Built-in Mathematica Symbol

RootIntervals

RootIntervals[{poly₁, poly₂, ...}]
gives a list of isolating intervals for the real roots of any of the poly_i, together with a list of which polynomials actually have each successive root.

RootIntervals[poly]
gives isolating intervals for real roots of a single polynomial.

RootIntervals[polys, Complexes]
gives bounding rectangles for complex roots.

MORE INFORMATION

The coefficients of poly must be integers or rationals.

An isolating interval for a root a of a polynomial poly is an interval where the only root of poly contained in the interval is a.

If a root is real, the isolating interval is an open real interval, or a point. If a root is not real, the isolating interval is an open rectangle, disjoint from the real axis.

Multiple roots give multiple entries in the second list generated by RootIntervals.

EXAMPLES

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Basic Examples (2)

Get isolating intervals, together with a list of which polynomial has which root:

The isolating intervals are always specified by exact rationals:

Get isolating intervals, together with a list of which polynomial has which root:

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The isolating intervals are always specified by exact rationals:

In[1]:=

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Scope (6)

Isolate the real roots of a polynomial:

Isolate the real roots of a list of polynomials:

Isolate the complex roots of a polynomial:

Isolate the complex roots of a list of polynomials:

Polynomials may have multiple roots; pairs of polynomials may have common roots:

Isolating intervals of rational roots may be single points:

Applications (1)

Find numeric approximations of real roots of a polynomial:

Find isolating intervals:

Find root approximations:

Reduce uses a similar approach, but factoring the polynomial for Root objects takes time:

Compute approximations of the Root objects:

Properties & Relations (1)

Find real and complex roots of polynomials:

Isolate the real roots; multiple roots are indicated in the second part of the output:

Use CountRoots to count the real roots; multiple roots are counted with multiplicities:

Use Reduce to find the real roots; multiple roots are given once:

Isolate the complex roots; multiple roots are indicated in the second part of the output:

Use Reduce to find the complex roots; multiple roots are given once:

Use Solve to find the complex roots with multiplicities: