Counting and Isolating Polynomial Roots
Counting Roots of Polynomials
| CountRoots[poly,x] | give the number of real roots of the polynomial poly in x |
| CountRoots[poly,{x,a,b}] | give the number of roots of the polynomial poly in x with  |
Counting roots of polynomials.
CountRoots accepts polynomials with Gaussian rational coefficients. The root count includes multiplicities.
This gives the number of real roots of
.
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This counts the roots of
in the closed interval
.
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The roots of
in the vertical axis segment between
and
consist of a triple root at
and a single root at
.
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This counts 17
-degree roots of unity in the closed unit square.
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The coefficients of the polynomial can be Gaussian rationals.
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Isolating Intervals
A set
, where
is
or
, is an
isolating set for a root
of a polynomial
if
is the only root of
in
. Isolating roots of a polynomial means finding disjoint isolating sets for all the roots of the polynomial.
| RootIntervals[{poly1,poly2,...}] | give a list of disjoint isolating intervals for the real roots of any of the  , together with a list of which polynomials actually have each successive root |
| RootIntervals[poly] | give disjoint isolating intervals for real roots of a single polynomial |
| RootIntervals[polys,Complexes] | give disjoint isolating intervals or rectangles for complex roots of polys |
| IsolatingInterval[a] | give an isolating interval for the algebraic number a |
| IsolatingInterval[a,dx] | give an isolating interval of width at most dx |
Functions for isolating roots of polynomials.
RootIntervals accepts polynomials with rational number coefficients.
For a real root
the returned isolating interval is a pair of rational numbers
, such that either
or
. For a nonreal root
the isolating rectangle returned is a pair of Gaussian rational numbers
, such that
and either
or
.
Here are isolating intervals for the real roots of
.
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The second list shows which interval contains a root of which polynomial.
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This gives isolating intervals for all complex roots of
.
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Here are isolating intervals for the third- and fourth-degree roots of unity. The second interval contains a root common to both polynomials.
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Here is an isolating interval for a root of a polynomial of degree seven.
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This gives an isolating interval of width at most
.
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All numbers in the interval have the first 10 decimal digits in common.
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