Euler Estimation

Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation.

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Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation. We will jump straight to exercises – it is assumed that you’re already familiar with Euler estimation from calculus.

Exercise: Single-Variable Euler Estimator

To start, build a single-variable Euler estimation class as follows:


>>> def derivative(t):
        return t+1
        
>>> euler = EulerEstimator(derivative)

>>> initial_point = (1,4)
>>> euler.eval_derivative(initial_point) # evaluates derivative at point (1,4)
2

>>> step_size = 0.5
>>> num_steps = 4
>>> euler.estimate_points(initial_point, step_size, num_steps)
[
    (1,   4   ),   # starting point
    (1.5, 5   ),   # after 1st step
    (2,   6.25),   # after 2nd step
    (2.5, 7.75),   # after 3rd step
    (3,   9.5 )    # after 4th step
]

Then, use your Euler estimator to plot several solution curves to the following differential equation on the interval $x \in [0, 5].$ (Your Euler estimator generates a list of points, and then you can use that list of points to generate a plot.)

$\begin{align*} \dfrac{\textrm dy}{\textrm dx} = x - 2 \end{align*}$


For one curve, use the initial condition $y(0)=-2.$ For another curve, use $y(0)=-1.$ Then another curve with $y(0)=0,$ another with $y(0) = 1,$ and another with $y(0)=2.$ All $5$ of these curves can go on the same plot.

Based on your knowledge of calculus, you should be able to tell if your plots look right.

Exercise: Multivariable Euler Estimator

Once you’ve implemented a single-variable Euler estimator, you can generalize it to simulate systems of differential equations. For example, consider the following system:

$\begin{align*} a'(t) &= a(t) + 1 \\ b'(t) &= a(t) + b(t) \\ c'(t) &= 2b(t) + 3t \end{align*}$


To simulate this system starting with the initial state $a(-0.4) = -0.45,$ $b(-0.4) = -0.05,$ $c(-0.4) = 0,$ construct a multivariable Euler estimator as follows:


>>> initial_state = {'a': -0.45, 'b': -0.05, 'c': 0}

>>> initial_point = (-0.4, initial_state) # points take form (t, state)

>>> def da_dt(t, state):
        return state['a'] + 1

>>> def db_dt(t, state):
        return state['a'] + state['b']
        
>>> def dc_dt(t, state):
        return 2 * state['b'] + 3 * t
        
>>> derivatives = {
    'a': da_dt,
    'b': db_dt,
    'c': dc_dt
}
        
>>> euler = EulerEstimator(derivatives)

>>> euler.eval_derivative_at_point(initial_point)
{'a': 0.55, 'b': -0.5, 'c': -1.3}

>>> step_size = 2
>>> num_steps = 3
>>> euler.estimate_points(initial_point, step_size, num_steps)
[
   (-0.4, {'a': -0.45, 'b': -0.05, 'c': 0   }),
   (1.6,  {'a': 0.65,  'b': -1.05, 'c': -2.6}),
   (3.6,  {'a': 3.95,  'b': -1.85, 'c': 2.8 }),
   (5.6,  {'a': 13.85, 'b': 2.35,  'c': 17  })
]

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