= Solution
{c}
The following <R (programming language)> function uses the factored polynomial for evaluation, handles endpoint and midpoint roots, and returns a midpoint only once its error bound is at most `eps`:
``
bisect <- function(a, b, eps = 1e-8, maxiter = 1000L) {
f <- function(x) x * (x - 1) * (x - 2) * (x - 5)
stopifnot(length(a) == 1L, length(b) == 1L,
is.finite(a), is.finite(b), a < b,
length(eps) == 1L, is.finite(eps), eps > 0)
fa <- f(a)
fb <- f(b)
if (fa == 0) return(a)
if (fb == 0) return(b)
if (sign(fa) == sign(fb)) stop("Endpoints must bracket a sign change")
for (i in seq_len(maxiter)) {
m <- a + (b - a) / 2
fm <- f(m)
if (fm == 0 || (b - a) / 2 <= eps) return(m)
if (sign(fm) == sign(fa)) {
a <- m
fa <- fm
} else {
b <- m
}
}
stop("Iteration limit reached")
}
``
Each update preserves the sign-changing bracket, and the stopping bound is the half-width of that bracket. Thus <continuity> and the <intermediate value theorem> justify the returned root approximation.
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