Issues with the normal equations

Copyright (C) 2020 Andreas Kloeckner

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In [3]:
import numpy as np

import numpy.linalg as la

Here's an example matrix to use with the normal equations:

In [4]:
eps = 1e-2  # set to 1e-5, 1e-10



A = np.array([

        [1, 1],

        [eps, 0],

        [0, eps],

        ])



np.set_printoptions(precision=20)

print(A)

print(A.T @ A)

[[ 1.                      1.                    ]
[ 0.01000000000000000021  0.                    ]
[ 0.                      0.01000000000000000021]]
[[ 1.00009999999999998899  1.                    ]
[ 1.                      1.00009999999999998899]]
  • What do you notice about the entries of $A^T A$?
In [3]:
n = 5



A = np.random.randn(5, 5) * 10**-np.linspace(0, -5, n)

la.cond(A)
Out[3]:
60358.505352514032
In [4]:
la.cond(np.dot(A.T, A))
Out[4]:
3643149168.4817424

  • What do you notice about the condition number?

  • What's a general bound? $\operatorname{cond}(AB)\le \dots$?