Algorithms And Least-square Solutions Of Inverse Problems For The Circulant Matrices

Circulant matrices have become one of the most important and active research fields of applied mathematics increasingly.In the paper, we introduce our some work in algorithms and the least-square solutions of inverse problems. For detail, we conclude them as follows:1. The least-square solutions of inverse problems and the optimal approximation problems for the r-circulant matrix are discussed. Then, a fast and parallel algorithm for determining whether symmetric r-circulant matrix linear system is solvable or not is presented.2. A fast algorithm for calculating the inverse and Moore-Penrose inverse of the scaled factor circulant matrix is presented. Then, a fast algorithm for the products of the scaled factor circulant matrix is given. Finally, a fast and parallel algorithm for determining whether scaled factor circulant matrix linear systems is solvable or not is presented.3. A fast algorithm for calculating the inverse and Moore-Penrose inverse of the permutation factor circulant matrix is presented. Then, a fast algorithm for the product of the permutation factor circulant matrix is given. Finally, The least-square solutions of inverse problems and the optimal approximation problems for the permutation factor circulant matrix are discussed.