Homotopic residual correction processes

Authors:
V. Y. Pan, M. Kunin, R. E. Rosholt and H. Kodal

Journal:
Math. Comp. **75** (2006), 345-368

MSC (2000):
Primary 65F10, 65F30

DOI:
https://doi.org/10.1090/S0025-5718-05-01771-0

Published electronically:
July 25, 2005

MathSciNet review:
2176403

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Abstract | References | Similar Articles | Additional Information

Abstract: We present and analyze homotopic (continuation) residual correction algorithms for the computation of matrix inverses. For complex indefinite Hermitian input matrices, our homotopic methods substantially accelerate the known nonhomotopic algorithms. Unlike the nonhomotopic case our algorithms require no pre-estimation of the smallest singular value of an input matrix. Furthermore, we guarantee rapid convergence to the inverses of well-conditioned structured matrices even where no good initial approximation is available. In particular we yield the inverse of a well-conditioned matrix with a structure of Toeplitz/Hankel type in flops. For a large class of input matrices, our methods can be extended to computing numerically the generalized inverses. Our numerical experiments confirm the validity of our analysis and the efficiency of the presented algorithms for well-conditioned input matrices and furnished us with the proper values of the parameters that define our algorithms.

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Additional Information

**V. Y. Pan**

Affiliation:
Mathematics and Computer Science Department, Lehman College, CUNY, Bronx, New York 10468; Ph. D. Program in Mathematics, Graduate Center, CUNY, New York, New York 10016

Email:
victor.pan@lehman.cuny.edu

**M. Kunin**

Affiliation:
Ph.D. Program in Computer Science, Graduate Center, CUNY, New York, New York 10016

**R. E. Rosholt**

Affiliation:
Mathematics and Computer Science Department, Lehman College, CUNY, Bronx, New York 10468

**H. Kodal**

Affiliation:
University of Kocaeli, Department of Mathematics, 41300 Izmit, Kocaeli, Turkey

DOI:
https://doi.org/10.1090/S0025-5718-05-01771-0

Keywords:
Residual correction,
Newton's iteration,
homotopic (continuation) algorithms,
(generalized) inverse matrix

Received by editor(s):
December 20, 2001

Received by editor(s) in revised form:
March 10, 2004

Published electronically:
July 25, 2005

Additional Notes:
This work was supported by NSF Grant CCR9732206, PSC CUNY Awards 63383-0032 and 66406-0033, and a Grant from the CUNY Institute for Software Design and Development (CISDD)

The results of this paper were presented at the Second Conference on Numerical Analysis and Applications, Rousse, Bulgaria, in June 2000, and at the AMS/IMS/SIAM Summer Research Conference on Fast Algorithms in Mathematics, Computer Science, and Engineering in South Hadley, Massachusetts, in August 2001.

Article copyright:
© Copyright 2005
American Mathematical Society

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