scholarly journals Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
N. Akhondi ◽  
F. Toutounian
Keyword(s):  
Convergence Property ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Optimal Parameters ◽  
Skew Circulant ◽  
Systems Of Linear Equations ◽  
Toeplitz Systems ◽  
Two Parameters ◽  
Two Parameter

We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrix . In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method.

scholarly journals A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhijun Luo ◽  
Lirong Wang
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Descent Direction ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Globally Convergent ◽  
Variable Distribution ◽  
Systems Of Linear Equations

A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method

2014 ◽  
Vol 26 (5) ◽  
pp. 566-572 ◽  
Author(s):  
Ailan Liu ◽  
◽  
Dingguo Pu ◽  
◽  
Keyword(s):  
Constraint Qualification ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Ncp Function ◽  
Line Searches ◽  
Complementarity Condition ◽  
Linear Independence Constraint Qualification ◽  
Systems Of Linear Equations

<div class=""abs_img""><img src=""[disp_template_path]/JRM/abst-image/00260005/04.jpg"" width=""300"" />Algorithm flow chart</div> We propose a nonmonotone QP-free infeasible method for inequality-constrained nonlinear optimization problems based on a 3-1 piecewise linear NCP function. This nonmonotone QP-free infeasible method is iterative and is based on nonsmooth reformulation of KKT first-order optimality conditions. It does not use a penalty function or a filter in nonmonotone line searches. This algorithm solves only two systems of linear equations with the same nonsingular coefficient matrix, and is implementable and globally convergent without a linear independence constraint qualification or a strict complementarity condition. Preliminary numerical results are presented. </span>

scholarly journals Shape Preserving Interpolation UsingC2Rational Cubic Spline

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Samsul Ariffin Abdul Karim ◽  
Kong Voon Pang
Keyword(s):  
Spline Interpolation ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Cubic Spline ◽  
Shape Preserving ◽  
Cubic Spline Interpolation ◽  
Positive Data ◽  
Shape Preserving Interpolation ◽  
Systems Of Linear Equations ◽  
Two Parameters

This paper discusses the construction of newC2rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parametersαi,βi, andγi. The sufficient conditions for the positivity are derived on one parameterγiwhile the other two parametersαiandβiare free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation withC2continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion andC2continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivativesdi,i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the newC2rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated isft∈C3t0,tnis also investigated in detail.

A Superfast Algorithm for Toeplitz Systems of Linear Equations

2008 ◽  
Vol 29 (4) ◽  
pp. 1247-1266 ◽  
Author(s):  
S. Chandrasekaran ◽  
M. Gu ◽  
X. Sun ◽  
J. Xia ◽  
J. Zhu
Keyword(s):  
Linear Equations ◽  
Systems Of Linear Equations ◽  
Toeplitz Systems ◽  
Superfast Algorithm

AN INFEASIBLE SSLE FILTER ALGORITHM FOR GENERAL CONSTRAINED OPTIMIZATION WITHOUT STRICT COMPLEMENTARITY

2011 ◽  
Vol 28 (03) ◽  
pp. 361-399 ◽  
Author(s):  
CHUNGEN SHEN ◽  
WENJUAN XUE ◽  
DINGGUO PU
Keyword(s):  
Linear Independence ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Active Set ◽  
Strict Complementarity ◽  
Complementarity Condition ◽  
Sequential Systems ◽  
Computational Effect ◽  
Systems Of Linear Equations ◽  
One Step

In this paper, we propose a new sequential systems of linear equations (SSLE) filter algorithm, which is an infeasible QP-free method. The new algorithm needs to solve a few reduced systems of linear equations with the same nonsingular coefficient matrix, and after finitely many iterations, only two linear systems need to be solved. Furthermore, the nearly active set technique is used to improve the computational effect. Under the linear independence condition, the global convergence is proved. In particular, the rate of convergence is proved to be one-step superlinear without assuming the strict complementarity condition. Numerical results and comparison with other algorithms indicate that the new algorithm is promising.

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