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Toby Bartels (talk | contribs) →Variations: Find real roots of odd-degree polynomials. |
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and that the roots are found simultaneously rather than one at a time.
== Variations ==
This iteration procedure, like the [[Gauss–Seidel method]] for linear equations,
computes one number at a time based on the already computed numbers.
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which may increasingly become a concern
as the degree of the polynomial increases.
If the coefficients are real and the polynomial has odd degree, then it must have at least one real root. To find this, use a real value of <i>p</i><sub>0</sub> as the initial guess and make <i>q</i><sub>0</sub> and <i>r</i><sub>0</sub>, etc, [[complex conjugate]] pairs. Then the iteration will preserve these properties; that is, <i>p</i><sub><i>n</i></sub> will always be real, and <i>q</i><sub><i>n</i></sub> and <i>r</i><sub><i>n</i></sub>, etc, will always be conjugate. In this way, the <i>p</i><sub><i>n</i></sub> will converge to a real root <i>P</i>. Alternatively, make all of the initial guesses real; they will remain so.
== Example ==
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