Journal of Computational and Applied Mathematics
A completed theory of the unsymmetric Lanczos process and related algorithms, part I
SIAM Journal on Matrix Analysis and Applications
A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms, Part II
SIAM Journal on Matrix Analysis and Applications
How well can the concept of Padé approximant be generalized to the multivariate case?
Proceedings of the conference on Continued fractions and geometric function theory
Numerical factorization of multivariate complex polynomials
Theoretical Computer Science - Algebraic and numerical algorithm
The Mathematics of Phylogenomics
SIAM Review
A new algorithm for sparse interpolation of multivariate polynomials
Theoretical Computer Science
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Journal of Computational and Applied Mathematics
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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.