Laguerre,s Method
المؤلف:
Acton, F. S
المصدر:
Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., 1990.
الجزء والصفحة:
...
12-12-2021
1518
Laguerre's Method
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,
Now consider the logarithm and logarithmic derivatives of 
Now make "a rather drastic set of assumptions" that the root 
being sought is a distance 
from the current best guess, so
 |
(11)
|
while all other roots are at the same distance 
, so
 |
(12)
|
for 
, 3, ..., 
(Acton 1990; Press et al. 1992, p. 365). This allows 
and 
to be expressed in terms of 
and 
as
Solving these simultaneously for 
gives
![a=n/(max[G+/-sqrt((n-1)(nH-G^2))]),](https://mathworld.wolfram.com/images/equations/LaguerresMethod/NumberedEquation3.gif) |
(15)
|
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.
REFERENCES:
Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., 1990.
Adams, D. A. "A Stopping Criterion for Polynomial Root Finding." Comm. ACM 10, 655-658, 1967.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 365-366, 1992.
Ralston, A. and Rabinowitz, P. §8.9-8.13 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978.
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