 
 
 
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 | ⎤ ⎦
 | ⎤ ⎦
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
 
 
