Iterative Solutions For Several Classes Of Matrix Extension

The matrix extension problem, also known as matrix equation problem under sub-matrix constraint. When the matrix equation(equations) is diff- erent, or the constraint matrix is different, or the sub-matrix constraint is different, we can get a different matrix extension problem. The matrix extension problem has been widely used in structural design, dynamic model updating, vibration theory, and many other fields. Its research has become a very popular one of the topics in computational mathematics. So far, many research results have been achieved.This thesis mainly discusses the following problems.Problemâ… Given A∈R^m×n, B∈R^m×n, X∈R^p×q, S (?) R^n×n, find X∈S , such that AX = B, where X ( p₁ :p₂,q₁:q₂)= X , p₂ - p₁+1=p, q₂ - q₁+1=q.Problemâ…¡Given , find X∈S , such that AXB = C, where X ( p₁ :p₂,q₁:q₂)= X , p₂ - p₁+1=p, q₂ - q₁+1=q.Problemâ…¢Given find X∈S , such that where X( p₁ :p₂,q₁:q₂)= X , p₂- p₁+1=p, q₂ - q₁+1=q.Problemâ…£Suppose Problemâ… orâ…¡orâ…¢is compatible, let S E denote the set of solutions, for given matrix X 0∈S, find X∈SE, such thatwhere‖is Frobenius norm, S is the matrix set satisfying some con- straint conditions.The main works on this dissertation are as follows.1. When S are reflexive matrices, anti-reflexive matrices, anti-sym- metric symmetric matrices, bi-anti-symmetric matrices, symmetric ortho- symmetric matrices, symmetric ortho-anti-symmetric matrices, by the generalized conjugate gradient method, we construct the corresponding iterative algorithm in this thesis. 2. We have proved limited termination of the corresponding algorithm. For any initial matrix, in the absence of round off errors, when the equation(equations) is consistent, it converges the solutions of the equa- tions within finite iteration steps; when the equation(equations) is incon- sistent, it converges the least-squares solutions of the equations within finite iteration steps.3. If a special kind of initial matrix is chosen, the solution with least norm can be obtained within finite iteration steps. So we solve the iterative solution of the corresponding best approximation problem. Finally, we carried out numerical experiments, and verify the correctness of the results.