# 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)
```

- 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? $\operatorname{cond}(AB)\le \dots$?