On approximate solutions of linear equations in Banach spaces

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

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On the construction of iteration methods for linear equations in Banach spaces by summation methods

Aequationes Mathematicae ◽

10.1007/bf01818449 ◽

1970 ◽

Vol 5 (2-3) ◽

pp. 273-284 ◽

Cited By ~ 8

Author(s):

W. Niethammer ◽

W. Schempp

Keyword(s):

Banach Spaces ◽

Linear Equations ◽

Summation Methods ◽

Iteration Methods

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Iterative methods for generalized split feasibility problems in Banach spaces

Carpathian Journal of Mathematics ◽

10.37193/cjm.2017.01.02 ◽

2017 ◽

Vol 33 (1) ◽

pp. 09-26

Author(s):

QAMRUL HASAN ANSARI ◽

◽

AISHA REHAN ◽

◽

Keyword(s):

Weak Convergence ◽

Banach Spaces ◽

Iterative Methods ◽

Hilbert Spaces ◽

Equilibrium Problems ◽

Iterative Algorithms ◽

Approximate Solutions ◽

Split Feasibility Problems ◽

Convergence Results ◽

Split Feasibility

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

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A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

Thermal Science ◽

10.2298/tsci181128041b ◽

2019 ◽

Vol 23 (Suppl. 1) ◽

pp. 275-283

Author(s):

Kubra Bicer ◽

Mehmet Sezer

Keyword(s):

Differential Equations ◽

Linear Equations ◽

Bernoulli Polynomials ◽

Approximate Solutions ◽

Mean Value ◽

Computational Method ◽

Mean Value Theorem ◽

Residual Functions ◽

Non Linear ◽

Linear Differential

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

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