How this calculation works
Choose a valid sign-changing bracket for bisection or an initial guess for Newton. Inspect the termination reason, iteration trace and polynomial residual. A returned last iterate after a stop or iteration limit is different from a result that reached its selected tolerance.
Bisection preserves opposite signs; Newton uses x(next)=x−f(x)/f′(x)A worked example
For x³−2 on [1,2], bisection approaches 1.2599210499. Newton starting at 1 reaches the same real root through tangent-based updates, but Newton is not guaranteed to converge from every guess.
Good to know
One real approximation only, degree ≤12. Bisection can miss even-multiplicity roots without a sign change. Newton can stall, cycle, encounter zero derivative or leave the supported range. Its scaled residual is not a root-location error bound, and floating-point zero is not exact symbolic certification.
Reference: OpenStax Calculus: Newton’s method and its limitations (opens in a new tab) ↗
A bracket and a guess make different promises
Bisection starts from opposite polynomial signs at two endpoints and repeatedly halves that bracket. For a continuous polynomial, the sign change establishes that at least one real root lies inside. It does not establish that only one root is present.
Newton uses the tangent update from one initial guess. It can converge rapidly near a suitable simple root, but a poor guess can lead to a zero derivative, a cycle or a large step. The termination message is part of the answer, not a detail to ignore.
The rational polynomial zero finder checks exact rational candidates. This numerical polynomial root calculator instead follows a selected real iteration with an explicit stopping condition.
A small residual is not always a small root error
The residual is the polynomial value at the final approximation. A flat or ill-conditioned root can have a tiny residual while its x coordinate is still inaccurate. This page shows the residual and a coefficient-scale-normalized residual separately.
For x³−2 on [1,2], both methods can approach the real cube root, about 1.2599210499. Change Newton’s starting point or reduce the iteration limit to see why returning a number alone would conceal meaningful differences in solver behavior.
Check which stopping rule was reached
Bisection stops using the half-width of its current bracket relative to 1+|x|, unless an endpoint or midpoint evaluates to floating-point zero. Newton uses the scaled residual. The same numeric tolerance therefore does not describe identical stopping quantities in the two methods.
An iteration limit, a zero derivative or an out-of-range update leaves a last approximation without a convergence claim. The rational zero calculator handles a different task: exact rational-root candidates under its own coefficient limits. This numerical page does not enumerate all real or complex roots.
A few questions, answered
Why does bisection reject x² on [−1,1]?
Both endpoints are positive. The root at zero has even multiplicity and does not create the required endpoint sign change.
Is “exact zero” a symbolic proof?
No. It means the selected floating-point evaluation returned zero. Cancellation and rounding still apply.
Is an iteration-limited result useless?
It can show progress or a poor setup, but it must not be presented as having reached the requested stopping tolerance. Inspect the trace and residuals.