广义逆理论、应用及计算

The generalized inverses have been used in a wide variety of applications, for.example, the application in the system and control theory, in statistics, in numerical mathematics and in optimization.The theory, the applications and the computational.methods of the generalized inveres have been developing rapidly during the last 50 years. In this project,16 papers on the theory and the computational methods of generalized inverses are published. The main results are as follows. (1) Some additive perturbation for Drazin inverse are given. In particular a formula is given for the Drazin inverses of the sum of two matrices when the product of these matrices vanishes [4]. (2) The necessary and sufficient conditions are given for the reverse order laws of the weighted M-P inverse of a triple matrix product and the Drazin inverse of multiple matrix product to hold [15,1]..(3) The expressions of the minors of the weighted M-P inverse and the.{2}inverse having the prescribed range T and null space S of a matrix A are.given. From these expressions ,we can calculate any minors of the.generalized inverses without calculating the generalized inverses at first.[6,5].(4) A perturbation theory for the generalized inverse ) 2 (.,S T A is developed. The.theory is based on a useful decomposition ) 2 (.,S T B - ) 2 (.,S T A under (W) condition.[16]. A new representation of the generalized inverse ) 2 (.,S T A is derived by.using the relationship between ) 2 (.,S T A and the group inverse g A [9]. The.singular value decomposition of the generalized inverse ) 2 (.,S T A is presented..The SVD for some other generalized inverse are also presented [13]. Given.two free modules T and S over commutative rings with identity 1, the.necessary and sufficient condition is m R S AT = ⊕ for the generalized.inverse ) 2 (.,S T A to exist [11]..(5) When D is a symmetric matrix, the general symmetric solution pair (X,Y).and the general bisymmetric solution pairs of the equation XA=YAD are.presented. The necessary and sufficient conditions for the existence of the.bisymmetric solution pair of the simultaneous matrix equation XA=YAD,.ATXA=D and the general forms of such solution are derived [8]..(6) Leverrier-Chebyshev and Leverrier-Hermite algorithms are presented for.simultaneous computations of B( μ)=adj( μE-A) and a( μ)=det( μE-A) of.the singular pencil μE-A, where E is singular,but det( μE-A)≠0..Leverrier-Laguerre algorithms is presented for simultaneous computations of.the adjoint G(s) and the determinant d(s) of the matrix polynomial.2 1.2 A sA J s . . ,where J is singular,but det( 2 1.2 A sA J s . . )≠0 [12,2,7]..(7) An improved parallel algorithm for computing the weighted Moore-Penrose.inverse +.MN A and a new highly parallel algorithm for computing the.minimum norm(T) least-squares(S) solution of inconsistent linear equations.Ax=b are presented [14,3]..(8) A finite algorithm for the computation of the Drazin inverse of a polynomial.matrix is given, and it is also implemented with the symbolic computational.package Matlab. A two-dimensional recursive algorithm for computing the.Drazin inverse is also presented [10]..(B) In the nature there are many nonlinear problems with different symmetry. The.relation between the possible symmetries of solution and the group of symmetries of.the equation is always the interesting subject of mathematicians and physicists..We analyse and compute the symmetry-breaking bifurcation for the nonlinear.equations with 2 Z , m O , )2 ( O symmetries. Bifurcation of the reaction diffusion.equations and Kuramoto-Sivashinsky equation is analyzed by Liapunov-Schmidt.reduction technique. The bifurcation equations and asymptotic expressions of.nontrivial solutions near the bifurcation points is obtained. The bifurcations and chaos.in a class of planar 3 D - equivariant mapping are studied. The figures by.computations show the whole procedure from periodic points to chaos and from 2symmetry chaotic attractors to 3 D symmetry chaotic attractors of the mapping wi

研究约束矩阵方程求解和系统论及控制论中产生的矩阵和多项式矩阵的各种广义逆的理论、算法和应用软件，有良好的应用前景。