Experimental evaluation of new SLICOT solvers for linear matrix equations based on the matrix sign function

2008 ◽  
Author(s):  
Vasile Sima ◽  
Peter Benner
Keyword(s):  
Experimental Evaluation ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Matrix Sign Function ◽  
Sign Function ◽  
The Matrix ◽  
Linear Matrix Equations

Parallel solution of Riccati matrix equations with the matrix sign function

Automatica ◽  
1998 ◽  
Vol 34 (2) ◽  
pp. 151-156 ◽  
Author(s):  
Enrique S. Quintana-Orti ◽  
Vicente Hernandez
Keyword(s):  
Matrix Equations ◽  
Matrix Sign Function ◽  
Sign Function ◽  
Parallel Solution ◽  
The Matrix ◽  
Riccati Matrix

PARALLEL DISTRIBUTED SOLVERS FOR LARGE STABLE GENERALIZED LYAPUNOV EQUATIONS

1999 ◽  
Vol 09 (01) ◽  
pp. 147-158 ◽  
Author(s):  
PETER BENNER ◽  
JOSÉ M. CLAVER ◽  
ENRIQUE S. QUINTANA-ORTI
Keyword(s):  
Large Scale ◽  
Iterative Algorithms ◽  
Direct Methods ◽  
Matrix Equations ◽  
Distributed Architectures ◽  
Matrix Sign Function ◽  
Sign Function ◽  
The Matrix ◽  
Qz Algorithm ◽  
Lyapunov Matrix

In this paper we study the solution of stable generalized Lyapunov matrix equations with large-scale, dense coefficient matrices. Our iterative algorithms, based on the matrix sign function, only require scalable matrix algebra kernels which are highly efficient on parallel distributed architectures. This approach avoids therefore the difficult parallelization of direct methods based on the QZ algorithm. The experimental analtsis reports a remarkable performance of our solvers on an IBM SP2 platform.

scholarly journals The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

2021 ◽  
Vol 7 (1) ◽  
pp. 384-397
Author(s):  
Yinlan Chen ◽  
◽  
Lina Liu
Keyword(s):  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
The Matrix ◽  
The Common ◽  
Necessary And Sufficient ◽  
Nonnegative Definite ◽  
Linear Matrix Equations

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

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.

scholarly journals The Solutions of Second-Order Linear Matrix Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Kefeng Li ◽  
Chao Zhang
Keyword(s):  
Time Scales ◽  
Characteristic Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Existence Of A Solution ◽  
Order Linear ◽  
The Matrix ◽  
Linear Matrix Equations

This paper studies the solutions of second-order linear matrix equations on time scales. Firstly, the necessary and sufficient conditions for the existence of a solution of characteristic equation are introduced; then two diverse solutions of characteristic equation are applied to express general solution of the matrix equations on time scales.

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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