import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
Let's make two particle collections: sources
and targets
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.)
c = np.sum(targets, axis=1) / targets.shape[1]
c
pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")
pt.plot(c[0], c[1], "bo")
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)
First, obtain an idea of what $\rho$ is:
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:
_, sigma, V = la.svd(interaction_mat)
pt.semilogy(sigma)
eps_values = 10**(-np.linspace(1, 12))
For the given precisions $\varepsilon$, find the associated numerical rank:
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:
pt.semilogx(eps_values, ranks)
pt.semilogx(eps_values, (np.log(eps_values)/np.log(rho)-1)**2)