Abstract: Frequency transformation with Pascal matrix
equations is a method for transforming an electronic filter (analogue
or digital) into another filter. The technique is based on frequency
transformation in the s-domain, bilinear z-transform with pre-warping
frequency, inverse bilinear transformation and a very useful
application of the Pascal’s triangle that simplifies computing and
enables calculation by hand when transforming from one filter to
another. This paper will introduce two methods to transform a filter
into a digital filter: frequency transformation from the s-domain into
the z-domain; and frequency transformation in the z-domain. Further,
two Pascal matrix equations are derived: an analogue to digital filter
Pascal matrix equation and a digital to digital filter Pascal matrix
equation. These are used to design a desired digital filter from a given
filter.
Abstract: In this paper, a system of linear matrix equations
is considered. A new necessary and sufficient condition for the
consistency of the equations is derived by means of the generalized
singular-value decomposition, and the explicit representation of the
general solution is provided.
Abstract: In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XBT +CDT = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projection subspaces is outlined. We show that the approximate solution is an exact solution of a perturbed Sylvester matrix equation. Moreover, exact expression for the norm of residual is derived and results on finite termination and convergence are presented. Some numerical examples are presented to illustrate the effectiveness of the proposed method.
Abstract: In the present work, we propose a new method for
solving the matrix equation AXB=F . The new method can
be considered as a generalized form of the well-known global full
orthogonalization method (Gl-FOM) for solving multiple linear
systems. Hence, the method will be called extended Gl-FOM (EGl-
FOM). For implementing EGl-FOM, generalized forms of block
Krylov subspace and global Arnoldi process are presented. Finally,
some numerical experiments are given to illustrate the efficiency of
our new method.
Abstract: In this paper the gradient based iterative algorithms are presented to solve the following four types linear matrix equations: (a) AXB = F; (b) AXB = F, CXD = G; (c) AXB = F s. t. X = XT ; (d) AXB+CYD = F, where X and Y are unknown matrices, A,B,C,D, F,G are the given constant matrices. It is proved that if the equation considered has a solution, then the unique minimum norm solution can be obtained by choosing a special kind of initial matrices. The numerical results show that the proposed method is reliable and attractive.
Abstract: In the present paper, we propose numerical methods for solving the Stein equation AXC - X - D = 0 where the matrix A is large and sparse. Such problems appear in discrete-time control problems, filtering and image restoration. We consider the case where the matrix D is of full rank and the case where D is factored as a product of two matrices. The proposed methods are Krylov subspace methods based on the block Arnoldi algorithm. We give theoretical results and we report some numerical experiments.
Abstract: The real representation of the quaternionic matrix is
definited and studied. The relations between the positive (semi)define
quaternionic matrix and its real representation matrix are presented.
By means of the real representation, the relation between the positive
(semi)definite solutions of quaternionic matrix equations and those of
corresponding real matrix equations is established.