Condition number

In [8]: 
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
In [97]: 
np.random.seed(17)
A = np.random.randn(2, 2)
Ainv = la.inv(A)
In [98]: 
phi = np.linspace(0, 2*np.pi, 15)
xs = np.array([
    np.cos(phi),
    np.sin(phi)
])

pt.plot(xs[0], xs[1], "x")
pt.grid()
In [99]: 
Axs = np.einsum("ik,kj->ij", A, xs)
Axs.shape
Out[99]:
(2, 15)
In [100]: 
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")
In [104]: 
# ys has axes: XY x Npoints x Npoints

perturbation_size = 0.1
ys = perturbation_size * xs[:, :, np.newaxis] + xs[:, np.newaxis, :]

Ays = np.einsum("ik,kjl->ijl", A, ys)
Ays.shape
Out[104]:
(2, 15, 15)
In [102]: 
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")
In [103]: 
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))

2.18194885762
Out[103]:
<matplotlib.patches.Circle at 0x7f0e09cbca90>

Now we want a $\kappa$ with $\frac{\|\Delta b\|}{\|b\|}\le \kappa \frac{\|\Delta x\|}{\|x\|}$.

Consider $\|x\|=1$. Equivalent: $\|\Delta b\|\le \kappa \|\Delta x\|\|b\|$.

Which $\kappa$ does the job?

In [117]: 
kappa = la.norm(A, 2)*la.norm(Ainv, 2)
In [118]: 
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))
In []: