Least-square solutions for inverse problems of centrosymmetric matrices
Let CSRnxn = {A = (a(ij)) is an element of R-nxn \a(ij) = a(n+1-i,n+1-j), i, j = 1, 2,..., n}. In this paper, we mainly discuss solving the following problems. PROBLEM I-0. Given X, B is an element of R-nxx. Find A is an element of CSRnxn such that AX=B PROBLEM I. Given X, B is an element of R-nxm. Find A is an element of CSRnxn such that parallel toAX - Bparallel to = min. PROBLEM II Given (A) over tilde is an element of R-nxn. Find A* is an element of S-E such that parallel to(A) over tilde -A*parallel to = inf(Ais an element ofSE) parallel to(A) over bar -Aparallel to, where parallel to . parallel to is the Frobenius norm, and SE is the solution set of Problem I or I-0. The expressions of the general solution of Problems I and I-0 have been derived. The sufficient and,necessary conditions for the solvability of Problem I-0 have been provided. The numerical algorithm to find the optimal approximate solution and some numerical experiments have been given. (C) 2003 Elsevier Science Ltd. All rights reserved.
least-squares solution inverse problem optimal approximation centrosymmetric matrix