orderp1 = order + 1; double r;
	
	*y = 0;
	
	for (j=orderp1; j <= nKnots-order; j++)
	{
		if (x < aknot[j]) break;
	}
	j--;
	if (j < i) return 1;
	if (j > i + order || (j == nKnots - order && x == aknot[j]))
	{
		*y = 1; return 1;
	}
	/*
		Equation 5 in Ramsay's article contains some errors!!!
		1. the interval selection must be 'j-k <= i <= j' instead of
			'j-k+1 <= i <= j'
		2. the summation index m starts at 'i+1' instead of 'i'
	*/
	for (m=i+1; m <=j; m++)
	{
		if (! NUMmspline (aknot, nKnots, orderp1, m, x, &r)) return 0; 
		*y += (aknot[m+orderp1] - aknot[m]) * r;
	}
	*y /= orderp1;
	return 1;
}

double NUMwilksLambda (double *lambda, long from, long to)
{
	long i; double result = 1;
	for (i=from; i <= to; i++) result /= (1 + lambda[i]);
	return result;
}

double NUMfactln (int n)
{
	static double table[101];
	if (n < 0) return NUMundefined;
	if (n <= 1) return 0;
	return n > 100 ? NUMlnGamma (n + 1.0) : table[n] ? table[n] :
		(table[n] = NUMlnGamma (n + 1.0));
}

double NUMstudentP (double t, long df)
{
	double ib;
	if (df < 1) return NUMundefined;
	ib = 0.5 * NUMincompleteBeta (0.5 * df, 0.5, df / (df + t * t));
	return t < 0 ? ib : 1 - ib;
}

double NUMstudentQ (double t, long df)
{
	if (df < 1) return NUMundefined;
	return 1 - NUMstudentP (t, df);
}

double NUMfisherP (double f, long df1, long df2)
{
	if (f < 0 || df1 < 1 || df2 < 1) return NUMundefined;
	return 1 - NUMincompleteBeta (0.5 * df2, 0.5 * df1, df2 / (df2 + f * df1));
}

double NUMfisherQ (double f, long df1, long df2)
{
	if (f < 0 || df1 < 1 || df2 < 1) return NUMundefined;
	return NUMincompleteBeta (0.5 * df2, 0.5 * df1, df2 / (df2 + f * df1));
}

double NUMinvGaussQ (double p)
{
	double t, pc = p;
	if (p <= 0 || p >= 1.0) return NUMundefined;
	if (p > 0.5) pc = 1 - p;
	t = sqrt (- 2.0 * log (pc));
	t -= (2.515517 + (0.802853 + 0.010328 * t) * t) /
		(1.0 + (1.432788 + (0.189269 + 0.001308 * t) * t) * t);
	return p > 0.5 ? -t : t;
}

double NUMinvStudentQ (double p, long df)
{
	double pc = p > 0.5 ? 1 - p : p, xmin, xmax = 1, x, tol = 1e-10;
	
	if (p < 0 || p >= 1) return NUMundefined;
	
	/*
		Bracket the function f(x) = NUMstudentQ (x, df) - p.
	*/
	
	while (NUMstudentQ (xmax, df) > pc)
	{
		xmax *= 2;
	}
	
	xmin = xmax > 1 ? xmax / 2 : 0;
	
	/*
		Find zero of f(x) with Ridders' method.
	*/
	
	x = NUMridders2dw (NUMstudentQ, pc, df, xmin, xmax, tol);
	
	return p > 0.5 ? -x : x;
}

double NUMinvChiSquareQ (double p, long df)
{
	double xmin, xmax = 1, x, tol = 1e-10;
	
	if (p < 0 || p >= 1) return NUMundefined;
	
	/*
		Bracket the function f(x) = NUMchiSquareQ (x, df) - p.
	*/
	
	while (NUMchiSquareQ (xmax, df) > p)
	{
		xmax *= 2;
	}
	
	xmin = xmax > 1 ? xmax / 2 : 0;
	
	/*
		Find zero of f(x) with Ridders' method.
	*/
	
	x = NUMridders2dw (NUMchiSquareQ, p, df, xmin, xmax, tol);
		
	return x;
}

double NUMinvFisherQ (double p, long df1, long df2)
{
	double f, pw = p, tol = 1e-10;

	if (p < 0 || p >= 1 || df1 < 1 || df2 < 1) return NUMundefined;
	
	if (p == 0) return 0;
	
