11.3.6 Exact bounds for complex roots of a polynomial
The complexroot
command finds bounds for the complex roots of a polynomial.
-
complexroot takes two mandatory arguments and two
optional arguments:
-
P, a polynomial.
- ε, a postive real number.
- Optionally, α, β, two complex numbers.
- complexroot(P,ε)
returns a list of vectors, where the elements of each vector are one
of:
- complexroot(P,ε,α,β)
returns a list of vectors as above, but only for the roots lying in
the rectangle with sides parallel to the axis having α,β as
opposite vertices.
Examples
Find the roots of x3+1.
|
| ⎡
⎢
⎢
⎢
⎢
⎣ | −1 | 1 |
| ⎡
⎣ | 0.499999046325680..0.500000953674320 | ⎤
⎦ | − | ⎡
⎣ | 0.866024494171135..0.866026401519779 | ⎤
⎦ |
i |
| 1 |
| ⎡
⎣ | 0.499999046325680..0.500000953674320 | ⎤
⎦ | + | ⎡
⎣ | 0.866024494171135..0.866026401519779 | ⎤
⎦ |
i |
| 1
|
| ⎤
⎥
⎥
⎥
⎥
⎦ |
|
| | | | | | | | | | |
|
Hence, for x3+1, −1 is a root of multiplicity 1,
a+ib is a root of multiplicity 1 with
0.499999046325680 ≤ a ≤ 0.500000953674320 and
−0.866026401519779 ≤ b ≤ −0.866024494171135, and
c+id is a root of multiplicity 1 with
0.499999046325680 ≤ c ≤ 0.500000953674320 and
0.866024494171135 ≤ d ≤ 0.866026401519779.
Find the roots of x3+1 lying inside the rectangle
with opposite vertices −1,1+2i.
complexroot(x^3+1,0.1,-1,1+2*i) |
|
| ⎡
⎢
⎢
⎣ | −1 | 1 |
| ⎡
⎣ | 0.499999046325680..0.500000953674320 | ⎤
⎦ | + | ⎡
⎣ | 0.866024494171135..0.866026401519779 | ⎤
⎦ |
i |
| 1
|
| ⎤
⎥
⎥
⎦ |
|
| | | | | | | | | | |
|