Numerical Algorithms For Matrix Pencil Nearness Problems

Matrix pencil nearness problem arises in structural modification and finite element model updating of structural systems.In this thesis,we consider the problem of finding the least change adjustment to a given matrix pencil,which arises in simultaneously correcting mass and stiffness matrices of finite element model of undamped structural systems,and impose the desired matrix properties,including satisfaction of the characteristic equation,symmetric positive semi-definiteness and sparsity,as side constraints to form the optimal matrix pencil approximation problem and obtain the following results.Using the positive approximation of a matrix and the canonical correlations of matrix pairs,we present an alternating projection method(APM)for solving the matrix pencil nearness problem,and show its convergence.Applying the relaxation technique,we accelerate the convergence of the alternating projection method,and we get the relaxation alternating projection method.Using partial Lagrangian multipliers,we transform the matrix pencil optimization problem into an equivalent matrix linear variational inequality,apply proximal point-like method to solve the matrix linear variational inequality,present a proximal point-like algorithm(PPA)for solving the matrix pencil nearness problem,and analyze the convergence of the proposed method.The global convergence is proved under some mild conditions and the worst-case O(1/ t)convergence rate in ergodic sense is established for the proposed method.Combining the alternating projection with the proximal point-like method,an APM-PPA method for solving the matrix pencil nearness problem is presented,and its convergence is shown.Numerical results show that the obtained results are correct and the proposed methods are efficient.