Condition number

Copyright (C) 2020 Andreas Kloeckner

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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
Out[2]:
array([[ 3.,  0.],
       [ 0.,  1.]])

And its inverse as Ainv:

In [3]:
Ainv = la.inv(A)
Ainv
Out[3]:
array([[ 0.33333333,  0.        ],
       [ 0.        ,  1.        ]])

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
Out[5]:
(2, 30)

...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
Out[7]:
(2, 30, 30)

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))

3.0
Out[9]:
<matplotlib.patches.Circle at 0x7fb8c1b29f28>

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))
In [ ]: