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Newton, I. (1711) A Treatise of Algebra Both Historical and Practical, Published by John Wallis; De analysi per aequationes numero terminorum infinitas (Written in 1669, Published in 1711 by William Jones) and in De metodisfluxionum et serieruminfinitarum (Written in 1671, Translated and Published as Method of Fluxions in 1736 by John Colson), 1685.

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• TITLE: Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems

  AUTHORS: Anastasia-Dimitra Lipitakis

  KEYWORDS: Approximate Inverse Preconditioners, Iterative Methods, Second Order Iterative Schemes, Exact Inverse Methods, Approximate Inverse, Explicit Preconditioning, Conjugate Gradients, Convergence Analysis

  JOURNAL NAME: Applied Mathematics, Vol.11 No.4, April 22, 2020

  ABSTRACT: Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.