Isolating and refining real roots

Cost

Isolation: one evaluation of

f

and

f'

on an interval per examined subinterval,

O(k \cdot \mathrm{depth})

subintervals for

k

roots in favourable cases. Refinement to

p

bits:

O(p)

evaluations by bisection,

O(\log p)

Newton steps at doubling precisions, total cost about twice that of one evaluation at

p

bits.

Let $f$ be a real function that is analytic on an interval $[a, b]$ , and suppose that for any subinterval $I$ we can compute balls containing $f(I) = \{f(t) : t \in I\}$ and $f'(I)$ , as well as the value of $f$ at exact points. The root isolation problem asks for disjoint intervals that each contain exactly one root of $f$ and together contain all roots in $[a, b]$ . With interval evaluations, this can be solved for simple roots by bisection, and isolated roots can then be refined rapidly by Newton's method with rigorous error bounds [Moo1966; Neu1990].

Exclusion and isolation tests#

Lemma 1 (Exclusion). If the computed enclosure of $f(I)$ does not contain 0, then $f$ has no root in $I$ .

Lemma 2 (Isolation). Let $I = [u, v]$ . If $f(u)$ and $f(v)$ are nonzero with opposite signs and the enclosure of $f'(I)$ does not contain 0, then $f$ has exactly one root in $I$ , it lies in the interior, and it is simple.

Proof. $f'$ is continuous and does not vanish on $I$ , so it has constant sign and $f$ is strictly monotonic on $I$ ; by the intermediate value theorem it has a root in $(u, v)$ , which is unique by monotonicity and simple since $f' \ne 0$ there.

Algorithm 1: Isolation by bisection

Input: $f$ , $[a, b]$ , limits on the depth, the number of evaluations and the number of roots. Output: a sorted list of subintervals with flags.

Compute the signs $\sigma_a$ , $\sigma_b$ of $f(a)$ and $f(b)$ (0 if the computed value contains zero).
Process $[a, b]$ recursively: for a subinterval $I = [u, v]$ with known endpoint signs, if Lemma 1 applies, discard $I$ ; if Lemma 2 applies, output $I$ with flag 1; if a limit has been reached, output $I$ with flag 2 (unknown); otherwise compute the sign of $f$ at the midpoint $w$ and process $[u, w]$ and $[w, v]$ .

Every root of $f$ in $[a, b]$ lies in some output interval, because only intervals that certainly contain no root are discarded. The intervals are output from left to right and overlap at most in endpoints. If all roots are simple and none of them is a bisection point, then once the intervals are small enough, each contains at most one root and the enclosures of $f(I)$ and $f'(I)$ are accurate enough for the tests to succeed, so the algorithm terminates with all roots isolated. A multiple root, or a root at a bisection point (where the sign is zero), is never isolated and ends up in intervals of the minimal width with flag 2.

The cost is dominated by the evaluations on intervals. If the enclosures are tight, roots at distance $\delta$ from each other are separated after about $\log_2((b - a)/\delta)$ bisection levels; the dependency problem of interval evaluation (the enclosure of $f(I)$ can be much wider than the true range) makes the tests succeed only later. Evaluating $f$ through a Taylor expansion at the midpoint of $I$ (centred forms) reduces the overestimation from $O(|I|)$ to $O(|I|^2)$ .

Newton iteration with a quadratic error bound#

Theorem 3. Let $f$ be twice continuously differentiable on an interval $J$ , let $x = [m - r, m + r] \subseteq J$ contain a root $\xi$ of $f$ , and let

C \ge \sup_{s, t \in J} \frac{|f''(s)|}{2 |f'(t)|}.

Then $m' = m - f(m)/f'(m)$ satisfies $|m' - \xi| \le C r^2$ .

