#!/usr/local/bin/perl

# ----------------------------------------------------------------
# John Kerl
# kerl.john.r@gmail.com
# 2005-03-29
#
# This is an example of numerical path integration.  Here, the numerical
# estimation is done in terms of dx and dy, rather than in terms of dt.
#   C : x^2 + y^2 = 9, counterclockwise closed loop.
#   x  =  3 cos t   y  = 3 sin t
#   path integral -2y dx + x^2 dy
#   f(x,y) = -2y   g(x,y) = x^2
#   path integral f(x(t),y(t)) dx + g(x(t),y(t)) dy   for   t from 0 to 2 pi.
# Analytically, one expects 18 pi.
# ----------------------------------------------------------------

$pi  =  4.0 * atan2(1.0,1.0);
$tlo =  0.0;
$thi =  2.0 * $pi;

$nt  = 4000;
$nt  = $ARGV[0] if (@ARGV);
$dt  = ($thi - $tlo) / $nt;

# Alternatively:
#$dt  = 0.001;

$sum = 0.0;
for ($t = $tlo; $t < $thi; $t += $dt) {
	$t1 = $t;
	$t2 = $t + $dt;

	$x1 = _x($t1);
	$y1 = _y($t1); # Can't just write y($t1) in Perl.
	$x2 = _x($t2);
	$y2 = _y($t2);

	$dx = $x2 - $x1;
	$dy = $y2 - $y1;

	$sum += f($x1, $y1) * $dx + g($x1, $y1) * $dy;
}
printf "%11.6f\n", $sum;

# ----------------------------------------------------------------
sub _x {
	my ($t) = @_;
	return 3.0 * cos($t);
}

# ----------------------------------------------------------------
sub _y {
	my ($t) = @_;
	return 3.0 * sin($t);
}

# ----------------------------------------------------------------
sub f {
	my ($x, $y) = @_;
	return -2.0 * $y;
}

# ----------------------------------------------------------------
sub g {
	my ($x, $y) = @_;
	return $x * $x;
}