# Euler Estimation

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.)

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

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 }), # starting point
(1.6, {'a': 0.65, 'b': -1.05, 'c': -2.6}), # after 1st step
(3.6, {'a': 3.95, 'b': -1.85, 'c': 2.8 }), # after 2nd step
(5.6, {'a': 13.85, 'b': 2.35, 'c': 17 }) # after 3rd step
]
```

## Leave a Comment