Research On Iterative Methods For Solving Several Matrix Extension Problem

The matrix equation problem with a sub-matrix constraint is known as the Matrix Extension Problem, which comes from the expansion of subsystem and the partial revision of structural dynamic model. It has a broad application background and has been one of the hot research topics in the filed of numerical algebra.In this thesis, the orthogonal projection iteration method and the conjugate gradient iteration method of the following matrix extension problem and its optimal approximation are studied:Problem I Given A∈Rm×n, B∈Rm×l,X∈Rpxq, S(?)Rn×l, find X∈S, such that AX=B,X(p1:p2,q1:q2)=X. where p2-p1+1= p, q2-q1+1= q, Sis Rn×l, or SRn×n, or ASRn×n.Problem II Given X0∈Rn×l, find X∈SE, such that Where||·||J is Frobenius norm, SE is the solution set of Problem IWhen S is Rn×1, First the primitive equation AX=B is converted to two lower-order equations by partitioning matrices A、X、B, the iteration algorithms is constructed by using the ideas of the orthogonal projection iterative method. Then combining singular value decomposition and the invariance of F-norm orthogonal transformation, the convergence of the algorithm is proven and the estimation of the convergence rate is derived.When S is Rn×l, SRn×n or ASRn×n, After the primitive equation AX= B is converted to two lower-order equations, The iteration algorithms are constructed by using the ideas of the conjugate gradient method. The limited termination of these algorithms are proven by using the orthogonality of residual.Finally some numerical examples are given to verify the effectiveness of these methods.