	/*
		F-distribution functions:
			Q (F, df1, df2) = 1 - P (F, df1, df2) =
			incompleteBeta (df2/(df2+F.df1), df2/2,df1/2).
		F (p, df1, df2) and F (1-p, df2, df1) satisfy
			Q (F (p, df1, df2), df1, df2) = p
			Q (F (1-p, df2, df1), df2, df1) = 1-p
			F (p, df1, df2) = 1 / F (1-p, df2, df1)
		It turns out handy to find f in the interval (0, 1], the upper 
		limit is reached for p = 0.5.
		Use Ridders method, which is faster than bisection.
		When (p or 1-p) and tol are of comparable size accuracy is lost.
	*/
	
	if (p < 0.5)
	{
		long t = df1; df1 = df2; df2 = t; pw = 1 - p;
	}
		
	f = NUMridders3dw (NUMfisherQ, pw, df1, df2, 0, 1, tol);
	
	return p < 0.5 ? 1 / f : f;
}

/*************** Hz <--> other freq reps *********************/

double NUMmelToHertz3 (double mel)
{
	return mel < 1000 ? mel : 1000 * (exp (mel * log10(2) / 1000) - 1);
}

double NUMhertzToMel3 (double hz)
{
	return hz < 1000 ? hz : 1000 * log10 (1 + hz / 1000) / log10 (2);
}

double NUMmelToHertz2 (double mel)
{
	return 700 * (pow (10.0, mel / 2595.0) - 1);
}

double NUMhertzToMel2 (double f)
{
	return 2595 * log10 (1 + f/700);
}

double NUMhertzToBark_traunmueller (double hz)
{
	return 26.81 * hz /(1960 + hz) -0.53;
}

double NUMbarkToHertz_traunmueller (double bark)
{
	return 1960* (bark + 0.53) / (26.28 - bark);
}

double NUMbarkToHertz_schroeder (double bark)
{
	return 650.0 * sinh (bark / 7.0);
}

double NUMbarkToHertz_zwickerterhardt (double hz)
{
	return 13 * atan (0.00076 * hz) + 3.5 * atan (hz / 7500);
}

double NUMhertzToBark_schroeder (double hz)
{
	double h650 = hz / 650;
	return 7.0 * log (h650 + sqrt (1 + h650 * h650));
}

double NUMbarkToHertz2 (double bark)
{
	return 650.0 * sinh (bark / 7.0);
}

double NUMhertzToBark2 (double hz)
{
	double h650 = hz / 650;
	return 7.0 * log (h650 + sqrt (1 + h650 * h650));
}


double NUMbladonlindblomfilter_amplitude (double zc, double z)
{
	double dz = zc - z + 0.474;
	return pow (10, 1.581 + 0.75 * dz - 1.75 * sqrt (1 + dz * dz));
}

double NUMsekeyhansonfilter_amplitude (double zc, double z)
{
	double dz = zc - z - 0.215;
	return pow (10, 0.7 - 0.75 * dz - 1.75 * sqrt (0.196 + dz * dz));
}

double NUMtriangularfilter_amplitude (double fl, double fc, double fh, double f)
{
	double a = 0;
	if (f > fl && f < fh)
	{
		a = f < fc ? (f - fl) / (fc - fl) : (fh - f) / (fh - fl);
		
		/* Normalize such that area under the filter is always 1. ???
		
		a /= 2 * (fh - fl);*/
	}
	return a;
}

double NUMformantfilter_amplitude (double fc, double bw, double f)
{
	double dq = (fc * fc - f * f) / (bw * f);
	return 1.0 / (dq * dq + 1.0);
}

/* Childers (1978), Modern Spectrum analysis, IEEE Press, 252-255) */
int NUMburg (float x[], long n, float a[], int m, float *xms)
{
	long i = 1, j; int status = 0;
	float p = 0.0;
	float *b1 , *b2 = NULL, *aa = NULL;
	
	if (((b1 = NUMfvector (1, n)) == NULL) ||
		((b2 = NUMfvector (1, n)) == NULL) ||
		((aa = NUMfvector (1, m)) == NULL)) goto end;
	