Proof. By Taylor's theorem, $0 = f(\xi) = f(m) + f'(m)(\xi - m) + \tfrac12 f''(\eta)(\xi - m)^2$ for some $\eta$ between $m$ and $\xi$ , hence

m' - \xi = m - \xi - \frac{f(m)}{f'(m)} = \frac{f''(\eta)}{2 f'(m)} (\xi - m)^2,

and $|m' - \xi| \le C |\xi - m|^2 \le C r^2$ .

arb_calc_newton_step and _arb_poly_newton_step compute $m'$ in ball arithmetic at the working precision (the rounding errors are part of the resulting ball) and add $C r^2$ to the radius. The step is accepted if the new ball is contained in $J$ (so that the next step can use the same $C$ ) and has a smaller radius. Once $C r < 1$ , the radius is roughly squared at each step, so the number of correct bits doubles. arb_calc_refine_root_newton and _arb_poly_newton_refine_root exploit this by running the steps at precisions that also double: the last step at $p + O(\log C)$ bits, the one before at about half of that, and so on, down to the accuracy of the starting ball. The total cost is then about twice that of the last step.

The factor $C$ is computed by evaluating $f'(J)$ and $f''(J)$ on the whole interval $J$ (arb_calc_newton_conv_factor, _arb_poly_newton_convergence_factor). It is finite only if $f'$ is bounded away from zero on $J$ , which requires $J$ to be a narrow interval around a simple root. (In FLINT 3.6 and earlier arb_calc_newton_conv_factor returned only $C/2$ , for which Theorem 3 does not apply.)

The interval Newton operator#

A variant needs no bound for $f''$ . For an interval $X$ with midpoint $m$ , the interval Newton operator is

N(X) = m - \frac{f(m)}{f'(X)},

where $f'(X)$ is an enclosure of the range of $f'$ on $X$ .

Theorem 4 (Interval Newton). If $0 \notin f'(X)$ , then every root of $f$ in $X$ lies in $N(X)$ . If moreover $N(X)$ is contained in the interior of $X$ , then $f$ has exactly one root in $X$ .

Proof. For a root $\xi \in X$ , the mean value theorem gives $f(m) = f(m) - f(\xi) = f'(\eta)(m - \xi)$ with $\eta \in X$ , so $\xi = m - f(m)/f'(\eta) \in N(X)$ .

For existence, let $X = [u, v]$ and suppose that $f'(X) \subset (0, \infty)$ (the negative case is symmetric). By the mean value theorem, $f(v) = f(m) + d (v - m)$ with $d = f'(\eta) \in f'(X)$ . Since $m - f(m)/d \in N(X)$ lies in the interior of $X$ , we have $m - f(m)/d < v$ , that is $-f(m) < d (v - m)$ , so $f(v) > 0$ . In the same way, $f(u) = f(m) - d'(m - u)$ with $d' \in f'(X)$ and $m - f(m)/d' > u$ give $f(u) < 0$ . By the intermediate value theorem, $f$ has a root in $(u, v)$ , which is unique since $f$ is strictly increasing on $X$ [Moo1966; Neu1990].

When the root is well isolated, $N(X)$ has width $O(|X|^2)$ , so the interval Newton method also converges quadratically. arb_fmpz_poly_refine_root_arb combines it with ordinary Newton steps and bisection: it keeps an exact interval $[\ell, h]$ that contains the root, tries an interval Newton step (accepted if $N(X)$ lies in the interior of $X$ and gains at least 10% in accuracy), otherwise an ordinary Newton step whose result is accepted if a small interval around it has a sign change at its endpoints, and otherwise bisects. The working precision follows twice the current accuracy, so the cost is again dominated by the last steps.

Use in FLINT#

Algorithm 1: arb_calc_isolate_roots, with arb_calc_partition for the bisection step; arb_calc_refine_root_bisect refines by bisection.
Theorem 3: arb_calc_newton_step, arb_calc_refine_root_newton and arb_calc_newton_conv_factor for functions given by evaluators; _arb_poly_newton_step, _arb_poly_newton_refine_root and _arb_poly_newton_convergence_factor for polynomials.
Theorem 4: arb_fmpz_poly_refine_root_arb, used by arb_fmpz_poly_real_roots after exact isolation by Descartes' rule (Counting and isolating real roots over Z).
The zeros of the Riemann zeta function and of Dirichlet $L$ -functions on the critical line are isolated by sign changes of the Hardy $Z$ -function and refined in acb_dirichlet.h.

@misc{flint-doc-algorithms-ball-arithmetic-real-root-isolation, author = {{The FLINT team}}, title = {{Isolating and refining real roots}}, howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev}, year = {2026}, url = {https://flintlib.org/doc/algorithms/ball-arithmetic/real-root-isolation.html}, note = {Generated 2026-11-02} }