Reliable root detection with the qd-algorithm: When Bernoulli, Hadamard and Rutishauser cooperate

When using Rutishauser's qd-algorithm for the determination of the roots of a polynomial (originally the poles of a meromorphic function), or for related problems, conditions have been formulated for the interpretation of the computed q- and e-values. For a correct interpretation, the so-called...

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Bibliographic Details
Published in:Applied numerical mathematics Vol. 60; no. 12; pp. 1188 - 1208
Main Authors: Allouche, Hassane, Cuyt, Annie
Format: Journal Article
Language:English
Published: Elsevier B.V 01.12.2010
Subjects:
Arithmetic
Computation
Formally orthogonal Hadamard polynomials
Mathematical models
Meromorphic function
Meromorphic functions
Method of Bernoulli
Pole detection
Poles
qd-algorithm
Roots
Formally orthogonal Hadamard polynomials
Pole detection
qd-algorithm
Method of Bernoulli
Meromorphic function
ISSN:0168-9274, 1873-5460
Online Access:Get full text
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Summary:When using Rutishauser's qd-algorithm for the determination of the roots of a polynomial (originally the poles of a meromorphic function), or for related problems, conditions have been formulated for the interpretation of the computed q- and e-values. For a correct interpretation, the so-called critical indices play a crucial role. They index a column of e-values that tends to zero because of a jump in modulus among the poles. For more than 50 years the qd-algorithm in exact arithmetic was considered to be fully understood. In this presentation we push the detailed theoretical investigation of the qd-algorithm even further and we present a new aspect that seems to have been overlooked. We indicate a new element that makes a column of e-values tend to zero, namely a jump in multiplicity among equidistant poles. This result is obtained by combining the qd-algorithm with a deflation technique, and hence mainly relying on Bernoulli's method and Hadamard's formally orthogonal polynomials. Our results round up the theoretical analysis of the qd-algorithm as formulated in its original form, and are of importance in a variety of practical applications as outlined in the introduction.
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ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2010.04.006