Function reliableLimitSamples

src/compute-engine/numerics/numeric.ts:298–362

* Probe the geometric sample ladder that `extrapolate()` uses for a numeric * limit and report how many leading samples are *trustworthy*. * * Returns `Infinity` when the ladder is well-behaved (the common case — no cap * needed), or a finite count when the function crosses a floating-point * "

(
  f: (x: number) => number,
  x0: number,
  step: number
)

  if (n < 0) return NaN;
  if (n <= 1) return 1;

  let result = n;
  while (n > 2) {
    n -= 2;
    result *= n;
  }

  return result;
}

export function chop(n: number, tolerance = DEFAULT_TOLERANCE): 0 | number {
  if (typeof n === 'number' && Math.abs(n) <= tolerance) return 0;
  return n;
}

/**
 * An 8th-order centered difference approximation can be used to get a highly
 * accurate approximation of the first derivative of a function.
 * The formula for the 8th-order centered difference approximation for the
 * first derivative is given by:
 *
 * $$ f'(x) \approx \frac{1}{280h} \left[ -f(x-4h) + \frac{4}{3}f(x-3h) - \frac{1}{5}f(x-2h) + \frac{8}{5}f(x-h) - \frac{8}{5}f(x+h) + \frac{1}{5}f(x+2h) - \frac{4}{3}f(x+3h) + f(x+4h) \right]$$
 *
 * Note: Mathematica uses an 8th order approximation for the first derivative
 *
 * f: the function
 * x: the point at which to approximate the derivative
 * h: the step size
 *
 * See https://en.wikipedia.org/wiki/Finite_difference_coefficient
 */
export function centeredDiff8thOrder(
  f: (x: number) => number,
  x: number,
  h = 0.1
) {
  return (
    (f(x - 4 * h) / 280 -
      (4 * f(x - 3 * h)) / 105 +
      f(x - 2 * h) / 5 -
      (4 * f(x - h)) / 5 +
      (4 * f(x + h)) / 5 -
      f(x + 2 * h) / 5 +
      (4 * f(x + 3 * h)) / 105 -
      f(x + 4 * h) / 280) /
    h
  );
}

/**
 *
 * @param f
 * @param x
 * @param dir Direction of approach: > 0 for right, < 0 for left, 0 for both
 * @returns
 */
/**
 * Probe the geometric sample ladder that `extrapolate()` uses for a numeric