scholarly journals The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of a Class of Matrix Equations and Its Least Squares Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yifen Ke ◽  
Changfeng Ma
Keyword(s):  
Least Squares ◽  
General Expression ◽  
Numerical Experiments ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Least Squares Problem ◽  
The Matrix

The solvability conditions and the general expression of the generalized bisymmetric and bi-skew-symmetric solutions of a class of matrix equations(AX=B,XC=D)are established, respectively. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equations are considered. In addition, two algorithms are provided to compute the generalized bisymmetric and bi-skew-symmetric least squares solutions. Numerical experiments illustrate that the results are reasonable.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

Gravity interpretation with the aid of quadratic programming

Geophysics ◽  
1980 ◽  
Vol 45 (3) ◽  
pp. 403-419 ◽  
Author(s):  
N. J. Fisher ◽  
L. E. Howard
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Theoretical Data ◽  
Linear Constraints ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Least Squares Problem ◽  
The Matrix ◽  
Gravity Interpretation ◽  
Value Decomposition

The inverse gravity problem is posed as a linear least‐squares problem with the variables being densities of two‐dimensional prisms. Upper and lower bounds on the densities are prescribed so that the problem becomes a linearly constrained least‐squares problem, which is solved using a quadratic programming algorithm designed for upper and lower bound‐type constraints. The solution to any problem is smoothed by damping, using the singular value decomposition of the matrix of gravitational attractions. If the solution is required to be monotonically increasing with depth, then this feature can be incorporated. The method is applied to both field and theoretical data. The results are plotted for (1) undamped, nonmonotonic, (2) damped, nonmonotonic, and (3) damped, monotonic solutions; these conditions illustrate the composite approach of interpretation where both damping techniques and linear constraints are used in refining a solution which at first is unacceptable on geologic grounds while fitting the observed data well.

scholarly journals Condition numbers of the least squares problems with multiple right-hand sides

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1667-1676
Author(s):  
Lingsheng Meng ◽  
Bing Zheng
Keyword(s):  
Least Squares ◽  
Numerical Experiments ◽  
Upper Bounds ◽  
Condition Numbers ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Right Hand

In this paper, we investigate the normwise, mixed and componentwise condition numbers of the least squares problem min X?Rnxd ||X - B||F, where A ? Rmxn is a rank-deficient matrix and B ? Rmxd. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the least squares problem with single right-hand side (i.e. B ? b is an m-vector) of several authors. Numerical experiments are given to confirm our results.

AGGREGATING VARIABLES FOR ASYNCHRONOUS ITERATIONS

2005 ◽  
Vol 15 (01n02) ◽  
pp. 99-114
Author(s):  
YASEMIN YALÇINKAYA ◽  
TROND STEIHAUG
Keyword(s):  
Least Squares ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Relaxation Parameter ◽  
Linear Least Squares ◽  
Asynchronous Algorithm ◽  
Aggregation Of Variables ◽  
Least Squares Problem ◽  
Asynchronous Iterations ◽  
Linear Least Squares Problem

The deteriorating effect of old history in asynchronous implementations of Jacobi-type iterative methods applied to linear least squares problem is well documented. A partially asynchronous algorithm is developed which employs a combination of synchronization, a relaxation parameter and an aggregation of variables. It is shown by numerical experiments that this combined effort to decrease the effect of old history is effective.

scholarly journals Constrained Solutions of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Qing-Wen Wang ◽  
Juan Yu
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
System Of Matrix Equations ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Constrained Solutions

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equationsAX=BandXC=D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

On the matrix equations U_iXV_j W_{i j} for 1 \leq i; j \leq k with i +j \leq k

2016 ◽  
Vol 31 ◽  
pp. 465-475
Author(s):  
Jacob Van der Woude
Keyword(s):  
Linear Matrix ◽  
Matrix Equations ◽  
Kronecker Products ◽  
Solvability Conditions ◽  
Common Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Triangular Decoupling ◽  
The Given ◽  
Linear Matrix Equations

Conditions for the existence of a common solution X for the linear matrix equations U_iXV_j 􏰁 W_{ij} for 1 \leq 􏰃 i,j \leq 􏰂 k with i\leq 􏰀 j \leq 􏰃 k, where the given matrices U_i,V_j,W_{ij} and the unknown matrix X have suitable dimensions, are derived. Verifiable necessary and sufficient solvability conditions, stated directly in terms of the given matrices and not using Kronecker products, are also presented. As an application, a version of the almost triangular decoupling problem is studied, and conditions for its solvability in transfer matrix and state space terms are presented.

Sign in / Sign up

Export Citation Format

Share Document