	/* (3) */
		
	for (j = 1; j <= n; j++)
	{
		p += x[j] * x[j];
	}
	
	*xms = p / n;
	if (*xms <= 0) goto end;
	
	/* (9) */
	
	b1[1] = x[1];
	b2[n - 1] = x[n];
	for (j = 2; j <= n - 1; j++)
	{
		b1[j] = b2[j - 1] = x[j];
	}
	
	for (i = 1; i <= m; i++)
	{
		/* (7) */
		
		float num = 0.0, denum = 0.0;
		for (j = 1; j <= n - i; j++)
		{
			num += b1[j] * b2[j];
			denum += b1[j] * b1[j] + b2[j] * b2[j];
		}
		
		if (denum <= 0) goto end;
		
		a[i] = 2.0 * num / denum;
		
		/* (10) */
		
		*xms *= 1.0 - a[i] * a[i];
		
		/* (5) */
		
		for (j = 1; j <= i - 1; j++)
		{
			a[j] = aa[j] - a[i] * aa[i - j];
		}
		
		if (i < m)
		{
		
			/* (8) */
			/* Watch out: i -> i+1 */
		
			for (j = 1; j <= i; j++)
			{
				aa[j] = a[j];
			}
			for (j = 1; j <= n - i - 1; j++)
			{
				b1[j] -= aa[i] * b2[j]; 
				b2[j] = b2[j + 1] - aa[i] * b1[j + 1];
			}
		}
	}
	
	status = 1;
	
end:	
	NUMfvector_free (aa, 1);
	NUMfvector_free (b2, 1);
	NUMfvector_free (b1, 1);
	for (j = i; j <= m; j++) a[j] = 0;
	return status;
}

static int NUMfmatrix_to_dBs2 (float **m, long rb, long re, long cb, long ce, 
	double ref, double factor, double dynamic_range_db)
{
	double ref_db, min2, max_db, min_db, factor10 = factor * 10;
	float max = m[rb][cb], min = max;
	long i, j;
	
	Melder_assert (ref > 0 && factor > 0 && rb <= re && cb <= ce);
	
	for (i=rb; i <= re; i++)
	{
		for (j=cb; j <= ce; j++)
		{
			if (m[i][j] > max) max = m[i][j];
			else if (m[i][j] < min) min = m[i][j];
		}
	}
	
	if (max < 0 || min < 0) return Melder_error ("NUMdmatrix_to_dBs: all "
		"matrix elements must be positive.");
	
	ref_db = factor10 * log10 (ref);
	max_db = factor10 * log10 (max) - ref_db;
	min_db = max_db - fabs (dynamic_range_db);
	min2 = ref * pow (10, min_db / factor10);
	
	if (min2 < min)
	{
		min2 = min; min_db = factor10 * log10 (min2) - ref_db;
	} 
	
	for (i=rb; i <= re; i++)
	{
		for (j=cb; j <= ce; j++)
		{
			double mij = m[i][j];
			if (mij <= min2) mij = min_db;
			else mij = factor10 * log10 (mij) - ref_db;
			m[i][j] = mij;
		}
	}	
	return 1;
}

int NUMfmatrix_to_dBs (float **m, long rb, long re, long cb, long ce, 
	double ref, double factor, double floor)
{
	double ref_db, factor10 = factor * 10;
	float max = m[rb][cb], min = max;
	long i, j;
	
	Melder_assert (ref > 0 && factor > 0 && rb <= re && cb <= ce);
	
	for (i=rb; i <= re; i++)
	{
		for (j=cb; j <= ce; j++)
		{
			if (m[i][j] > max) max = m[i][j];
			else if (m[i][j] < min) min = m[i][j];
		}
	}
	
	if (max < 0 || min < 0) return Melder_error ("NUMdmatrix_to_dBs: all "
		"matrix elements must be positive.");

	ref_db = factor10 * log10 (ref);
		
	for (i=rb; i <= re; i++)
	{
		for (j=cb; j <= ce; j++)
		{
			double mij = floor;
			if (m[i][j] > 0)
			{
				mij = factor10 * log10 (m[i][j]) - ref_db;
				if (mij < floor) mij = floor;
			}
			m[i][j] = mij;
		}
	}	
	return 1;
}

double **NUMcosinesTable_d (long first, long last, long npoints)
{
	double **m;
	long i, j;
	
