#!/usr/local/bin/perl

# ----------------------------------------------------------------
# John Kerl
# kerl.john.r@gmail.com
# 2005-07-26
#
# This is an example of numerical series estimation.
#
# The Maclaurin series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! + ...
#
# Output:
#
#   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
#   0.0100000   0.0100000   0.0100000   0.0100000   0.0100000
#   0.0200000   0.0200000   0.0200000   0.0200000   0.0200000
#   0.0300000   0.0299999   0.0299999   0.0299999   0.0299999
#   ...
#   1.5400000   0.6025856   1.0085661   0.8100019   0.9443573
#   1.5500000   0.5879990   1.0183314   0.7981013   0.9560548
#   1.5600000   0.5729318   1.0289059   0.7848086   0.9703120
#   1.5700000   0.5573780   1.0403430   0.7699696   0.9876044
#
# The first column is x; the second and subsequent columns are increasingly
# higher-order approximations to sin(x).
#
# This illustrates the rapid convergence of the series near 0, with slower
# convergence farther away.
# ----------------------------------------------------------------

$pi = 4 * atan2(1,1);
$kmax = 10;
$kmax = $ARGV[0] if (@ARGV);
$xmin = 0.0;
$xmax = $pi/2;
$dx   = 0.01;

$k = 0;

for ($x = $xmin; $x <= $xmax; $x += $dx) {
	$expsum = 0.0;
	$kfact  = 1;
	$xpower = 1.0;
	$sign = 1.0;
	for ($k = 1; $k < $kmax; $k++) {
		$kfact  *= $k;
		next unless ($k & 1);
		$xpower *= $x ** $k;
		$expterm = $sign * $xpower / $kfact;
		$expsum += $expterm;

		printf " %11.7f", $expsum;
		$sign = -$sign;
	}
	printf "\n";
}