Numerical Methods For Solving Least Squares Problem With Quadratic Constraints And A Matrix Equation

In this paper,we first discuss the least squares problem with a quadraticinequality constraint (LSQI):subject towhere A∈R^m×n(m≥n),C∈R^p×n,b∈R^m,d∈R^p,andα>.we discuss the conditions that guarantee the LSQI problem has a solution,andvarious cases of the solutions.Then we discuss thoroughly the least squaresproblem with a quadratic equality constraint (LSQE):subject toUsing the method of Lagrange multiplier,we obtain the normal equations ofthe LSQE problem.By the relation of solutions between the normal equationsand the LSQE problem,we present a projection method for solving numericallythe LSQE problem,we prove the bounded-ness of the sequence generated bythe projection method,and if the initial value satisfies some condition then thesequence generated by the projection method converges monotonically to thesolution of the LSQE problem.In addition,we discuss the convergence rate ofthe projection method,theoretical results show the projection method has atleast quadratic convergence.We also give a disturbance analysis of the LSQEproblem,and corresponding results are obtained.Numerical example indicatesthat the projection method is more efficient than Newton's methods.Finally,in Chapter three we present an iterative algorithm for solving theleast Frobenius norm problem of an inconsistent matrix equation pair (AXB,CXD)=(E,F) with a real matrix X.By this algorithm,for any (special) initial matrixX₀,a solution (the minimal Frobenius norm solution) can be obtained within finite iteration steps in the absence of roundoff errors.The numerical examplesverify the efficiency of the algorithm.