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 Re (a)≤Re (r)≤Re (b) Im (a)≤Im (r)≤Im (b) |
Counting roots of polynomials.
CountRoots accepts polynomials with Gaussian rational coefficients. The root count includes multiplicities.
This gives the number of real roots of (x2-2) (x2-3) (x2-4).
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This counts the roots of (x2-2) (x2-3) (x2-4) in the closed interval [1, 2].
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The roots of (x2+1) x3 in the vertical axis segment between 0 and 2 consist of a triple root at 0 and a single root at .
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This counts 17 th-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
S
K, where
K is
or
, is an
isolating set for a root
a of a polynomial
f if
a is the only root of
f in
S. 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 polyi, 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
r the returned isolating interval is a pair of rational numbers
{a, b}, such that either
a<r<b or
a
br. For a nonreal root
r the isolating rectangle returned is a pair of Gaussian rational numbers
{a, b}, such that
Re (a)<Re (r)<Re (b)Im (a)<Im (r)<Im (b) and either
Im (a)≥0 or
Im (b)≤0.
Here are isolating intervals for the real roots of f.
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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 f+3.
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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 10-10.
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All numbers in the interval have the first ten decimal digits in common.
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