#!/usr/bin/env python
# coding: utf-8

# # Shooting Method
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# 
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
# 
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
# </details>

# In[7]:


import numpy as np
import matplotlib.pyplot as pt


# In[8]:


def rk4_step(y, t, h, f):
    k1 = f(t, y)
    k2 = f(t+h/2, y + h/2*k1)
    k3 = f(t+h/2, y + h/2*k2)
    k4 = f(t+h, y + h*k3)
    return y + h/6*(k1 + 2*k2 + 2*k3 + k4)


# Want to solve:
# 
# $$w''(t)=\frac 32w^2$$
# 
# with $w(0)=4$ and $w(1)=1$. (Example due to Stoer and Bulirsch)

# In[9]:


def f(t, y):
    w, w_prime = y
    return np.array([w_prime, 3/2*w**2])


# The following function carries out the shooting method for a given $w'(0)$ using RK4:

# In[20]:


def shoot(w_prime):
    times = [0]
    y_values = [np.array([4, w_prime])]

    h = 1/2**7
    t_end = 1

    while times[-1] < t_end:
        y_values.append(rk4_step(y_values[-1], times[-1], h, f))
        times.append(times[-1]+h)

    y_values = np.array(y_values)

    # actually floating-point-equal due to power-of-2 h
    assert times[-1] == t_end

    print("w'(0) = %g  ->  w(1)= %.5g" % (w_prime, y_values[-1,0]))

    pt.plot(times, y_values[:, 0], label="$w'(0)=%.2g$" % w_prime)


# Call `shoot` to see if you can solve the boundary value problem.
# 
# Start with $w'(0)=0$.
# 
# (You may call `pt.legend` to take advantage of automatic labeling.)

# In[19]:


shoot(0)
shoot(-5)
shoot(-7)

pt.grid()
pt.legend(loc="best")


# See if you can find another solution to the boundary value problem by starting with $w'(0)=-30$.
# 
# (You may call `pt.legend` to take advantage of automatic labeling.)

# In[18]:


shoot(-30)
shoot(-33)
shoot(-36)

pt.grid()
pt.legend(loc="best")


# In[ ]: