11.3.1 Exact bounds for real roots of a polynomial
Bounds for the real roots of a polynomial can be found by
realroot and VAS commands.
-
realroot takes two mandatory arguments and two
optional arguments:
-
P, a polynomial.
- ε, a postive real number.
- Optionally, a, b, two complex numbers.
- realroot(P,ε)
returns a list of vectors, where the elements of each vector are a
list containing one of:
-
an interval of length less than ε containing a real
root of the polynomial and the multiplicity of this root.
- the value of an exact real root of the polynomial and the
multiplicity of this root.
- realroot(P,ε,a,b)
returns a list of vectors as above, but only for the roots lying in
the interval [a,b].
The VAS command uses the
Vincent-Akritas-Strzebonski algorithm to find intervals
containing the real roots of polynomials.
-
VAS takes
P, a polynomial.
- VAS(P) returns a list of
intervals which contain the real roots of P, where each
interval contains exactly one root.
Examples
Find the real roots of x3+1.
Find the real roots of x3−x2−2x+2.
| realroot(x^3-x^2-2*x+2,0.1) |
|
| |
| ⎡
⎢
⎢
⎣ | | −[1.40624999999999..1.50000000000001] | 1 |
| 1 | 1 |
| [1.37499999999999..1.43750000000001] | 1
|
| ⎤
⎥
⎥
⎦ |
|
| | | | | | | | | | |
|
Find the real roots of x3−x2−2x+2 in the interval [0;2].
| realroot(x^3-x^2-2*x+2,0.1,0,2) |
|
| |
| ⎡
⎢
⎣ | | 1 | 1 |
| [1.37499999999999..1.43750000000001] | 1 |
| ⎤
⎥
⎦ |
|
| | | | | | | | | | |
|
|
| |
| ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
|
| | | | | | | | | | |
|
| VAS(x^5+2*x^4-6*x^3-7*x^2+7*x+7) |
|
| |
| ⎡
⎢
⎢
⎣ | ⎡
⎣ | −5,−1 | ⎤
⎦ | ,−1, | ⎡
⎢
⎢
⎣ | 1, | | ⎤
⎥
⎥
⎦ | , | ⎡
⎢
⎢
⎣ | | ,2 | ⎤
⎥
⎥
⎦ | ⎤
⎥
⎥
⎦ |
| | | | | | | | | | |
|
|
| |
| ⎡
⎣ | ⎡
⎣ | −3,0 | ⎤
⎦ | ,1, | ⎡
⎣ | 1,3 | ⎤
⎦ | ⎤
⎦ |
| | | | | | | | | | |
|