#!/usr/bin/env python
"""Simple trapezoid-rule integrator."""
import numpy as np
def trapz(x, y):
"""Simple trapezoid integrator for sequence-based innput.
Inputs:
- x,y: arrays of the same length (and more than one element). If the two
inputs have different lengths, a ValueError exception is raised.
Output:
- The result of applying the trapezoid rule to the input, assuming that
y[i] = f(x[i]) for some function f to be integrated.
Minimally modified from matplotlib.mlab."""
# Sanity checks.
#@
#@ Hint: if the two inputs have mismatched lengths or less than 2
#@ elements, we raise ValueError with an explanatory message.
if len(x) != len(y): #@
raise ValueError('x and y must have the same length') #@
if len(x) < 2: #@
raise ValueError('x and y must have > 1 element') #@
# Efficient application of trapezoid rule via numpy
#@
#@ Hint: think of using numpy slicing to compute the moving difference in
#@ the basic trapezoid formula.
return 0.5*((x[1:]-x[:-1])*(y[1:]+y[:-1])).sum() #@
def trapzf(f,a,b,npts=100):
"""Simple trapezoid-based integrator.
Inputs:
- f: function to be integrated.
- a,b: limits of integration.
Optional inputs:
- npts(100): the number of equally spaced points to sample f at, between
a and b.
Output:
- The value of the trapezoid-rule approximation to the integral."""
#@ Hint: you will need to apply the function f to easch element of the
#@ vector x. What are several ways to do this? Can you profile them to see
#@ what differences in timings result for long vectors x?
# Generate an equally spaced grid to sample the function.
x = np.linspace(a,b,npts)#@
# For an equispaced grid, the x spacing can just be read off from the first
# two points and factored out of the summation.
dx = x[1]-x[0]#@
# Sample the input function at all values of x
#@
#@ Hint: you need to make an array out of the evaluations, and the python
#@ builtin 'map' function can come in handy.
y = np.array(map(f,x))#@
# Compute the trapezoid rule sum for the final result
return 0.5*dx*(y[1:]+y[:-1]).sum()#@
#-----------------------------------------------------------------------------
# Tests
#-----------------------------------------------------------------------------
import nose, nose.tools as nt
import numpy.testing as nptest
# A simple function for testing
def square(x): return x**2
def test_err():
"""Test that mismatched inputs raise a ValueError exception."""
nt.assert_raises(ValueError,trapz,range(2),range(3))
def test_call():
"Test a direct call with equally spaced samples. "
x = np.linspace(0,1,100)
y = np.array(map(square,x))
nptest.assert_almost_equal(trapz(x,y),1./3,4)
def test_square():
"Test integrating the square() function."
nptest.assert_almost_equal(trapzf(square,0,1),1./3,4)
def test_square2():
"Another test integrating the square() function."
nptest.assert_almost_equal(trapzf(square,0,3,350),9.0,4)
# If called from the command line, run all the tests
if __name__ == '__main__':
# This call form is ipython-friendly
nose.runmodule(argv=['-s','--with-doctest'],
exit=False)
