[newton_method] Some suggestions on code and wording #1063

Hi @jstac

I was reading through the newton_method lecture with codex and here are some potential updates.

I'd be grateful if you could take a look when you have a chance.

Overview

• At lines 66–72, the descriptions use “the fixed point” and “the zero,” which can imply uniqueness. Use “a fixed point” and “a zero” instead.

Fixed-point computation using Newton's method

• At line 123, specify that the savings rate satisfies $s \in (0,1)$, rather than only $s>0$.

• At lines 283–291, define the linear approximation with equality:

$$ \hat g(x):=g(x_0)+g'(x_0)(x-x_0). $$

• At lines 293–306, state that the Newton update requires $1-g'(x_t)\neq 0$.

• At lines 336–367, plot_trajectories relies on the global value of k_star, which may have been calculated from different parameters. Calculate k_star = exact_fixed_point(params) inside the function instead.

Root-finding in one dimension

• [qe-writing-006] At line 378, change the heading from “Root-Finding in one dimension” to the sentence-case form “Root-finding in one dimension.”

• At lines 397–405, define the linear approximation with equality:

$$ \hat f(x):=f(x_0)+f'(x_0)(x-x_0). $$

• At lines 405–418, state that each Newton update requires $f'(x_t)\neq 0$.

• At line 460, correct “We can convert to this to a zero-finding problem” to “We can convert this to a zero-finding problem.”

Multivariate Newton's method

• At lines 485–486, change “5,000 goods” to “3,000 goods” to match the code implementation later.

• At line 492, change “A two-goods market equilibrium” to “A two-good market equilibrium.”

• [qe-writing-001] At lines 513, 606, and 701, split each two-sentence paragraph so that it contains one sentence per paragraph.

• At line 606, capitalize “Python.”

• At lines 621–643, plot_excess_demand reads A, b, and c from mutable global state. Pass these parameters explicitly so that the two-good plots can still be rerun after the variables are reassigned for the 3,000-good example.

• At line 624, update the mesh-grid comment from combinations of p_1 and p_2 to combinations of p_0 and p_1 so that it matches with the notation in the text.

• [qe-fig-003] At lines 620–643, remove the embedded ax.set_title(). It also causes the combined plot at lines 667–671 to end with the misleading title “Excess demand for good 1,” even though both goods are shown.

• At line 661, the prose describes a “black contour line,” but the code does not explicitly set the contour color. Currently it looks like purple not black. We can set the contour color to black.

• At lines 725–735, update the manually calculated Jacobian. For

$$ e_i(p)=\exp(-(Ap)_i)+c_i-b_i\sqrt{p_i}, $$

so each exponential derivative contain the complete row product $(Ap)_i$. The element-by-element implementation could be:

def jacobian_e(p, A, b, c):
    p_0, p_1 = p
    a_00, a_01 = A[0, :]
    a_10, a_11 = A[1, :]

    exp_0 = jnp.exp(-(a_00 * p_0 + a_01 * p_1))
    exp_1 = jnp.exp(-(a_10 * p_0 + a_11 * p_1))

    j_00 = -a_00 * exp_0 - (b[0] / 2) * p_0 ** (-1 / 2)
    j_01 = -a_01 * exp_0
    j_10 = -a_10 * exp_1
    j_11 = -a_11 * exp_1 - (b[1] / 2) * p_1 ** (-1 / 2)

    J = [[j_00, j_01], [j_10, j_11]]
    return jnp.array(J)

• At lines 760–768, state that the multivariate Newton update requires $J_e(p_n)$ to be nonsingular.

• At lines 777–799, the multivariate newton definition overwrites the scalar implementation from the preceding section. Give it a distinct name such as newton_nd and update all downstream multivariate calls in Section 7.4 and the exercise solutions.

Exercise 1

• At lines 904–909, denote the three initial vectors by $k_0^{(1)}$, $k_0^{(2)}$, and $k_0^{(3)}$, currently it looks like $k1_0$, $k2_0$, and $k3_0$.

• At lines 917–934, clarify that the three-dimensional analytical benchmark is $(k^,k^,k^)^\top$ , where the linked $k^$ is the scalar analytical solution.

Exercise 2

• At lines 1053–1058, denote the three initial price vectors by $p_0^{(1)}$, $p_0^{(2)}$, and $p_0^{(3)}$, currently the number is after the letter: $p1_0$, $p2_0$, and $p3_0$.

What do you think? Happy to put up a PR.

Best,
Longye