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

(1) | |||

(2) |

so the result is

(3) |

(4) |

(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.

© 1996-9

1999-05-25