A Generalized Conjugate Direction Method for Nonsymmetric Large III-Conditioned Linear Systems

2010 ◽  
pp. 210-221
Author(s):  
Edouard R. Boudinov ◽  
Arkadiy I. Manevich
Keyword(s):  
Linear Systems ◽  
Conjugate Direction ◽  
Conjugate Direction Method

scholarly journals Quantum Implementation of Powell’s Conjugate Direction Method

2019 ◽  
Vol 23 (4) ◽  
pp. 726-734
Author(s):  
Kehan Chen ◽  
Fei Yan ◽  
Kaoru Hirota ◽  
Jianping Zhao ◽  
◽  
...  
Keyword(s):  
Search Algorithm ◽  
Quantum Circuit ◽  
Data Fitting ◽  
Quadratic Equation ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Circuit Implementation ◽  
One Dimensional ◽  
Conjugate Direction ◽  
Conjugate Direction Method

A quantum circuit implementation of Powell’s conjugate direction method (“Powell’s method”) is proposed based on quantum basic transformations in this study. Powell’s method intends to find the minimum of a function, including a sequence of parameters, by changing one parameter at a time. The quantum circuits that implement Powell’s method are logically built by combining quantum computing units and basic quantum gates. The main contributions of this study are the quantum realization of a quadratic equation, the proposal of a quantum one-dimensional search algorithm, the quantum implementation of updating the searching direction array (SDA), and the quantum judgment of stopping the Powell’s iteration. A simulation demonstrates the execution of Powell’s method, and future applications, such as data fitting and image registration, are discussed.

A conjugate direction method for linear systems in Banach spaces

2017 ◽  
Vol 25 (5) ◽  
pp. 553-572 ◽  
Author(s):  
Roland Herzog ◽  
Winnifried Wollner
Keyword(s):  
Linear Systems ◽  
Hilbert Spaces ◽  
Reflexive Banach Space ◽  
Duality Mapping ◽  
Well Posedness ◽  
Short Term ◽  
Stopping Criteria ◽  
Limited Memory ◽  
Conjugate Direction Method ◽  
Riesz Isomorphism

AbstractIn this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

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