	Melder_assert (0 < first && first <= last && npoints > 1);
	
	if ((m = NUMdmatrix (first, last, 1, npoints)) == NULL) return NULL;
	
	for (i = first; i <= last; i++)
	{
		double f = i * NUMpi / npoints;
		for (j = 1; j <= npoints; j++)
		{
			m[i][j] = cos (f * (j - 0.5));
		}
	}
	return m;
}

int NUMspline_f (float x[], float y[], long n, float yp1, float ypn, float y2[])
{
	float p, qn, sig, un, *u;
	long i,k;

	if ((u = NUMfvector (1, n - 1)) == NULL) return 0;
	
	if (yp1 > 0.99e30)
	{
		y2[1] = u[1] = 0.0;
	}
	else
	{
		y2[1] = -0.5;
		u[1] = (3.0 / (x[2] - x[1])) * ((y[2] - y[1]) / (x[2] - x[1]) - yp1);
	}
	
	for (i=2; i <= n-1; i++)
	{
		sig = (x[i] - x[i-1]) / (x[i+1] - x[i-1]);
		p = sig * y2[i-1] + 2.0;
		y2[i] = (sig - 1.0) / p;
		u[i] = (y[i+1] - y[i]) / (x[i+1] - x[i]) - (y[i] - y[i-1]) /
			(x[i] - x[i-1]);
		u[i] = (6.0 * u[i] / (x[i+1] - x[i-1]) - sig * u[i-1]) / p;
	}
	
	if (ypn > 0.99e30)
	{
		qn = un = 0.0;
	}
	else
	{
		qn = 0.5;
		un = (3.0 / (x[n] - x[n-1])) * (ypn - (y[n] - y[n-1]) /
			(x[n] - x[n-1]));
	}
	
	y2[n] = (un - qn * u[n-1]) / (qn * y2[n-1] + 1.0);
	for (k=n-1; k >= 1; k--)
	{
		y2[k] = y2[k] * y2[k+1] + u[k];
	}
	
	NUMfvector_free (u, 1);
	
	return 1;
}

int NUMspline_d (double x[], double y[], long n, double yp1, double ypn,
	double y2[])
{
	double p, qn, sig, un, *u;
	long i,k;

	if ((u = NUMdvector (1, n - 1)) == NULL) return 0;
	
	if (yp1 > 0.99e30)
	{
		y2[1] = u[1] = 0.0;
	}
	else
	{
		y2[1] = -0.5;
		u[1] = (3.0 / (x[2] - x[1])) * ((y[2] - y[1]) / (x[2] - x[1]) - yp1);
	}
	
	for (i=2; i <= n-1; i++)
	{
		sig = (x[i] - x[i-1]) / (x[i+1] - x[i-1]);
		p = sig * y2[i-1] + 2.0;
		y2[i] = (sig - 1.0) / p;
		u[i] = (y[i+1] - y[i]) / (x[i+1] - x[i]) - (y[i] - y[i-1]) /
			(x[i] - x[i-1]);
		u[i] = (6.0 * u[i] / (x[i+1] - x[i-1]) - sig * u[i-1]) / p;
	}
	
	if (ypn > 0.99e30)
	{
		qn = un = 0.0;
	}
	else
	{
		qn = 0.5;
		un = (3.0 / (x[n] - x[n-1])) * (ypn - (y[n] - y[n-1]) /
			(x[n] - x[n-1]));
	}
	
	y2[n] = (un - qn * u[n-1]) / (qn * y2[n-1] + 1.0);
	for (k=n-1; k >= 1; k--)
	{
		y2[k] = y2[k] * y2[k+1] + u[k];
	}
	
	NUMdvector_free (u, 1);
	
	return 1;
}

int NUMsplint_f (float xa[], float ya[], float y2a[], long n, float x, float *y)
{
	long klo, khi, k;
	float h, b, a;

	klo = 1; khi = n;
	while (khi-klo > 1)
	{
		k = (khi + klo) >> 1;
		if (xa[k] > x) khi = k;
		else klo = k;
	}
	h = xa[khi] - xa[klo];
	if (h == 0.0) return Melder_error ("NUMsplint_f: bad input value");
	a = (xa[khi] - x) / h;
	b = (x - xa[klo]) / h;
	*y = a * ya[klo] + b * ya[khi]+((a * a * a - a) * y2a[klo] +
		(b * b * b - b) * y2a[khi]) * (h * h) / 6.0;
	return 1;
}

