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

# # Checking Rank Estimates

# In[1]:


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


# Let's make two particle collections: `sources` and `targets`

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sources = np.random.randn(2, 200)
targets = np.random.randn(2, 200) + 10

pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")


# What's our convergence ratio $\rho$:
# $$
# \rho=\frac{d(c,\text{furthest target})}{d(c,\text{closest source})}?
# $$
# 
# Well, what should $c$ be, really? (Technically, we can use any $c$, so we're free to choose whichever we think is best.)

# In[3]:


c = np.sum(targets, axis=1) / targets.shape[1]

c


# In[4]:


pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")
pt.plot(c[0], c[1], "bo")


# ## Assemble the interaction matrix

# In[5]:


all_distvecs = sources.reshape(2, 1, -1) - targets.reshape(2, -1, 1)
dists = np.sqrt(np.sum(all_distvecs**2, axis=0))
interaction_mat = np.log(dists)


# ## Estimate the rank depending on precision $\varepsilon$

# First, obtain an idea of what $\rho$ is:

# In[6]:


dist_tgt_to_c = np.sqrt(np.sum((c.reshape(2, 1) - targets)**2, axis=0))
dist_src_to_c = np.sqrt(np.sum((c.reshape(2, 1) - sources)**2, axis=0))

rho = np.max(dist_tgt_to_c) / np.min(dist_src_to_c)
rho


# Then plot the numerical rank depending on epsilon:

# In[7]:


_, sigma, V = la.svd(interaction_mat)
pt.semilogy(sigma)


# In[9]:


eps_values = 10**(-np.linspace(1, 12))


# For the given precisions $\varepsilon$, find the associated numerical rank:

# In[10]:


def numrank(eps):
    return np.sum(sigma > eps)

ranks = [numrank(e) for e in eps_values]
pt.semilogx(eps_values, ranks)


# Now compare with our estimate:

# In[11]:


pt.semilogx(eps_values, ranks)
pt.semilogx(eps_values, (np.log(eps_values)/np.log(rho)-1)**2)


# * We estimated that the rank would grow *quadratically*.
# * Comments on how good our estimate is?

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