Notes on Note G

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The operations

Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Each operation reads two variables and writes one, as in this table:

Operation	Reads	Writes
×	V2, V3	V4
−	V4, V1	V5
÷	V5, V4	V11
+	V11, V13	V13

In modern code

Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. Here is the same idea as a bernoulli function:

// Computes the nth Bernoulli number with the Akiyama–Tanigawa algorithm.
export function bernoulli(n: number): number {
  const a: number[] = [];
  for (let m = 0; m <= n; m++) {
    a[m] = 1 / (m + 1);
    for (let j = m; j >= 1; j--) {
      a[j - 1] = j * (a[j - 1] - a[j]);
    }
  }
  return a[0];
}

console.log(bernoulli(8)); // -0.0333…

Checking the result

Curabitur pretium tincidunt lacus. Nulla gravida orci a odio. Nullam varius, turpis et commodo pharetra, est eros bibendum elit:

from fractions import Fraction

def bernoulli(n: int) -> Fraction:
    """The nth Bernoulli number, exactly."""
    a = [Fraction(0)] * (n + 1)
    for m in range(n + 1):
        a[m] = Fraction(1, m + 1)
        for j in range(m, 0, -1):
            a[j - 1] = j * (a[j - 1] - a[j])
    return a[0]

assert bernoulli(8) == Fraction(-1, 30)

The Analytical Engine weaves algebraical patterns just as the Jacquard loom weaves flowers and leaves.

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