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A New and Efficient Program for Finding all Polynomial Roots

Markus Lang?, Electrical and Computer Engineering Department
Rice University, Houston, TX 77251, lang@dsp.rice.edu

Bernhard-Christian Frenzel, Sonnebergerstr. 11
90765 F?urth, Germany

Technical Report # 9308

April 15, 1994

Abstract

Finding polynomial roots rapidly and accurately is an important problem in many areas of signal processing. We present a new program which is a combination of Muller's and Newton's method. We use the former for computing a root of the deflated polynomial which is a good estimate for the root of the original polynomial. This estimate is improved by applying Newton's method to the original polynomial. Test polynomials up to the degree 10000 show the superiority of our program over the best methods to our knowledge regarding speed and accuracy, i.e., Jenkins/Traub program and the eigenvalue method.

Furthermore we give a simple approach to improve the accuracy for spectral factorization in the case there are double roots on the unit circle. Finally we briefly consider the inverse problem of root finding, i.e., computing the polynomial coefficients from the roots which may lead to surprisingly large numerical errors.

?This research was partially supported by Alexander von Humboldt-Stiftung and by AFOSR under grant F49620- 1-0006 funded by DARPA