Stability of QR-based fast system solvers for a subclass of quasiseparable rank one matrices

Authors:
Froilán M. Dopico, Vadim Olshevsky and Pavel Zhlobich

Journal:
Math. Comp. **82** (2013), 2007-2034

MSC (2010):
Primary 65F05, 65G50, 15A06, 15A23, 15B99

Published electronically:
May 14, 2013

MathSciNet review:
3073190

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Abstract: The development of fast algorithms to perform computations with quasiseparable matrices has received a lot of attention in the last decade. Many different algorithms have been presented by several research groups all over the world. Despite this intense activity, to the best of our knowledge, there is no rounding error analysis published for these fast algorithms. In this paper, we present error analyses for two fast solvers of quasiseparable linear systems when they are applied on order one quasiseparable matrices that include the diagonal in the lower triangular rank structure. Both solvers are based on computing first the QR factorization of the coefficient matrix, and their error analyses require novel structured techniques for proving rigorously that only one of the considered algorithms is backward stable, while the other one is not. Two fundamental consequences of this work are: (i) users should employ with caution fast algorithms for quasiseparable matrices since they may be unstable; and (ii) a lot of work has to be done to identify which fast algorithms for quasiseparable matrices are backward stable among the large family available in the literature.

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Additional Information

**Froilán M. Dopico**

Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

Email:
dopico@math.uc3m.es

**Vadim Olshevsky**

Affiliation:
Department of Mathematics, 196 Auditorium Road, University of Connecticut, Storrs, Connecticut 06269

Email:
olshevsky@uconn.edu

**Pavel Zhlobich**

Affiliation:
School of Mathematics, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom

Email:
P.Zhlobich@ed.ac.uk

DOI:
http://dx.doi.org/10.1090/S0025-5718-2013-02710-X

Keywords:
Quasiseparable,
semiseparable,
QR-decomposition,
backward error analysis

Received by editor(s):
May 30, 2011

Received by editor(s) in revised form:
January 25, 2012

Published electronically:
May 14, 2013

Additional Notes:
The work of F. M. Dopico was partially supported by the Ministerio de Economía y Competitividad of Spain through the research grant MTM-2009-09281.

This research was partially done while V. Olshevsky held a position as “Catedrático de Excelencia” at Universidad Carlos III de Madrid in the academic year 2009-10.

This research was partially done while P. Zhlobich visited the Department of Mathematics of Universidad Carlos III de Madrid in January–March 2010, partially funded by the Ministerio de Economía y Competitividad of Spain through the grant MTM-2009-09281

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.