Easier object-oriented calculations for numerical solvers.

## Project description

# npsolve

The *npsolve* package is a small, simple package built on *numpy* and
*fastwire* to make it easy to use object-oriented classes and methods for
the calculation step for numerical solvers.

Many numerical solvers (like those in *scipy*) provide candidate solutions as
a numpy ndarray. They often also require a numpy ndarray as a return value
(e.g. an array of derivatives) during the solution. These requirements can make
it difficult to use an object oriented approach to performing the calculations.
Usually, we end up with script-like code that looses many of the benefits
of object-oriented programming.

The npsolve framework links a solver with multiple classes that handle the calculations for each step in the algorithm. It allows different parts of the calculations to be encapsulated and polymorphic, and makes the code much easier to modify and maintain.

## Basic usage tutorial

Let's use npsolve to do some integration through time, like you would to solve an ODE. Instead of equations, though, we're using class methods. The code for all the tutorials is available in the repository under 'examples'.

First, setup some classes that you want to do calculations with. We do this
by using the `add_var`

method to setup variables and their initial values.

```
import numpy as np
import npsolve
class Component1(npsolve.Partial):
def __init__(self):
super().__init__() # Don't forget to call this!
self.add_var('position', init=0.1)
self.add_var('velocity', init=0.3)
class Component2(npsolve.Partial):
def __init__(self):
super().__init__() # Don't forget to call this!
self.add_var('force', init=-0.1)
```

All the variables are made available to all Partial instances automatically
through their `state`

attribute. It's a dictionary. The `add_var`

method
sets initial values into the instance's state dictionary. Later, the `Solver`

will ultimately replace the `state`

attribute with a new dictionary that
contains all variables from all the Partial classes.

Next, we'll tell these classes how to do some calculations during each time
step. The `step`

method is called automatically and expects a dictionary of
return values (e.g. derivatives). We'll use that one here. The state
dictionary is given again as the first argument, but we're going to use the
internal `state`

attribute instead. So, we'll add some more methods:

```
class Component1(npsolve.Partial):
def __init__(self):
super().__init__() # Don't forget to call this!
self.add_var('position', init=0.1)
self.add_var('velocity', init=0.3)
def step(self, state_dct, t, *args):
""" Called by the solver at each time step
Calculate acceleration based on the net force.
"""
acceleration = 1.0 * self.state['force']
derivatives = {'position': self.velocity,
'velocity': acceleration}
return derivatives
class Component2(npsolve.Partial):
def __init__(self):
super().__init__() # Don't forget to call this!
self.add_var('force', init=-0.1)
def calculate(self, t):
''' Some arbitrary calculations based on current time t
and the position at that time calculated in Component1.
This returns a derivative for variable 'c'
'''
dc = 1.0 * np.cos(2*t) * self.state['position']
derivatives = {'force': dc}
return derivatives
def step(self, state_dct, t, *args):
''' Called by the solver at each time step '''
return self.calculate(t)
```

Now, we'll set up the solver. For this example, we'll use the odeint solver from Scipy. Here's what it looks like:

```
from scipy.integrate import odeint
class Solver(npsolve.Solver):
def solve(self):
self.npsolve_init() # Initialise
self.t_vec = np.linspace(0, 10, 1001)
result = odeint(self.step, self.npsolve_initial_values, self.t_vec)
return result
```

Let's look at what's going on in the `solve`

method. By default, Solvers
have a `step`

method that's ready to use. (They also have a `one_way_step`

method that doesn't expect return values from the Partials, and a `tstep`

method that expects a time value as the first argument.) After initialisation,
the initial values set by the Partial classes are captured in the
`npsolve_initial_values`

attribute. By default, the Solver's `step`

method
returns a vector of all the return values, the same size as the Solver's
`npsolve_initial_values`

array. So most of the work is done for us here
already.

Note here that we don't need to know anything about the model or the elements in the model. This allows us to decouple the model and Partials from the solver. We can pass in different models, or pass models to different solvers. We can make models with different components. It's flexible and easy to maintain!

To run, we just have to instantiate the Solver and Partial instances,
then pass a list or dictionary of the Partial instances to the `connect`

method of the Solver. They'll link up automatically through *fastwire*.

```
def run():
solver = Solver()
partials = [Component1(), Component2()]
solver.connect(partials)
res = solver.solve()
return res, solver
```

Let's set up a plot to see the results. Use the `npsolve_slices`

attribute of
the Solver to get the right columns.

```
import matplotlib.pyplot as plt
def plot(res, s):
slices = s.npsolve_slices
plt.plot(s.t_vec, res[:,slices['position']], label='position')
plt.plot(s.t_vec, res[:,slices['velocity']], label='velocity')
plt.plot(s.t_vec, res[:,slices['force']], label='force')
plt.legend()
```

Run it and see what happens!

```
res, s = run()
plot(res, s)
```

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