/**
 * Returns a refined root given a root bracketed in the interval `(a,b)` of the
 * given polynomial using Brent's Method - modified slightly to allow for
 * error certified bounds.
 *
 * * also returns (as the 3rd element of the returned array) the current Brent
 * iterate `b` as the best point estimate within the interval
 *
 * * near exact implementation of the original Brent Dekker Method (also known
 * as Brent's Method), except that it is specialized to polynomial evaluation.
 *
 * * Brent's Method is an excellent root-refinement choice since:
 *  * guaranteed converge (unlike the Newton and other so-called single-point
 *    methods),
 *  * converges in a reasonable number of iterations even for highly contrived
 *    functions (unlike Dekker's Method) and
 *  * nearly always converges fast, i.e. super-linearly (unlike the Secant and
 *    Regula-Falsi methods).
 * * unfortunately the algorithm given on [Wikipedia](https://en.wikipedia.org/wiki/Brent%27s_method)
 *   works but is not precisely Brent's method and runs about 2x or more slower
 *   due to it not implementing the critically important 'micro-step' (Aug 2020).
 *
 * * see [Brent (page 47)](https://maths-people.anu.edu.au/~brent/pd/rpb011i.pdf)
 * * [c++ implementation of Brent's Method](https://people.sc.fsu.edu/~jburkardt/cpp_src/brent/brent.cpp)
 *
 * @param p a polynomial with coefficients given densely as an array of double-double
 * floating point numbers from highest to lowest power, e.g. `[[0,5],[0,-3],[0,0]]`
 * represents the polynomial `5x^2 - 3x`. If `exact` is `true` then this is allowed
 * to be `undefined`.
 * @param p_ an error polynomial that provides a coefficientwise error bound on
 * the input polynomial; all coefficients must be positive. If `exact` is `true`
 * then this is allowed to be `undefined`.
 * @param lb the lower limit of the search interval.
 * @param ub the upper limit of the search interval.
 * @param fa the result of evaluating the input polynomial at `a`
 * @param fb the result of evaluating the input polynomial at `b`
 * @param getPolyExact a function that returns the exact polynomial coefficients
 * @param exact defaults to `false`; set to true if you need to do exact evaluations from the start
 *
 * @internal
 */
declare function refineCertified(p: number[][] | undefined, p_: number[] | undefined, lb: number, ub: number, fa: number, fb: number, getPolyExact: () => number[][], exact?: boolean): [number, number, number];
export { refineCertified };