Durand–Kerner method: Difference between revisions

Revision as of 15:04, 3 November 2020

Revision as of 15:07, 3 November 2020

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Both variant are effective root-finding algorithms.

One could also choose the initial values for ''p'', ''q'', ''r'', ''s''

by some other procedure, even randomly, but in a way that

* they are inside some not-too-large circle containing also the roots of &fnof;''f''(''x''), e.g. the circle around the origin with radius <math>1 + \max\big(|a|, |b|, |c|, |d|\big)</math>, (where 1, ''a'', ''b'', ''c'', ''d'' are the coefficients of &fnof;''f''(''x''))

and that

* they are not too close to each other,

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 ''p''₀ as the initial guess and make ''q''₀ and ''r''₀, etc, [[complex conjugate]] pairs. Then the iteration will preserve these properties; that is, ''p''_''n'' will always be real, and ''q''_''n'' and ''r''_''n'', etc, will always be conjugate. In this way, the ''p''_''n'' will converge to a real root ''P''. Alternatively, make all of the initial guesses real; they will remain so.

== Example ==