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

# # Gauss-Newton
# 
# 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:
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# The above copyright notice and this permission notice shall be included in
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# 
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# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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# THE SOFTWARE.
# </details>

# In[8]:


import numpy as np
import scipy as sp
import matplotlib.pyplot as pt
import scipy.linalg as la


# We would like to fit the model $f(t) = x_0 e^{x_1 t}$ to the following data using Gauss-Newton:

# In[13]:


t = np.array([0.0, 1.0, 2.0, 3.0])
y = np.array([2.0, 0.7, 0.3, 0.1])


# First, define a residual function (as a function of $\mathbf x=(x_0, x_1)$)
# 
# **NOTE:** $\mathbf x$ has *model coefficients, not data points*.

# In[14]:


def residual(x):
    return y - x[0] * np.exp(x[1] * t)


# Next, define its Jacobian matrix:

# In[15]:


def jacobian(x):
    return np.array([
        -np.exp(x[1] * t),
        -x[0] * t * np.exp(x[1] * t)
        ]).T


# Here are two initial guesses. Try both:

# In[56]:


#x = np.array([1, 0])
x = np.array([0.4, 2])


# Here's a plotting function to judge the quality of our guess:

# In[57]:


def plot_guess(x):
    pt.plot(t, y, 'ro', markersize=20, clip_on=False)
    T = np.linspace(t.min(), t.max(), 100)
    Y = x[0] * np.exp(x[1] * T)
    pt.plot(T, Y, 'b-')
    
    print("Residual norm:", la.norm(residual(x), 2))

plot_guess(x)


# Code up one iteration of Gauss-Newton. Use `numpy.linalg.lstsq()` to solve the least-squares problem, noting that that function returns a tuple--the first entry of which is the desired solution.
# 
# Also print the residual norm. Use `plot_guess` to visualize the current guess.
# 
# Then evaluate this cell in-place many times (Ctrl-Enter):

# In[79]:


x = x + la.lstsq(jacobian(x), -residual(x))[0]

plot_guess(x)


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