Find a root of every row's function within its bracket

Regula falsi with the Illinois modification, run on every row at once. Each row keeps an interval whose ends give its function opposite signs, so the root is never lost, and whenever the same end survives twice in a row its function value is halved for the next interpolation, which restores superlinear convergence where plain regula falsi would stall. A point the interpolation cannot place strictly inside the interval, as with an infinite end, is replaced by the midpoint. A row stops when its interval is narrower than tolerance times the larger of one and its magnitude, or when its function is exactly zero, which collapses the interval onto the root.

Usage

vectorised_bracketed_root(
  f,
  lower,
  upper,
  f_lower,
  f_upper,
  tolerance = 1e-13,
  max_iterations = 100L
)

Arguments

f

Function of the points and the indices of the rows they belong to, returning one value per point.

lower, upper

Ends of each row's interval.

f_lower, f_upper

The function at those ends, of opposite signs or zero.

tolerance

Relative width at which a row stops.

max_iterations

Iteration cap, far above what convergence needs.

Value

A list with the final lower and upper ends, the function's sign there as f_lower and f_upper (its magnitude is rescaled by the Illinois step), and root, the midpoint of the final interval.