“Graeffe's Method”的概念、定义、习题解题方法及翻译-数学辞典

单词

Graeffe's Method

释义

Graeffe's Method

A Root-finding method which proceeds by multiplying a Polynomial by and noting that

			(1)
			(2)

so the result is

(3)

Repeat

times, then write this in the form

(4)

where

. Since the coefficients are given by Newton's Relations

			(5)
			(6)
			(7)

and since the squaring procedure has separated the roots, the first term is larger than rest. Therefore,

			(8)
			(9)
			(10)

giving

			(11)
			(12)
			(13)

Solving for the original roots gives

			(14)
			(15)
			(16)

This method works especially well if all roots are real.

References

von Kármán, T. and Biot, M. A. ``Squaring the Roots (Graeffe's Method).'' §5.8.c in Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems. New York: McGraw-Hill, pp. 194-196, 1940.

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