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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 [a1 + ib1, a2 + ib2] then | a1 - a2| < , | b1 - b2| < and the root a + ib verifies a1 a a2 and b1 b b2.
• 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 x3 + 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 x3 + 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 x3 + 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]]

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giac documentation written by Renée De Graeve and Bernard Parisse