# RealZeroPackageQ Pol¶

This package provides functions for finding the real zeros of univariate polynomials over the rational numbers to arbitrary user-specified precision. The results are returned as a list of isolating intervals, expressed as records with “left” and “right” rational number components.

realZeros: (Pol, Fraction Integer) -> List Record(left: Fraction Integer, right: Fraction Integer)
realZeros(pol, eps) returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial pol.
realZeros: (Pol, Record(left: Fraction Integer, right: Fraction Integer)) -> List Record(left: Fraction Integer, right: Fraction Integer)
realZeros(pol, range) returns a list of isolating intervals for all the real zeros of the univariate polynomial pol which lie in the interval expressed by the record range.
realZeros: (Pol, Record(left: Fraction Integer, right: Fraction Integer), Fraction Integer) -> List Record(left: Fraction Integer, right: Fraction Integer)
realZeros(pol, int, eps) returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial pol which lie in the interval expressed by the record int.
realZeros: Pol -> List Record(left: Fraction Integer, right: Fraction Integer)
realZeros(pol) returns a list of isolating intervals for all the real zeros of the univariate polynomial pol.
refine: (Pol, Record(left: Fraction Integer, right: Fraction Integer), Fraction Integer) -> Record(left: Fraction Integer, right: Fraction Integer)
refine(pol, int, eps) refines the interval int containing exactly one root of the univariate polynomial pol to size less than the rational number eps.
refine: (Pol, Record(left: Fraction Integer, right: Fraction Integer), Record(left: Fraction Integer, right: Fraction Integer)) -> Union(Record(left: Fraction Integer, right: Fraction Integer), failed)
refine(pol, int, range) takes a univariate polynomial pol and and isolating interval int which must contain exactly one real root of pol, and returns an isolating interval which is contained within range, or “failed” if no such isolating interval exists.