#import "@preview/physica:0.9.6": *

== a)

given the explicit midpoint method
$
  y_(n+1) = y_n + h f(t_n + 0.5h, y_n + 0.5h f(t_n, y_n)),
$
notice $k_1 = f(t_n, y_n)$ appears at depth 1 meaning the funcion is only nested
twice, such that our number of stages $s = 2$. furthermore, we have $c_2 = 0.5$
from the time step and $a_21 = 0.5$ from the function step. we can also observe
that $b_1 = 1$ and $b_2 = 0$. this can be visualized in a mnemonic butcher
tableau.

#let butcher(s, nums) = {
  let nums = nums.map(x => $#x$)
  table(
    stroke: (x, y) => if x == 0 and y == s {
      (right: 0.7pt + black, top: 0.7pt + black)
    } else if x == 0 {
      (right: 0.7pt + black)
    } else if y == s {
      (top: 0.7pt + black)
    },
    align: (x, y) => (
      if x > 0 { center } else { left }
    ),
    columns: s + 1,
    $0$, ..range(s).map(x => none), // first row
    ..range(2, s + 1)
      .map(i => {
        let p(i) = calc.floor(i * i / 2 + i / 2 - 1)
        (
          nums.slice(p(i - 1), p(i - 1) + i),
          range(i, s + 1).map(x => none),
        ).flatten()
      })
      .flatten(),
    none, ..nums.rev().slice(0, s).rev() // last row
  )
}

#align(center)[#butcher(2, (0.5, 0.5, 1, 0))]

#pagebreak()

== b)

I don't wanna write python right now, so I'm gonna solve this in typst. yes, I'm
coding this in the typesetting language I use to write this text.

#let code = ```typc
  let h = 0.01;
  let f(t, y) = calc.exp(-y * y);
  let y_1(t_0, y_0) = y_0
      + h * f(t_0 + 0.5 * h,
              y_0 + 0.5 * h * f(t_0, y_0));
  let t = 0;
  let y = 1;
  while t < 1 {
    y = y_1(t, y);
    t += h;
  };
  [the value for $y$ at $t = 1$ is
  #calc.round(y, digits: 4)]
```

#eval(code.text.split("\n").join())

#code