RootIntervals

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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.

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RootIntervals[poly]

gives isolating intervals for real roots of a single polynomial.

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RootIntervals[polys,Complexes]

gives bounding rectangles for complex roots.

Details

The coefficients of poly must be integers or rationals.
An isolating interval for a root of a polynomial poly is an interval where the only root of poly contained in the interval is .
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

open allclose all

Basic Examples (2)Summary of the most common use cases

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

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-le271

Out[1]=1

The isolating intervals are always specified by exact rationals:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-c9r06z

Out[1]=1

Scope (6)Survey of the scope of standard use cases

Isolate the real roots of a polynomial:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-jyy5i

Out[1]=1

Isolate the real roots of a list of polynomials:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-jzdn21

Out[1]=1

Isolate the complex roots of a polynomial:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-lu2g3i

Out[1]=1

Isolate the complex roots of a list of polynomials:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-4jscs

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-c1dk5o

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/05fuof7dac-2x5j8

Out[2]=2

Isolating intervals of rational roots may be single points:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-jb1noo

Out[1]=1

Applications (1)Sample problems that can be solved with this function

Find numeric approximations of real roots of a polynomial:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-bq4wo4

Find isolating intervals:

In[2]:=2

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https://wolfram.com/xid/05fuof7dac-cn71sp

Out[2]=2

Find root approximations:

In[3]:=3

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https://wolfram.com/xid/05fuof7dac-dafgmq

Out[3]=3

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

In[4]:=4

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https://wolfram.com/xid/05fuof7dac-det4oo

Out[4]=4

Compute approximations of the Root objects:

In[5]:=5

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https://wolfram.com/xid/05fuof7dac-g7yqbf

Out[5]=5

Properties & Relations (1)Properties of the function, and connections to other functions

Find real and complex roots of polynomials:

In[1]:=1

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https://wolfram.com/xid/05fuof7dac-cikx4x

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

In[2]:=2

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https://wolfram.com/xid/05fuof7dac-c4ai2m

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/05fuof7dac-cze7ae

Out[3]=3

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

In[4]:=4

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https://wolfram.com/xid/05fuof7dac-drxqyf

Out[4]=4

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

In[5]:=5

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https://wolfram.com/xid/05fuof7dac-d3hwjd

Out[5]=5

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

In[6]:=6

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https://wolfram.com/xid/05fuof7dac-b2a4v

Out[6]=6

Use Solve to find the complex roots with multiplicities:

In[7]:=7

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https://wolfram.com/xid/05fuof7dac-cwhf2m

Out[7]=7

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