> <\body> <\tmdoc-title> Sleeve univariate solver \; A first family of methods correspond to the scheme. It starts with a representation of a polynomial (on an interval >) and according to some criterion , it applies remove the interval, keep it on split it:\ <\framed-fragment> <\named-algorithm-old> solve_binary(f,I) <|named-algorithm-old> \; : Isolation of the roots of on the interval >. : A representation , |)>> associate with .\ <\itemize> |)>> of the coefficient sequence >>. ;a,b|)>=0>, remove the interval. > ;a,b|)>=1>, output the interval > and |)>>.> ;a,b|)>\1> subdivide the representation into two subrepresentations, corresponding to the two halves of the input interval and apply recursively the algorithm to them>. : a list of subintervals of >. \; Typically, one can use the representation of a polynomial in the Bernstein basis on >: = f B>, where =>|>>>>>. The subdivision step will be based on de Casteljau algorithm. For ;a,b|)>>, one can take the number of sign variation of the control coeffcients =,\,f|]>>, not taking into account the of this sequence. By Descartes' rule, if the polynomial has no multiple root, the algorithm outputs a list of intervals containing one and only one root.\ Two variants are implemented. <\itemize-dot> If we take for ;a,b|)>> the number of sign variation of sequence =,\,f|]>>, not taking into account the of this sequence. By Descartes' rule, if the polynomial has no multiple root, the algorithm outputs in this casea list of intervals containing one and only one root.\ If we asked in addition that \\>, where > is the precision attached to the solver, we approximate the roots within the precision >. \; <\initial> <\collection>