int NUMsplint_d (double xa[], double ya[], double y2a[], long n,
	double x, double *y)
{
	long klo, khi, k;
	double h, b, a;

	klo = 1; khi = n;
	while (khi-klo > 1)
	{
		k = (khi + klo) >> 1;
		if (xa[k] > x) khi = k;
		else klo = k;
	}
	h = xa[khi] - xa[klo];
	if (h == 0.0) return Melder_error ("NUMsplint_d: bad input value");
	a = (xa[khi] - x) / h;
	b = (x - xa[klo]) / h;
	*y = a * ya[klo] + b * ya[khi]+((a * a * a - a) * y2a[klo] +
		(b * b * b - b) * y2a[khi]) * (h * h) / 6.0;
	return 1;
}

#define MACRO_NUMvector_extrema(TYPE) \
{ \
	long i; \
	TYPE tmin = v[lo]; \
	TYPE tmax = tmin; \
	for (i = lo + 1; i <= hi; i++) \
	{ \
		if (v[i] < tmin) tmin = v[i]; \
		else if (v[i] > tmax) tmax = v[i]; \
	} \
	*min = tmin; *max = tmax; \
}

void NUMlvector_extrema (long v[], long lo, long hi, double *min, double *max)
MACRO_NUMvector_extrema (long)

void NUMfvector_extrema (float v[], long lo, long hi, double *min, double *max)
MACRO_NUMvector_extrema (float)

void NUMivector_extrema (int v[], long lo, long hi, double *min, double *max)
MACRO_NUMvector_extrema (int)

void NUMdvector_extrema (double v[], long lo, long hi, double *min, double *max)
MACRO_NUMvector_extrema (double)

void NUMsvector_extrema (short v[], long lo, long hi, double *min, double *max)
MACRO_NUMvector_extrema (short)

#undef MACRO_NUMvector_extrema

#define MACRO_NUMclip(TYPE)\
{\
	long i; \
	TYPE tmin = min, tmax = max;\
	for (i = lo; i <= hi; i++)\
	{\
		if (v[i] < tmin) v[i] = tmin;\
		else if (v[i] > tmax) v[i] = tmax;\
	}\
}


void NUMlvector_clip (long v[], long lo, long hi, double min, double max)
MACRO_NUMclip (long)

void NUMfvector_clip (float v[], long lo, long hi, double min, double max)
MACRO_NUMclip (float)

void NUMivector_clip (int v[], long lo, long hi, double min, double max)
MACRO_NUMclip (int)

void NUMdvector_clip (double v[], long lo, long hi, double min, double max)
MACRO_NUMclip (double)

void NUMsvector_clip (short v[], long lo, long hi, double min, double max)
MACRO_NUMclip (short)

#undef MACRO_NUMclip

#define MACRO_NUMmatrix_extrema(TYPE)\
{\
	long i, j;\
	TYPE xmin, xmax;\
	xmax = xmin = x[rb][cb];\
	for (i = rb; i <= re; i++)\
	{\
		for (j = cb; j <= ce; j++)\
		{\
			TYPE t = x[i][j];\
			if (t < xmin) xmin = t; \
			else if (t > xmax) xmax = t;\
		}\
	}\
	*min = xmin; *max = xmax;\
}

void NUMdmatrix_extrema (double **x, long rb, long re, long cb, long ce, 
	double *min, double *max)
MACRO_NUMmatrix_extrema(double)

void NUMfmatrix_extrema (float **x, long rb, long re, long cb, long ce, 
	double *min, double *max)
MACRO_NUMmatrix_extrema(float)

void NUMlmatrix_extrema (long **x, long rb, long re, long cb, long ce, 
	double *min, double *max)
MACRO_NUMmatrix_extrema(long)

void NUMimatrix_extrema (int **x, long rb, long re, long cb, long ce, 
	double *min, double *max)
MACRO_NUMmatrix_extrema(int)

void NUMsmatrix_extrema (short **x, long rb, long re, long cb, long ce, 
	double *min, double *max)
MACRO_NUMmatrix_extrema(short)

