a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

the general solution to a system of real quaternion matrix equations in this paper we consider the system of matrix equations a 1x c 1 a 2x c 2 a 3xb 3 c 3 and a 4xb 4 c 4 over the real quaternion algebra a necessary and sufficient condition for the existence and the expression of the general solution to the system are given as particular cases the corresponding results on other systems over are also obtained

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

explicit time reversible adaptive step size control adaptive step size control is difficult to combine with geometric numerical integration as classical step size control is based on past information only time symmetry is destroyed and with it the qualitative properties of the method in this paper we develop completely explicit reversible symmetry preserving adaptive step size selection algorithms for geometric numerical integrators such as the stormer verlet method a new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory for integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and in particular the total energy for hamiltonian systems it has the same excellent long term behavior as that obtained when constant steps are used with variable steps however both accuracy and efficiency are greatly improved

doubly quasi consistent parallel explicit peer methods with built in global error estimation recently kulikov presented the idea of double quasi consistency which facilitates global error estimation and control considerably more precisely a local error control implemented in such methods plays a part of global error control at the same time however kulikov studied only nordsieck formulas and proved that there exists no doubly quasi consistent scheme among those methods here we prove that the class of doubly quasi consistent formulas is not empty and present the first example of such sort this scheme belongs to the family of superconvergent explicit two step peer methods constructed by weiner schmitt podhaisky and jebens we present a sample of s stage doubly quasi consistent parallel explicit peer methods of order s 1 when s 3 the notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi consistent peer method and hence its global error at the same time numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi consistent numerical integration techniques is capable of producing numerical solutions for user supplied accuracy conditions in automatic mode

algebraic systems of matrices and grobner basis theory abstract the problem of finding all the n n complex matrices a b c such that for all real t e ta e tb e tc is a scalar matrix reduces to the study of a symmetric system s in the form a b c i n a 2 b 2 c 2 i n a 3 b 3 c 3 i n where are given complex numbers except in a special case we solve explicitly these systems depending on the values of the parameters for this purpose we use grobner basis theory a nilpotent algebra is associated to the special case the method used for solving s leads to complete solution of the original problem we study also similar systems over the n n real matrices and over the skew field of quaternions

constant coefficient linear multistep methods with step density control in linear multistep methods with variable step size the method s coefficients are functions of the step size ratios the coefficients therefore need to be recomputed on every step to retain the method s proper order of convergence an alternative approach is to use step density control to make the method adaptive if the step size sequence is smooth the method can use constant coefficients without losing its order of convergence the paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems the technique works in perfect agreement with theory for a given step density function for practical use however the density must be generated with data computed from the numerical solution we introduce a local error tracking controller which automatically adapts the density to computed data and demonstrate in computational experiments that the technique works well at least up to fourth order methods

solving a kind of restricted matrix equations and cramer rule the problem of finding determinantal formula for solutions of some restricted linear systems ax b has been discussed in linear multilinear algebra 15 1984 319 linear algebra appl 116 1989 27 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 this paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsax d r x r a k 1 xb d n x n b k 2 axb d r x r a k 1 n x n b k 2 where a c n x n with ind a k 1 b c m x m with ind b k 2 and d c n x m the results in linear multilinear algebra 15 1984 319 linear algebra appl 74 1986 213 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 are partially the special cases in our paper the classic cramer rule is also a special case of our results

extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications abstract assume that the linear quaternion matrix expression f x 1 x 2 a a 3 x 1 b 3 a 4 x 2 b 4 where x 1 x 2 are variant quaternion matrices in this paper we derive the maximal and minimal ranks of f x 1 x 2 subject to consistent systems of quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 and a 2 x 2 c 3 x 2 b 2 c 4 moreover corresponding results on some special cases are presented as applications we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations some previous known results can be regarded as the special cases of this paper

the p q skew symmetric extremal rank solutions to a system of quaternion matrix equations abstract let h m n denote the set of all m n matrices over the quaternion algebra h and p h m m q h n n be involutions we say that a h m n is p q symmetric or p q skewsymmetric if a paq or a paq we in this paper mainly investigate the p q skew symmetric maximal and minimal rank solutions to the system of quaternion matrix equations ax b xc d we present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with p q symmetry and p q skewsymmetry to the system the expressions of such solutions to this system are also given when the solvability conditions are satisfied a numerical example is presented to illustrate our results the findings of this paper extend some known results in this literature

p skew symmetric common solutions to a pair of quaternion matrix equations abstract an n n quaternion matrix a is termed p symmetric or p skewsymmetric if a pap or a pap where p is an n n nontrivial quaternion involution in this paper we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations a 1 x 1 c 1 a 2 x 2 c 2 a 3 x 1 b 1 a 4 x 2 b 2 c b then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of p symmetric and p skewsymmetric solutions to the system of quaternion matrix equations a a x c a and a b xb b c b furthermore we establish representations of p symmetric and p skewsymmetric quaternion matrices a numerical example is presented to illustrate the results of this paper

ranks and the least norm of the general solution to a system of quaternion matrix equations we in this paper first establish a new expression of the general solution to the consistent system of linear quaternion matrix equations a1x1 c1 a2x2 c2 a3x1b1 a4x2b2 c3 which was investigated recently q w wang h x chang c y lin p skew symmetric common solutions to a pair of quaternion matrix equations appl math comput 195 2008 721 732 then derive the maximal and minimal ranks and the least norm of the general solution to the system mentioned above some previous known results can be viewed as special cases of the results of this paper

right eigenvalues for quaternionic matrices a topological approach abstract we apply the lefschetz fixed point theorem to show that every square matrix over the quaternions has right eigenvalues we classify them and discuss some of their properties such as an analogue of jordan canonical form and diagonalization of elements of the compact symplectic group sp n

