daviddoji/CalculusWithJuliaNotes.jl
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jverzani
2024-04-26 18:26:12 -04:00
parent 6e807edb46
commit 4f924557ad
45 changed files with 326 additions and 296 deletions
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10
quarto/derivatives/newtons_method.qmd
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@@ -403,8 +403,8 @@ A plot shows us roughly where the value lies:
#| hold: true
f(x) = exp(x)
g(x) = x^6
plot(f, 0, 25, label="f")
plot!(g, label="g")
plot(f, 0, 25; label="f")
plot!(g; label="g")
```
Clearly by $20$ the two paths diverge. We know exponentials eventually grow faster than powers, and this is seen in the graph.
@@ -858,8 +858,10 @@ The calculation that produces the quadratic convergence now becomes:
$$
x_{i+1} - \alpha = (x_i - \alpha) - \frac{1}{k}(x_i-\alpha - \frac{f''(\xi)}{2f'(x_i)}(x_i-\alpha)^2) =
\frac{k-1}{k} (x_i-\alpha) + \frac{f''(\xi)}{2kf'(x_i)}(x_i-\alpha)^2.
\begin{align*}
x_{i+1} - \alpha &= (x_i - \alpha) - \frac{1}{k}(x_i-\alpha - \frac{f''(\xi)}{2f'(x_i)}(x_i-\alpha)^2)
&= \frac{k-1}{k} (x_i-\alpha) + \frac{f''(\xi)}{2kf'(x_i)}(x_i-\alpha)^2.
\end{align*}
$$
As $k > 1$, the $(x_i - \alpha)$ term dominates, and we see the convergence is linear with $\lvert e_{i+1}\rvert \approx (k-1)/k \lvert e_i\rvert$.