#!/usr/bin/python

# ----------------------------------------------------------------
# John Kerl
# kerl.john.r@gmail.com
# 2005-07-06
#
# Plots a function along with its estimated and exact first and second
# derivatives.
# ----------------------------------------------------------------
# This software is released under the terms of the GNU GPL.
# Please see LICENSE.txt in the same directory as this file.
# ================================================================

from __future__ import division # 1/2 = 0.5, not 0.
from math import *

# ----------------------------------------------------------------
def f(x):
	#return  sin(x)
	#return exp(x)
	return  (1 + x**2)**0.5
def fp(x):
	#return  cos(x)
	#return exp(x)
	return  x / ((1 + x**2)**0.5)
def fpp(x):
	#return -sin(x)
	#return exp(x)
	return  1 / ((1 + x**2)**1.5)

# ----------------------------------------------------------------
xlo = -4.0
xhi =  4.0
nx  =  800
dx  = (xhi - xlo) / nx

x = xlo
for i in range(0, nx):

	# Exact
	y   = f(x)
	yp  = fp(x)
	ypp = fpp(x)

	# Estimated
	h = 0.5 * dx
	xf = x + h
	xb = x - h

	ypest = (f(xf) - f(xb)) / (2.0*h)
	yppest = (f(xf) - 2*f(x) + f(xb)) / (h*h)

	diffp  = yp  - ypest
	diffpp = ypp - yppest


	#print "%11.6f %11.6f %11.6f %11.6f %11.6f %11.6f" % (x, y, yp, ypest, ypp, yppest)
	print "%11.6f  %11.6f  %11.6f %11.6f %8.2e  %11.6f %11.6f %8.2e" % (x, y, yp, ypest, diffp, ypp, yppest, diffpp)

	x += dx