# The Laguerre-and-sums-of-powers algorithm for the efficient and reliable approximation of all polynomial roots

The Laguerre-and-sums-of-powers algorithm for the efficient and reliable approximation of all... We prove the first sufficient convergence criterion for Laguerre’s root-finding algorithm, which by empirical evidence is highly efficient. The criterion is applicable to simple roots of polynomials with degree greater than 3. The “Sums of Powers Algorithm” (SPA), which is a reliable iterative root-finding method, can be used to fulfill the condition for each root. Therefore, Laguerre’s method together with the SPA is now an efficient and reliable algorithm (LaSPA). In computational mathematics these results solve a central task which was first attacked by L. Euler 266 years ago. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# The Laguerre-and-sums-of-powers algorithm for the efficient and reliable approximation of all polynomial roots

, Volume 51 (4) – Jan 5, 2016
10 pages

/lp/springer_journal/the-laguerre-and-sums-of-powers-algorithm-for-the-efficient-and-qEI8mx9vxr
Publisher
Springer Journals
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946015040055
Publisher site
See Article on Publisher Site

### Abstract

We prove the first sufficient convergence criterion for Laguerre’s root-finding algorithm, which by empirical evidence is highly efficient. The criterion is applicable to simple roots of polynomials with degree greater than 3. The “Sums of Powers Algorithm” (SPA), which is a reliable iterative root-finding method, can be used to fulfill the condition for each root. Therefore, Laguerre’s method together with the SPA is now an efficient and reliable algorithm (LaSPA). In computational mathematics these results solve a central task which was first attacked by L. Euler 266 years ago.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jan 5, 2016

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