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Journal of Applied Mathematics
Volume 2013, Article ID 154358, 15 pages
Research Article

An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems

Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan

Received 25 August 2013; Revised 29 October 2013; Accepted 29 October 2013

Academic Editor: Hui-Shen Shen

Copyright © 2013 Chein-Shan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It is known that the steepest-descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and descent direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm with -vector descent direction in a Krylov subspace is constructed, of which the optimal weighting parameters are solved in closed-form to accelerate the convergence speed in solving ill-posed linear problems. The optimally generalized steepest-descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems.