Nonrecursive Equivalent of the Conjugate Gradient Method without the Need to Restart

A simple alternative to the conjugate gradient (CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy; i.e., it is not necessary to maintain overall o...

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Bibliographic Details
Published in:Advances in civil engineering 2019-01, Vol.2019 (2019), p.1-5
Main Authors: Demsic, Marija, Jaguljnjak Lazarevic, Antonia, Lazarevic, Damir, Dvornik, Josip
Format: Article
Language:English
Subjects:
Algorithms
Applied mathematics
Civil engineering
Conjugate gradient method
Conjugates
Energy
Iterative methods
Linear algebra
Linear equations
Mathematical analysis
Mathematical problems
Matrix algebra
Matrix methods
Methods
Multiplication
Orthogonality
Preconditioning
Restarting
Ritz method
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Summary:A simple alternative to the conjugate gradient (CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy; i.e., it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the A-orthogonality in n-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.
ISSN:1687-8086
1687-8094
DOI:10.1155/2019/7527590