Exercise sheet \(2\)

The aim of this exercise sheet is to provide guidance for implementing several quadrature formulas for numerical integration. We are interested in approximating

(1)\[\int_{a}^{b}f(x)\,\mathrm{d}x\]

where \(a,b\in\mathbb{R}\).

For a given \(N\in\mathbb{N}_{>0}\), \(h=\frac{b-a}{N}\) be the width of subintervals. We call \(x_{i}=a+hi\) the nodal points.

  1. Open an empty file and name it as quadrature.py

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import numpy as np

class Integrator():
    """A class to integrate functions numerically.
    Create an object by Integrator(f, a, b, N)
    f = Integrand function
    a = Interval lowerbound
    b = Interval upperbound
    N = Number of subintervals
    """

    def __init__(self, f, a, b, N):
        """(Integrator, f, a, b, N) -> NoneType
        Cretate an Integrator with N intervals.
        """
        self.a = a
        self.b = b
        self.N = N
        self.x = 0   # TODO: Part (a)
        self.h = 0   # TODO: Part (b)
        self.f = f   # TODO: Part (c)

>>> import quadrature
>>> help(quadrature)
Help on module quadrature:
NAME
    quadrature
CLASSES
    builtins.object
        Integrator
    class Integrator(builtins.object)
     |  Integrator(f, a, b, N)
     |
     |  A class to integrate functions numerically.
     |  Create an object by Integrator(f, a, b, N)
     |  f Integrand function
     |  a Interval lowerbound
     |  b Interval uperbound
     |  N Number of subintervals
     |
     |  Methods defined here:
     |
     |  __init__(self, f, a, b, N)
     |      (Integrator, f, a, b, N) -> NoneType
     |      Create an Integrator with N intervals
     |
     |  midpoint(self)
     |
     |  ----------------------------------------------------------------------
     |  Data descriptors defined here:
     |
     |  __dict__
     |      dictionary for instance variables (if defined)
     |
     |  __weakref__
     |      list of weak references to the object (if defined)
FILE
    /home/carlosal1015/Git_Projects/Python/Scientific_Python/source/exercises/quadrature.py
>>> quad = quadrature.Integrator(0, 0, 1, 10)
>>> print(f'The interval ({quad.a}, {quad.b}) is \
divided into {quad.N} subintervals.\nThe nodal \
points are {quad.x}.')
The interval (0, 1) is divided into 10 subintervals.
The nodal points are [0.  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. ].
(2)\[M_{h}\left(f\right) = h\sum_{i=1}^{N}f\left(\frac{x_{i}+x_{i-1}}{2}\right).\]
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import numpy as np

class Integrator():
    """
    The same code as above.
    """

    def __init__(self, f, a, b, N):
        """
        The same code as above.
        """

     def midpoint(self):
        """(Integrator) -> float
        Numerical integration by midpoint rule
        """
        s = # TODO: Part (a)
        y = # TODO: Part (b)
        T = # TODO: Part (c)
        return T

>>> import quadrature
>>> import numpy as np
>>> quad = quadrature.Integrator(np.exp, 0, 1, 10)
>>> result = quad.midpoint()
>>> print(result)

>>> exactintegral = np.exp(1)-1
>>> quad = quadrature.Integrator(np.exp, 0, 1, 10)
>>> errmidpoint[0] = np.fbs(exactintegral - quad.midpoint())
>>> print(result)

>>> convratemid = np.log(errmid[:-1] / errmid[1:]) / np.log(2)
>>> print(convratemid)
(3)\[\begin{aligned} \int\limits_{0}^{1}x\exp(x)\mathrm{d}x &={x\cdot\left.\exp\left(x\right)\right|}^{1}_{0}-\int_{0}^{1}\exp\left(x\right)\\ &=\left[1\cdot\exp\left(1\right)-0\cdot\exp\left(0\right)\right]-{\left.\exp\left(x\right)\right|}^{1}_{0}\\ &=\left[\exp\left(1\right)\right]-\left[\exp\left(1\right)-\exp\left(0\right)\right]\\ \int\limits_{0}^{1}x\exp(x)\mathrm{d}x &=1. \end{aligned}\]