Page - 81 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3.7 DoubleandTriple Integrals 81 def test_midpoint_double(): """Test that a linear function is integrated exactly.""" def f(x, y): return 2*x + y a = 0; b = 2; c = 2; d = 3 import sympy x, y = sympy.symbols(’x y’) I_expected = sympy.integrate(f(x, y), (x, a, b), (y, c, d)) # Test three cases: nx < ny, nx = ny, nx > ny for nx, ny in (3, 5), (4, 4), (5, 3): I_computed1 = midpoint_double1(f, a, b, c, d, nx, ny) I_computed2 = midpoint_double2(f, a, b, c, d, nx, ny) tol = 1E-14 #print I_expected, I_computed1, I_computed2 assert abs(I_computed1 - I_expected) < tol assert abs(I_computed2 - I_expected) < tol Lettestfunctionsspeakup? Ifwecall theabovetest_midpoint_doublefunctionandnothinghappens,our implementationsare correct. However, it is somewhat annoying tohave a func- tion that is completely silentwhen itworks–arewesureall thingsareproperly computed? During development it is therefore highly recommended to insert a print statement such that we canmonitor the calculations and be convinced that the test functiondoeswhatwewant. Since a test function should not have anyprint statement,we simply comment it out aswehavedone in the function listedabove. The trapezoidalmethodcanbeusedasalternative for themidpointmethod.The derivationof a formula for thedouble integral and the implementations followex- actly the same ideas asweexplainedwith themidpointmethod,but therearemore terms to write in the formulas. Exercise 3.13 asks you to carry out the details. That exercise is a very good test on your understanding of themathematical and programmingideas in thepresent section. 3.7.2 TheMidpointRuleforaTripleIntegral Theory Once amethod thatworks for a one-dimensional problem is generalized to twodimensions, it is usuallyquite straightforward to extend themethod to three dimensions. Thiswillnowbedemonstratedforintegrals.Wehavethetriple integral bZ a dZ c fZ e g.x;y;z/dzdydx