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

on quasi consistent integration by nordsieck methods in this paper we study advantages of numerical integration by quasi consistent nordsieck formulas all quasi consistent numerical methods possess at least one important property for practical use which has not attracted attention yet i e the global error of a quasi consistent method has the same order as its local error this means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms in other words the global error control can be as cheap as the local error control in the methods under discussion here we apply the above mentioned idea to nordsieck adams moulton methods which are known to be quasi consistent moreover some nordsieck adams moulton methods are even super quasi consistent the latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods in turn this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control we also introduce implicitly extended nordsieck methods such that in some sense they form pairs of embedded formulas with their source nordsieck methods this facilitates the local error control in quasi consistent nordsieck schemes numerical examples presented in this paper confirm clearly the power of quasi consistent integration in practice

hypercomplex correlation techniques for vector images correlation techniques have been applied to almost every area of signal processing over the past century yet their use has in general been limited to scalar signals while there have been implementations in multichannel applications these can be characterized as a combination of single channel processes true vector correlation techniques with global and interchannel measures have only recently been demonstrated and are still in their infancy by comparison this paper describes our work on vector correlation based on the use of hypercomplex fourier transforms and presents for the first time a unified theory behind the information contained in the peak of a vector correlation response by using example applications for color images we also demonstrate some of the practical implications together with our latest results

characterization for the general solution to a system of matrix equations with quadruple variables in this paper we give some necessary and sufficient conditions for the solvability to the system of matrix equations 0 1 a 1x 1 c 1 x 2b 1 d 1a 2x 3 c 2 x 3b 2 d 2 a 3x 4 c 3 x 4b 3 d 3 a 4x 1 x 2b 4 c 4x 3d 4 c 5x 4d 5 e 1and provide an expression of the general solution to 0 1 furthermore we obtain the maximal and minimal ranks of x 3 and x 4 in 0 1 the findings of this paper extend the known results in the literatures

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

construction of variable stepsize multistep formulas a systematic way of extending a general fixed stepsize multistep formula to a minimum storage variable stepsize formula has been discovered that encompasses fixed coefficient interpolatory variable coefficient variable step and fixed leading coefficient as special cases in particular it is shown that the interpolatory stepsize changing technique of nordsieck leads to a truly variable stepsize multistep formula which has implications for local error estimation and formula changing and it is shown that the variable step stepsize changing technique applicable to the adams and backward differentiation formulas has a reasonable generalization to the general multistep formula in fact it is shown how to construct a variable order family of variable coefficient formulas finally it is observed that the first dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

littlewood s algorithm and quaternion matrices abstract a strengthened form of schur s triangularization theorem is given for quaternion matrices with real spectrum for complex matrices it was given by littlewood it is used to classify projectors a 2 a and self annihilating operators a 2 0 on a quaternion unitary space and examples of unitarily wild systems of operators on such a space are presented littlewood s algorithm for reducing a complex matrix to a canonical form under unitary similarity is extended to quaternion matrices whose eigenvalues have geometric multiplicity 1

on the matrix equation ax b with applications to the generators of a controllability matrix abstract given two matrices a and b over an arbitrary field for which ax b is consistent we determine all possible ranks for solutions x of ax b given one solution x we furthermore determine all possible ranks of other solutions of this matrix equation we then apply the results to determine when a given n nm matrix a with first m columns b occurs as a controllability matrix a b ab a n 1 b for some a we determine all matrices a that generate a and find all their possible ranks and some bounds on their sparsity structure

adaptive time stepping and computational stability we investigate the effects of adaptive time stepping and other algorithmic strategies on the computational stability of ode codes we show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability a series of computational experiments with the standard implementation of dassl and a modified version including stepsize control based on digital filters is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense the inherent performance and stability of dassl are therefore much greater than the standard implementation seems to suggest

completing triangular block matrices with maximal and minimal ranks abstract this paper considers two completion problems on block matrices with maximal and minimal ranks let a a ij be an n n block matrix where a ij n i j 1 is given and a ij 1 i is a variant block entry then determine all these variant block entries such that a a ij has maximal and minimal possible ranks respectively by making use of the theory of generalized inverses of matrices we present complete solutions to these two problems as applications we also determine maximal and minimal ranks of the matrix expression a bxc when x is a variant triangular block matrix and then present a necessary and sufficient condition for the matrix equation bxc a to have a triangular block solution