BUG: Fix Samuelson quadratic formula

[blkboxeconomist]

Two small bugs.

One is a typo in the documentation for the class that uses the formulas for the old version of the Samuelson lecture and the second is a typo in finding the roots of the characteristic equation.

First is self-explanatory once you look at it.

For the second

 The three bad lines are the -\ρ1 prefixes:

discriminant = ρ1**2 + 4 * ρ2> if discriminant == 0:
     roots.append(-ρ1 / 2)
elif discriminant > 0:>     roots.append((-ρ1 + sqrt(discriminant).real) / 2)
      roots.append((-ρ1 - sqrt(discriminant).real) / 2)>     ...
else:
     roots.append((-ρ1 + sqrt(discriminant)) / 2)
     roots.append((-ρ1 - sqrt(discriminant)) / 2)

The polynomial in the lecture is z² − ρ₁ z − ρ₂ = 0, so those should be ρ₁ / 2 and (ρ₁ ± √⋅)/2.

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[jstac]

Thanks very much, Chase, for bringing this to our attention, and for the PR!

Your two fixes look right to us:

1. Characteristic roots in y_nonstochastic. The lecture's characteristic polynomial is $z^2 - \rho_1 z - \rho_2$, so the roots are $(\rho_1 \pm \sqrt{\rho_1^2 + 4\rho_2})/2$. The by-hand calculation used $-\rho_1$, so every root it printed had the wrong sign. It also disagreed with the np.roots([1, -ρ1, -ρ2]) calls elsewhere in the lecture.
2. Samuelson class docstring. The recurrence came from an older version of the lecture. It now matches the model in the text.

I've added one small follow-up commit to the docstring:

• It's now a raw string (r"""). Otherwise Python reads \a and \b in \alpha and \beta as escape characters.
• The equation now includes the $\sigma \epsilon_t$ term, since the class supports the stochastic case: $Y_t = (\alpha + \beta) Y_{t-1} - \beta Y_{t-2} + (\gamma + G_t) + \sigma \epsilon_t$.

I ran the lecture end to end on the branch. The by-hand roots now agree with np.roots.

[jstac]

@mmcky This looks good to me, pls merge when ready.

[mmcky]

thanks @blkboxeconomist and @jstac.

[mmcky]

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• lectures/samuelson.md