Find multiple roots of a function within an interval

Searches a numeric interval for all roots (zeros) of a function f by subdividing it into n sub-intervals, detecting sign changes, and refining each candidate with uniroot().

Usage

unirootAll(
  f,
  interval,
  lower = min(interval),
  upper = max(interval),
  tol = .Machine$double.eps^0.5,
  maxiter = 1000,
  n = 100,
  ...
)

Arguments

f

a function for which roots are sought. Must accept a numeric first argument; additional arguments are passed via ....

interval

a numeric vector of length 2 specifying the search interval. Either interval or both lower and upper must be supplied.

lower

lower bound of the search interval. Default: min(interval).

upper

upper bound of the search interval. Default: max(interval).

tol

convergence tolerance passed to uniroot(), and also used as the threshold for (i) treating grid-point values as exact zeros and (ii) collapsing near-duplicate roots. Default: .Machine$double.eps^0.5.

maxiter

maximum number of iterations for uniroot(). Default: 1000.

n

number of sub-intervals used for the initial grid search. Increase n if roots may be close together or the function oscillates rapidly. Default: 100.

...

additional arguments passed to f.

Value

a numeric vector of roots found in [lower, upper], sorted in ascending order. Returns numeric(0) if no roots are found.

Details

The function f is called as f(x, ...) where x is a numeric vector. If f does not accept vector input, it is called element-wise via sapply.

Grid points at which |f(x)| < tol are returned directly as roots. Sign changes are detected using sign(), which avoids numerical overflow that can occur with product-based approaches. Non-finite function values are silently ignored when detecting sign changes. If uniroot fails on a sub-interval, that interval is skipped with a warning rather than aborting the entire search.

Limitations: Roots within the same sub-interval of width (upper - lower) / n may be missed. Roots of even multiplicity that do not produce a sign change will not be found unless they happen to fall on a grid point. A warning is issued if no roots are found at all despite finite function values being present.

See also

stats::uniroot() for the underlying single-root solver.

Other math.basic: closest(), crossProd(), crossProdN(), dotProd(), roundTo()

Examples

f <- function(x) cos(2 * x)^3
roots <- unirootAll(f, c(0, 10))
stopifnot(all(abs(f(roots)) < 1e-6))

# Non-vectorized function
g <- Vectorize(function(x) integrate(function(t) t^x, 0, 1)$value - 0.5)
unirootAll(g, c(0.1, 5))
#> [1] 1