scholarly journals TWO-STEP, user's guide. Solution of a linear system of equations using the exact two-step method. [Linear matrix equations, for CDC-6400]

1977 ◽  
Author(s):  
R. Kobbe ◽  
E.H. Bareiss ◽  
A. Antolak
Keyword(s):  
Linear System ◽  
Linear Matrix ◽  
System Of Equations ◽  
Matrix Equations ◽  
Step Method ◽  
Linear System Of Equations ◽  
Linear Matrix Equations

Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3181-3198
Author(s):  
Arezo Ameri ◽  
Fatemeh Beik
Keyword(s):  
Linear System ◽  
Coefficient Matrix ◽  
System Of Equations ◽  
Matrix Equations ◽  
Linear System Of Equations ◽  
System Of Matrix Equations ◽  
Gradient Based ◽  
Richardson Method ◽  
Singular Linear System ◽  
Symmetric Systems

Recently in [Journal of Computational Physics, 321 (2016), 829-907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper, we first prove semi-convergence of iterative methods with DORS to solve singular linear system of equations. In particular, we propose applying the DORS in the Modified HSS (MHSS) to solve singular complex symmetric systems and in the Richardson method to solve normal equations. Moreover, based on the obtained results, an algorithm is developed for solving coupled matrix equations. It is seen that the proposed method outperforms the relaxed gradient-based (RGB) method [Comput. Math. Appl. 74 (2017), no. 3, 597-604] for solving coupled matrix equations. Numerical results are examined to illustrate the validity of the established results and applicability of the presented algorithms.

scholarly journals A generalized inverse for matrices

1955 ◽  
Vol 51 (3) ◽  
pp. 406-413 ◽  
Author(s):  
R. Penrose
Keyword(s):  
Unique Solution ◽  
Spectral Decomposition ◽  
Generalized Inverse ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Singular Matrix ◽  
Rectangular Matrix ◽  
New Type ◽  
Linear Matrix Equations

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.

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