We provide necessary and sufficient conditions for the generalized ⋆Sylvester matrix equation, AXB+CX ⋆ D = E, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices A,B,C,D (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the ex...

We provide necessary and sufficient conditions for the generalized ?Sylvester matrix equation, AXB +CX ? D = E, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices A,B,C,D (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the e...

We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their exten...

If k = 1, then X is called the group inverse of A, and is denoted by X = Ag. The Drazin inverse is very useful in various applications (see, e.g. [1]–[4]; applications in singular differential and difference equations, Markov chains and iterative methods). In 1980, Cline and Greville [5] extended the Drazin inverse of square matrix to rectangular matrix, which can be generalized to the quaterni...

The GMRES method is one of the most popular iterative schemes for the solution of large linear systems of equations with a square nonsingular matrix. GMRES-type methods also have been applied to the solution of linear discrete ill-posed problems. Computational experience indicates that for the latter problems variants of the standard GMRES method, that require the solution to live in the range ...

In this paper, bending analysis of concentric and eccentric beam stiffened square and rectangular plate using the meshless collocation method has been investigated. For detecting the governing equations of plate and beams, Mindlin plate theory and Timoshenko beam theory have been used, respectively, with the stiffness matrices of the plate and the beams obtained separately. The stiffness matric...

A square system of linear equations is ‘ill-conditioned’ when the norm of the corresponding inverse matrix is large. This norm bounds the size of the solution, and measures how close the system is to being inconsistent: it is thus of fundamental computational significance. We generalize this idea from linear equations to inclusions governed by closed convex processes, and hence to ‘conic linear...

For solving systems of linear algebraic equations with blockfivediagonal matrices arising in geoelectrics and diffusion problems, the parallel matrix square root method, conjugate gradient method with preconditioner, conjugate gradient method with regularization, and parallel matrix sweep algorithm are proposed and some of them are implemented numerically on multi-core CPU Intel. Investigation ...

This paper describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (eg square matrices) and implementations of several tree types (eg Braun trees, 2-3 trees). Consider, for instance, representing square /102/ matrices. The usual representation...

The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for reaction-diffusion systems (three equations) with a tridiagonal matrix of diffusion coefficients and with nonhomogeneous boundary conditions after the work of Kouachi (2004) on the system of reaction diffusion with a full 2-square matrix. Our techniques are based on inv...