__author__ = "Marie E. Rognes (meg@simula.no)"
__copyright__ = "Copyright (C) 2012 Marie Rognes"
__license__  = "GNU LGPL version 3 or any later version"

import math

# Define e_h as a function of f, the derivative of f: df, x and h
def e_h(f, df, x, h):
    return df(x) - (f(x+h) - f(x))/h

# Define E_h as a function of f, the derivative of f: df, x and h
def E_h(f, df, x, h):
    return df(x) - (f(x+h) - f(x-h))/(2*h)

# Define a function that prints a pretty LaTeX table of the results
def print_table(f, df, approx, x, hs, name):
    print "\\begin{table}[h]"
    print "\\begin{tabular}{lllll}"
    print "\\toprule"
    title = "$%s$ & $%s$ & $%s$ & $%s$ & $%s$ \\\\" \
        % ("h", name, "%s/h" % name, "|%s|/h" % name, "|%s|/h^2" % name)
    print title
    print "\\midrule"
    for h in hs:
        d_h = approx(f, df, x, h)
        print "%.4f&" % h,
        print "%.4f& " % d_h,
        print "%.4f&" % (d_h/h),
        print "%.4f&" % (abs(d_h)/h),
        print "%.4f \\\\" % (abs(d_h)/(h**2))
    print "\\bottomrule"
    print "\\end{tabular}"
    print "\\end{table}"

# Define the sample function f = e^x
def f(x):
    return math.exp(x)

# Define the derivative of f: df = e^x
def df(x):
    return math.exp(x)

# Define a series of h's:
hs = [1.0/n for n in (2, 4, 8)]

# Let
x = 1.

print "f = e^x:"
print_table(f, df, e_h, x, hs, "e_h")
print_table(f, df, E_h, x, hs, "E_h")

# Define sample function g = ln(x)
def g(x):
    return math.log(x)

# Define derivative of g: dg = 1.0/x
def dg(x):
    return 1.0/x


print
print "g = ln(x):"
print_table(g, dg, e_h, x, hs, "e_h")
print_table(g, dg, E_h, x, hs, "E_h")