Projection method for solving a singular system of linear equations and its applications

1971 ◽  
Vol 17 (3) ◽  
pp. 203-214 ◽  
Author(s):  
Kunio Tanabe
Keyword(s):  
Projection Method ◽  
Linear Equations ◽  
Singular System ◽  
System Of Linear Equations

scholarly journals Introducing a Conditional Ghost Correction Into the Vector Method

1984 ◽  
Vol 6 (2) ◽  
pp. 117-123 ◽  
Author(s):  
H. Schaeben
Keyword(s):  
Mathematical Model ◽  
Quadratic Programming ◽  
Optimization Problem ◽  
Linear Equations ◽  
Singular System ◽  
Orientation Distribution ◽  
Mathematical Approach ◽  
System Of Linear Equations ◽  
Vector Method ◽  
The Mathematical Model

The concept of conditional ghost correction is introduced into the vector method of quantitative texture analysis. The mathematical model actually chosen here reduces the texture problem to one of quadratic programming. Thus, a well defined optimization problem has to be solved, the singular system of linear equations governing the correspondence between pole and orientation distribution being reduced to a set of equality constraints of the restated texture problem. This new mathematical approach in terms of the vector method reveals the modeling character of the solution of the texture problem provided by the vector method completely.

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

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.

scholarly journals Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Khalid ◽  
M. N. Naeem ◽  
P. Agarwal ◽  
A. Ghaffar ◽  
Z. Ullah ◽  
...  
Keyword(s):  
Numerical Approximation ◽  
Analytic Solution ◽  
Linear Equations ◽  
Computational Cost ◽  
Spline Method ◽  
System Of Linear Equations ◽  
Approximation Solution ◽  
B Spline ◽  
Novel Technique ◽  
Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

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