# coding: utf-8

# # Condition number

# In[1]:


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


# Let's grab a $2\times 2$ matrix $A$:

# In[2]:


if 0:
    np.random.seed(17)
    A = np.random.randn(2, 2)
else:
    A = np.array([[3, 0], [0,1]], dtype=np.float64)

A


# And its inverse as `Ainv`:

# In[3]:


Ainv = la.inv(A)
Ainv


# Now we would like to figure out where that matrix puts all the vectors with 2-norm 1.
# 
# To do so, let's make an array of vectors with vectors with norm 1:

# In[4]:


phi = np.linspace(0, 2*np.pi, 30)
xs = np.array([
    np.cos(phi),
    np.sin(phi)
])

pt.gca().set_aspect("equal")
pt.plot(xs[0], xs[1], "x")
pt.grid()


# Now apply $A$ to all those vectors...:

# In[5]:


Axs = A.dot(xs)
Axs.shape


# ...and plot:

# In[6]:


pt.figure(figsize=(10, 5))

pt.subplot(121)
pt.title("$x$")
pt.plot(xs[0], xs[1], "x")
pt.gca().set_aspect("equal")

pt.subplot(122)
pt.title("$Ax$")
pt.plot(Axs[0], Axs[1], "v")
pt.gca().set_aspect("equal")


# -------------
# 
# Next, let's see what happens to small perturbations at each of the $x$ and $Ax$ points.
# 
# To that end, let's make an array `ys` of shape $2\times N_p\times N_p$, where $N_p$ is the number of points above.

# In[7]:


# ys has axes: XY x Npoints x Npoints

perturbation_size = 0.1
ys = perturbation_size * xs.reshape(2, -1, 1) + xs.reshape(2, 1, -1)

Ays = np.tensordot(A, ys, axes=1)
Ays.shape


# Side note: What does the argument `-1` to reshape do?

# -----------------------
# Let's plot what we've just made

# In[8]:


pt.figure(figsize=(10, 5))

pt.subplot(121)
pt.title("$y$")
pt.plot(ys[0], ys[1])
pt.gca().set_aspect("equal")

pt.subplot(122)
pt.title("$Ax$")
pt.plot(Ays[0], Ays[1])
pt.gca().set_aspect("equal")


# -------------------
# Let's compare this with $\|A\|$:

# In[9]:


norm = la.norm(A, 2)
print(norm)

pt.plot(Ays[0], Ays[1])

ax = pt.gca()
ax.set_aspect("equal")
ax.add_artist(pt.Circle([0, 0], norm, alpha=0.3, lw=0))


# ------------------
# What we want now is a circle around each of the $Ax$ that says,
# 
# "Because of the $\Delta x$ variation, $b$ is at most going to wiggle by this much,
# i.e. $\Delta b$ will be at most this big."

# ------
# 
# Now we want a $\kappa$ with $\frac{\|\Delta b\|}{\|b\|}\le \kappa \frac{\|\Delta x\|}{\|x\|}$.
# 
# Assume $\|x\|=1$. Equivalent: $\|\Delta b\|\le \kappa \|\Delta x\|\|b\|$.
# 
# Which $\kappa$ does the job?

# In[10]:


kappa = la.norm(A, 2)*la.norm(Ainv, 2)


# In[11]:


pt.plot(Ays[0], Ays[1])

ax = pt.gca()
ax.set_aspect("equal")
for i in range(Ays.shape[2]):
    b = Axs[:, i]
    norm_delta_y = kappa * perturbation_size * la.norm(b)
    ax.add_artist(pt.Circle(b, norm_delta_y, alpha=0.3, lw=0))