#undef MACRO_NUMmatrix_extrema

#define AREA2(ax,ay,bx,by,px,py)  ((bx-ax) * (py-ay) - (px-ax) * (by-ay))

int NUMpointLinePosition (double ax, double ay, double bx, double by,
	double px, double py)
{
	double area2 = AREA2(ax, ay, bx, by, px, py);
	/*
		Not very sophisticated.
	*/
	return area2 < 0 ? -1 : (area2 > 0 ? 1 : 0); 
}

int NUMsegmentsIntersection (double ax, double ay, double bx, double by, 
	double cx, double cy, double dx, double dy, double *ix, double *iy)
{
	double ba_min, ba_max, dc_min, dc_max;
	double ba_dx, ba_dy, dc_dx, dc_dy;
	double area2, s, t, num, denom;

	/*
		Prescan: can segments possibly intersect?
	*/
	
	if (ax < bx)
	{
		ba_min = ax; ba_max = bx;
	}
	else
	{
		ba_min = bx; ba_max = ax;
	}
	if (cx < dx)
	{
		dc_min = cx; dc_max = dx;
	}
	else
	{
		dc_min = dx; dc_max = cx;
	}
	if (dc_min >= ba_max || dc_max <= ba_min) return 0;
	
	if (ay < by)
	{
		ba_min = ay; ba_max = by;
	}
	else
	{
		ba_min = by; ba_max = ay;
	}
	if (cy < dy)
	{
		dc_min = cy; dc_max = dy;
	}
	else
	{
		dc_min = dy; dc_max = cy;
	}
	if (dc_min >= ba_max || dc_max <= ba_min) return 0;

	/*
		Find out where they intersect.
	*/
		
	ba_dx = bx - ax;
	ba_dy = by - ay;
	dc_dx = dx - cx;
	dc_dy = dy - cy;
	denom = ba_dy * dc_dx - dc_dy * ba_dx;
		
	if (denom == 0) /* Not sophisticated for small scale */
	{
		/*
			lines ba and dc are parallel.
			Are they also collinear?
		*/
		
		area2 = AREA2(ax, ay, bx, by, cx, cy);		
		if (area2 == 0)
		{
			/* Yes: collinear */
		
		}
		
		/*
			No: no intersection
		*/
		
		return 0;
	}
	
	num = ax * dc_dy + cx * (ay - dy) + dx * (cy - ay);
	if (num == 0)
	{
		*ix = ax; *iy = ay;
		return 2;	
	}
	else if (num == denom)
	{
		*ix = bx; *iy = by;
		return 2;	
	}
	
	s = num / denom;
	
	num = ax * (cx - bx) + bx * (ay -cy) + cx * ba_dy;
	if (num == 0)
	{
		*ix = cx; *iy = cy;
		return 2;	
	}
	else if (num == denom)
	{
		*ix = dx; *iy = dy;
		return 2;	
	}
	
	t = num / denom;
	*ix = ax + s * ba_dx;
	*iy = ay + s * ba_dy;
	
	return (s > 0 && s < 1 && t > 0 && t < 1) ? 1 : 0;	
}

#undef AREA2
/* lines_intersect:  AUTHOR: Mukesh Prasad 
  * 
  *   This function computes whether two line segments, 
  *   respectively joining the input points (x1,y1) -- (x2,y2) 
  *   and the input points (x3,y3) -- (x4,y4) intersect. 
  *   If the lines intersect, the output variables x, y are 
  *   set to coordinates of the point of intersection. 
  * 
  *   All values are in integers.  The returned value is rounded 
  *   to the nearest integer point. 
  * 
  *   If non-integral grid points are relevant, the function 
  *   can easily be transformed by substituting floating point 
  *   calculations instead of integer calculations. 
  * 
  *   Entry 
  *        x1, y1,  x2, y2   Coordinates of endpoints of one segment. 
  *        x3, y3,  x4, y4   Coordinates of endpoints of other segment. 
  * 
  *   Exit 
  *        x, y              Coordinates of intersection point. 
  * 
  *   The value returned by the function is one of: 
  * 
  *        DONT_INTERSECT    0 
  *        DO_INTERSECT      1 
  *        COLLINEAR         2 
  * 
  * Error conditions: 
  * 
  *     Depending upon the possible ranges, and particularly on 16-bit 
  *     computers, care should be taken to protect from overflow. 
  * 
  *     In the following code, 'long' values have been used for this 
  *     purpose, instead of 'int'. 
  * 
  */ 

 #define DONT_INTERSECT    0 
 #define DO_INTERSECT      1 
 #define COLLINEAR         2 

/************************************************************** 
  *                                                            * 
  *    NOTE:  The following macro to determine if two numbers  * 
  *    have the same sign, is for 2's complement number        * 
  *    representation.  It will need to be modified for other  * 
  *    number systems.                                         * 
  *                                                            * 
  **************************************************************/ 

#define SAME_SIGNS(a,b) \
   (((long) ((unsigned long) a ^ (unsigned long) b)) >= 0 ) 

static int lines_intersect(long x1, long y1, long x2, long y2, 
                     long x3, long y3, long x4, long y4, 
                                         long *x, long *y) 
 { 
     long a1, a2, b1, b2, c1, c2; /* Coefficients of line eqns. */ 
     long r1, r2, r3, r4;         /* 'Sign' values */ 
     long denom, offset, num;     /* Intermediate values */ 

    /* Compute a1, b1, c1, where line joining points 1 and 2 
      * is "a1 x  +  b1 y  +  c1  =  0". 
      */ 

    a1 = y2 - y1; 
    b1 = x1 - x2; 
    c1 = x2 * y1 - x1 * y2; 

    /* Compute r3 and r4. */ 
     r3 = a1 * x3 + b1 * y3 + c1; 
     r4 = a1 * x4 + b1 * y4 + c1; 

    /* Check signs of r3 and r4.  If both point 3 and point 4 lie on 
      * same side of line 1, the line segments do not intersect. 
      */ 

    if ( r3 != 0 && r4 != 0 && SAME_SIGNS( r3, r4 )) 
         return ( DONT_INTERSECT ); 

    /* Compute a2, b2, c2 */ 

    a2 = y4 - y3; 
    b2 = x3 - x4; 
    c2 = x4 * y3 - x3 * y4; 

    /* Compute r1 and r2 */ 

    r1 = a2 * x1 + b2 * y1 + c2; 
    r2 = a2 * x2 + b2 * y2 + c2; 

    /* Check signs of r1 and r2.  If both point 1 and point 2 lie 
      * on same side of second line segment, the line segments do 
      * not intersect. 
      */ 

    if ( r1 != 0 && r2 != 0 && SAME_SIGNS( r1, r2 )) 
         return ( DONT_INTERSECT ); 

    /* Line segments intersect: compute intersection point.  
      */ 

    denom = a1 * b2 - a2 * b1; 
    if ( denom == 0 ) 
         return ( COLLINEAR ); 
    offset = denom < 0 ? - denom / 2 : denom / 2; 

    /* The denom/2 is to get rounding instead of truncating.  It 
      * is added or subtracted to the numerator, depending upon the 
      * sign of the numerator. 
      */ 

    num = b1 * c2 - b2 * c1; 
    *x = ( num < 0 ? num - offset : num + offset ) / denom; 

    num = a2 * c1 - a1 * c2; 
    *y = ( num < 0 ? num - offset : num + offset ) / denom; 

    return ( DO_INTERSECT );
 } 

void NUMgetEllipseBoundingBox (double a, double b, double cospsi, 
	double *width, double *height)
{

	Melder_assert (cospsi>= -1 && cospsi <= 1);
		
	if (cospsi == 1)
	{
		/*
			a-axis along x-axis
		*/
		
		*width = a;
		*height = b;
	}
	else if (cospsi == 0)
	{
		/*
			a-axis along y-axis
		*/
		
		*width = b;
		*height = a;
	}
	else
	{
		double psi = acos (cospsi), sn = sin (psi);
		double phi = atan2 (-b * sn, a * cospsi);
		*width = fabs (a * cospsi * cos (phi) - b * sn * sin (phi));
		phi = atan2 (b *cospsi , a * sn);
		*height = fabs (a * sn * cos (phi) + b * cospsi * sin (phi));
	}
}

static void NUMgetLinearTransform (double x1, double x2, double x3, double x4,
	double *a, double *b)
{
	Melder_assert (x1 < x2 && x3 < x4);
	*a = (x3 - x4) / (x1 - x2);
	*b = x3 - *a * x1;
}

static double NUMtransformLinear (double x, double a, double b)
{
	return a * x + b;
}

#undef MAX
#undef MIN

/* End of file NUM2.c */