CDS: Numerical Methods Assignments¶

See lecture notes and documentation on Brightspace for Python and Jupyter basics. If you are stuck, try to google or get in touch via Discord.
Solutions must be submitted via the Jupyter Hub.
Make sure you fill in any place that says YOUR CODE HERE or "YOUR ANSWER HERE".

Submission¶

Name all team members in the the cell below
make sure everything runs as expected
restart the kernel (in the menubar, select Kernel$\rightarrow$Restart)
run all cells (in the menubar, select Cell$\rightarrow$Run All)
Check all outputs (Out[*]) for errors and resolve them if necessary
submit your solutions in time (before the deadline)

team_members = "Koen Vendrig, Kees van Kempen"

Composite Numerical Integration: Trapezoid and Simpson Rules¶

In the following we will implement the composite trapezoid and Simpson rules to calculate definite integrals. These rules are defined by

\begin{align} \int_a^b \, f(x)\, dx &\approx \frac{h}{2} \left[ f(a) + 2 \sum_{j=1}^{n-1} f(x_j) + f(b) \right] &\text{trapezoid} \\ &\approx \frac{h}{3} \left[ f(a) + 2 \sum_{j=1}^{n/2-1} f(x_{2j}) + 4 \sum_{j=1}^{n/2} f(x_{2j-1}) + f(b) \right] &\text{Simpson} \end{align}

with $a = x_0 < x_1 < \dots < x_{n-1} < x_n = b$ and $x_k = a + kh$. Here $k = 0, \dots, n$ and $h = (b-a) / n$ is the step size.

In [1]:

import numpy as np
import scipy.integrate

# And for printing the lambdas:
import inspect

Task 1¶

Implement both integration schemes as Python functions $\text{trapz(yk, dx)}$ and $\text{simps(yk, dx)}$. The argument $\text{yk}$ is an array of length $n+1$ representing $y_k = f(x_k)$ and $\text{dx}$ is the step size $h$. Compare your results with Scipy's functions $\text{scipy.integrate.trapz(yk, xk)}$ and $\text{scipy.integrate.simps(yk, xk)}$ for a $f(x_k)$ of your choice.

Try both even and odd $n$. What do you see? Why?

Hint: go to the Scipy documentation. How are even and odd $n$ handled there?

In [2]:

def trapz(yk, dx):
    a, b = yk[0], yk[-1]
    h = dx
    integral = h/2*(a + 2*np.sum(yk[1:-1]) + b)
    return integral
    
def simps(yk, dx):
    a, b = yk[0], yk[-1]
    h = dx
    # Instead of summing over something with n/2, we use step size 2,
    # thus avoiding any issues with 2 ∤ n.
    integral = h/3*(a + 2*np.sum(yk[2:-1:2]) + 4*np.sum(yk[1:-1:2]) + b)
    return integral

In [3]:

def compare_integration(f, a, b, n):
    # Let's check whether f is callable.
    # TODO: Improve checks on f, or not.
    assert callable(f)
    
    h    = (b - a)/n
    xk   = np.linspace(a, b, n + 1)
    yk   = f(xk)
    
    print("For function", inspect.getsource(f))
    print("for boundaries a =", a, ", b =", b, "and steps n =", n, "the algorithms say:")
    print("trapezoid:\t\t", trapz(yk, h))
    print("Simpson:\t\t", simps(yk, h))
    print("scipy.integrate.trapz:\t", scipy.integrate.trapz(yk, xk))
    print("scipy.integrate.simps:\t", scipy.integrate.simps(yk, xk))
    print()

In [4]:

# We need a function to integrate, so here we go.
f = lambda x: x**2

n    = 100001
a, b = 0, 1

In [5]:

compare_integration(f, a, b, n)
compare_integration(f, a, b, n + 1)

For function f = lambda x: x**2

for boundaries a = 0 , b = 1 and steps n = 100001 the algorithms say:
trapezoid:		 0.33333333334999976
Simpson:		 0.3333300000666658
scipy.integrate.trapz:	 0.33333333334999965
scipy.integrate.simps:	 0.3333333333333335

For function f = lambda x: x**2

for boundaries a = 0 , b = 1 and steps n = 100002 the algorithms say:
trapezoid:		 0.3333333333499994
Simpson:		 0.33333333333333337
scipy.integrate.trapz:	 0.3333333333499993
scipy.integrate.simps:	 0.3333333333333333

Task 2¶

Implement at least one test function for each of your integration functions.

In [6]:

# In the comparison of n even and n odd, and the testing of the integrations,
# we have already tested the functions, but as it is asked, here we go again.

def test_trapz():
    fun  = lambda x: x**3 + 6*x
    a, b = 2, 16
    n    = 82198
    
    h    = (b - a)/n
    xk   = np.linspace(a, b, n + 1)
    yk   = f(xk)

    trapz_our   = trapz(yk, h)
    trapz_scipy = scipy.integrate.trapz(yk, xk)
    
    print("For function f(x) = x^3 + 6x")
    print("for boundaries a =", a, ", b =", b, "and steps n =", n, "the algorithms say:")
    print("trapezoid:\t\t", trapz_our)
    print("scipy.integrate.trapz:\t", trapz_scipy)
    print("with difference trapz(yk, h) - scipy.integrate.trapz(yk, xk) =", trapz_our - trapz_scipy)
    print()
    
def test_simps():
    fun  = lambda x: -x**3 + 6*x
    a, b = 2, 17
    n    = 82228
    
    h    = (b - a)/n
    xk   = np.linspace(a, b, n + 1)
    yk   = f(xk)

    simps_our   = simps(yk, h)
    simps_scipy = scipy.integrate.simps(yk, xk)
    
    print("For function f(x) = -x^3 + 6x")
    print("for boundaries a =", a, ", b =", b, "and steps n =", n, "the algorithms say:")
    print("Simpson:\t\t", simps_our)
    print("scipy.integrate.simps:\t", simps_scipy)
    print("with difference simps(yk, h) - scipy.integrate.simps(yk, xk) =", simps_our - simps_scipy)
    print()
    
test_trapz()
test_simps()

For function f(x) = x^3 + 6x
for boundaries a = 2 , b = 16 and steps n = 82198 the algorithms say:
trapezoid:		 1362.6666667343538
scipy.integrate.trapz:	 1362.6666667343543
with difference trapz(yk, h) - scipy.integrate.trapz(yk, xk) = -4.547473508864641e-13

For function f(x) = -x^3 + 6x
for boundaries a = 2 , b = 17 and steps n = 82228 the algorithms say:
Simpson:		 1635.0
scipy.integrate.simps:	 1635.0000000000002
with difference simps(yk, h) - scipy.integrate.simps(yk, xk) = -2.2737367544323206e-13

Task 3¶

Study the accuracy of these integration routines by calculating the following integrals for a variety of step sizes $h$:

$\int_0^1 \, x\, dx$
$\int_0^1 \, x^2\, dx$
$\int_0^1 \, x^\frac{1}{2}\, dx$

The integration error is defined as the difference (not the absolute difference) between your numerical results and the exact results. Plot the integration error as a function of $h$ for both integration routines and all listed functions. Comment on the comparison between both integration routines. Does the sign of the error match your expectations? Why?

In [7]:

f1 = lambda x: x
f2 = lambda x: x**2
f3 = lambda x: x**(1/2)

n    = 100000
a, b = 0, 1

print("As we are calculus geniuses, we know that the first integral gives 1/2.")
compare_integration(f1, a, b, n)

print("As we are calculus geniuses, we know that the second integral gives 1/3.")
compare_integration(f2, a, b, n)

print("As we are calculus geniuses, we know that the third integral gives 2/3.")
compare_integration(f3, a, b, n)

print("For all three cases, the results are very close, and the functions work quickly.")

As we are calculus geniuses, we know that the first integral gives 1/2.
For function f1 = lambda x: x

for boundaries a = 0 , b = 1 and steps n = 100000 the algorithms say:
trapezoid:		 0.5000000000000001
Simpson:		 0.5000000000000001
scipy.integrate.trapz:	 0.5000000000000001
scipy.integrate.simps:	 0.5

As we are calculus geniuses, we know that the second integral gives 1/3.
For function f2 = lambda x: x**2

for boundaries a = 0 , b = 1 and steps n = 100000 the algorithms say:
trapezoid:		 0.33333333335000004
Simpson:		 0.3333333333333335
scipy.integrate.trapz:	 0.33333333335
scipy.integrate.simps:	 0.3333333333333333

As we are calculus geniuses, we know that the third integral gives 2/3.
For function f3 = lambda x: x**(1/2)

for boundaries a = 0 , b = 1 and steps n = 100000 the algorithms say:
trapezoid:		 0.6666666600968939
Simpson:		 0.6666666640993837
scipy.integrate.trapz:	 0.6666666600968938
scipy.integrate.simps:	 0.6666666640993836

For all three cases, the results are very close, and the functions work quickly.

In [ ]:

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