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• W1539062885 abstract "The most common class of methods for solving linear systems is the class of gradient algorithms,the most famous of which being the steepest descent (SD) algorithm. Linear systems equivalentto the quadratic optimization problems where the coefficient matrix of the linear system is theHessian of the corresponding quadratic function. In recent years, the development of a particulargradient algorithm, namely the Barzilai-Borwein (BB) algorithm, has instigated a lot of researchin the area of optimization and many algorithms now exist which have faster rates of convergencethan those possessed by the steepest descent algorithm.In this thesis, some well known gradient algorithms (which are originally developed for solvinglinear symmetric problems) are applied to non-symmetric systems of equations. Their efficiencyis demonstrated through an implementation on a constructed a class of random slightlynon-symmetric matrices E (with positive and real eigenvalues) as well as on the symmetric part ofE, ES = 12 (E +ET ), the performance of the algorithm is then investigated.Also several gradient algorithms are created that have a better performance than the Cauchy-Barzilai-Borwein (CBB) method for a non-symmetric problem when applied to another constructedclass of random slightly non-symmetric matrices A (having eigenvalues with a positivereal part). Numerically, it is established that the asymptotic rate of convergence of these algorithmsis faster when the roots of their polynomials have a Beta(2,2) distribution rather than a uniformdistribution.In fact, there is a strong dependence of the asymptotic rate of convergence on the distributionof the spectrum, this has been proven from numerical results. Most of the created algorithmsachieve faster convergence than other methods, especially with the non-clustered distribution ofthe spectrum.The modified Cauchy- Barzilai-Borwein (MCBB) algorithm is the first created algorithm whichsplits the two equal polynomial roots of the CBB algorithm into two different roots at each iteration.This means, moving the roots of the polynomial of CBB method from the real-axis to thecomplex plane where the eigenvalues of the objective matrix would lie. To attain further improvementon the convergence rates, another gradient algorithm is proposed by using theMCBB methodor CBB method at each iteration depending on two parameters g and x . Furthermore, some differentmethods are then created by utilizing different step-sizes of gradient algorithms which alreadyexist for finding the solution of symmetric problems.The effectiveness of the constructed algorithms, which depend on several parameters, have beenverified by various numerical experiments and comparisons with other approaches. The motivationfor choosing different gradient methods stems from the fact that simple examples can beconstructed in which each of these gradient methods will show superiority over the other. Mostof these algorithms have faster rates of convergence than CBB for non-symmetric matrices butslower for symmetric matrices.A new efficient gradient-based method to solve non-symmetric linear systems of equations ispresented. The simplicity of the new method, guaranteed convergent property, along with the miniiiimum memory requirement where it has almost the same number of gradient evaluations as theCBB method per iteration. This improved method is very attractive compared to alternative methodsfor solving slightly non-symmetric linear problems. Efficiency and feasibility of the presentedmethod are supported by numerical experiments." @default.
• W1539062885 created "2016-06-24" @default.
• W1539062885 creator A5081974773 @default.
• W1539062885 date "2013-01-01" @default.
• W1539062885 modified "2023-09-27" @default.
• W1539062885 title "Gradient optimization algorithms with fast convergence" @default.
• W1539062885 hasPublicationYear "2013" @default.
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