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##

Exact bounds for complex roots of a polynomial :

`complexroot`

`complexroot` takes 2 or 4 arguments : a polynomial and a real
number and optionnally two complex numbers
,.

`complexroot` returns a list of vectors.
- If
`complexroot` has 2 arguments,
the elements of each vector are
- either an interval (the
boundaries of this interval are the opposite vertices of a rectangle with sides
parallel to the axis and containing a complex root of the polynomial) and the
multiplicity of this root.

Let the interval be
[*a*_{1} + *ib*_{1}, *a*_{2} + *ib*_{2}] then
| *a*_{1} - *a*_{2}| < ,
| *b*_{1} - *b*_{2}| < and the root *a* + *ib* verifies
*a*_{1} *a* *a*_{2} and
*b*_{1} *b* *b*_{2}.
- or the value of an exact complex root of
the polynomial and the multiplicity of this root

- If
`complexroot` has 4 arguments, `complexroot` returns a list of
vectors as above, but only for the roots lying in
the rectangle with sides parallel to the axis having
, as
opposite vertices.

To find the roots of *x*^{3} + 1, input:
`complexroot(x``^`

3+1,0.1)

Output :
`[[-1,1],[[(4-7*i)/8,(8-13*i)/16],1],[[(8+13*i)/16,(4+7*i)/8],1]]`

Hence, for *x*^{3} + 1 :
- -1 is a root of multiplicity 1,
- 1/2+i*
*b* is a root of multiplicity 1 with
-7/8 *b* - 13/16,
- 1/2+i*
*c* is a root of multiplicity 1 with
13/1 *c* 7/8.

To find the roots of *x*^{3} + 1 lying inside the rectangle
of opposite vertices -1, 1 + 2**i*, input:
`complexroot(x``^`

3+1,0.1,-1,1+2*i)

Output :
`[[-1,1],[[(8+13*i)/16,(4+7*i)/8],1]]`

** suivant:** Exact bounds for real
** monter:** Exact roots of a
** précédent:** Exact roots of a
** Table des matières**
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giac documentation written by Renée De Graeve and Bernard Parisse