scholarly journals A New Method for the Bisymmetric Minimum Norm Solution of the Consistent Matrix EquationsA1XB1=C1,A2XB2=C2

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Aijing Liu ◽  
Guoliang Chen ◽  
Xiangyun Zhang
Keyword(s):  
Iterative Method ◽  
New Method ◽  
Frobenius Norm ◽  
Minimum Norm ◽  
Matrix Equations ◽  
Minimum Norm Solution ◽  
New Iterative Method

We propose a new iterative method to find the bisymmetric minimum norm solution of a pair of consistent matrix equationsA1XB1=C1,A2XB2=C2. The algorithm can obtain the bisymmetric solution with minimum Frobenius norm in finite iteration steps in the absence of round-off errors. Our algorithm is faster and more stable than Algorithm 2.1 by Cai et al. (2010).

scholarly journals New Iterative Method for Solving Nonlinear Equations

2014 ◽  
Vol 11 (4) ◽  
pp. 1649-1654 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
New Method ◽  
New Iterative Method

The aim of this paper is to propose an efficient three steps iterative method for finding the zeros of the nonlinear equation f(x)=0 . Starting with a suitably chosen , the method generates a sequence of iterates converging to the root. The convergence analysis is proved to establish its five order of convergence. Several examples are given to illustrate the efficiency of the proposed new method and its comparison with other methods.

scholarly journals LSMR Iterative Method for General Coupled Matrix Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
F. Toutounian ◽  
D. Khojasteh Salkuyeh ◽  
M. Mojarrab
Keyword(s):  
Iterative Method ◽  
Symmetric Matrix ◽  
Matrix Group ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Matrix Groups ◽  
Special Cases ◽  
Least Frobenius Norm Solution ◽  
General Coupled Matrix Equations

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group(X¯1,X¯2,…,X¯q)in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

The Mixed-Restriction Solution and its Application in Augmented Reality

2013 ◽  
Vol 380-384 ◽  
pp. 1439-1443
Author(s):  
Yan Feng ◽  
Ming Hui Wang ◽  
Chun Yun Sheng
Keyword(s):  
Augmented Reality ◽  
Iterative Method ◽  
Matrix Equation ◽  
Minimum Norm ◽  
Minimum Norm Solution ◽  
Initial Value ◽  
True Solution ◽  
Numerical Examples ◽  
The Matrix

In this paper, an iterative method is proposed to obtain the mixed-restriction solutions of . By the iterative method, the solvability of the matrix equation can be determined automatically. And if the matrix equation is consistent, then, for any initial value, the sequence generated by the iterative method, converges to the true solution within finite iteration steps in the absence of round off errors. Also, for the special initial value, the minimum norm solution can be obtained. Finally, two numerical examples are presented to demonstrate the efficiency of the iterative method.

scholarly journals Modifikasi Metode Householder Tiga Parameter yang Bebas Turunan Kedua dengan Orde Konvergensi Optimal

2021 ◽  
Vol 18 (1) ◽  
pp. 62-74
Author(s):  
Wartono ◽  
M Zulianti ◽  
Rahmawati
Keyword(s):  
Numerical Simulation ◽  
Iterative Method ◽  
Iterative Methods ◽  
Efficiency Index ◽  
New Method ◽  
Third Order ◽  
The Third ◽  
New Iterative Method ◽  
Better Than ◽  
Taylor’S Series

The Householder’s method is one of the iterative methods with a third-order convergence that used to solve a nonlinear equation. In this paper, the authors modified the iterative method using the expansion of second order Taylor’s series and approximated its second derivative using equality of two the third-order iterative methods. Based on the results of the study, it was found that the new iterative method has a fourth-order of convergence and requires three evaluations of function with an efficiency index of 1,587401. Numerical simulation is given by using several functions to compare the performance between the new method with other iterative methods. The results of numerical simulation show that the performance of the new method is better than other iterative methods.

scholarly journals Modified CGLS iterative algorithm for solving the generalized Sylvester-conjugate matrix equation

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1329-1346
Author(s):  
Caiqin Song ◽  
Qing-Wen Wang
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Inner Product ◽  
Frobenius Norm ◽  
Finite Convergence ◽  
Numerical Examples ◽  
Roundoff Errors ◽  
New Iterative Method

By introducing the real inner product, this paper offers an modified conjugate gradient least squares iterative algorithm (MCGLS)for solving the generalized Sylvester-conjugate matrix equation. The properties of this algorithm are discussed and the finite convergence of this algorithm is proven. This new iterative method can obtain the symmetric least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.

scholarly journals Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yajun Xie ◽  
Changfeng Ma
Keyword(s):  
Iterative Method ◽  
Linear Matrix ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Initial Value ◽  
Least Frobenius Norm Solution ◽  
Centrally Symmetric ◽  
Solution Pair ◽  
Matrix Pair ◽  
Linear Matrix Equations

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair(X,Y)solutions of the generalized coupled Sylvester-conjugate matrix equationsA1X+B1Y=D1X¯E1+F1,A2Y+B2X=D2Y¯E2+F2. On the condition that the coupled matrix equations are consistent, we show that the solution pair(X*,Y*)can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient.

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