A root-finding algorithm which converges to a complex root from any starting position. To motivate the formula, consider
an 
th
order polynomial and its derivatives,
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(1)
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(2)
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(3)
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(4)
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Now consider the logarithm and logarithmic derivatives of 
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(5)
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(6)
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(7)
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(8)
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(9)
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 |  | ![[(P_n^'(x))/(P_n(x))]^2-(P_n^('')(x))/(P_n(x))=H(x).](/images/equations/LaguerresMethod/Inline31.svg) |
(10)
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Now make "a rather drastic set of assumptions" that the root 
being sought is a distance 
from the current best guess, so
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(11)
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while all other roots are at the same distance 
, so
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(12)
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for 
,
3, ..., 
(Acton 1990; Press et al. 1992, p. 365). This allows 
and 
to be expressed in terms of 
and 
as
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(13)
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(14)
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Solving these simultaneously for 
gives
![a=n/(max[G+/-sqrt((n-1)(nH-G^2))]),](/images/equations/LaguerresMethod/NumberedEquation3.svg) |
(15)
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where the sign is taken to give the largest magnitude for the denominator.
To apply the method, calculate 
for a trial value 
, then use 
as the next trial value, and iterate until 
becomes sufficiently small. For example, for the polynomial
with starting point 
, the algorithmic converges to
the real root very quickly as (
, 
, 
).
Setting 
gives Halley's irrational formula.
