Fast Algorithm For Some Special Block Matrices And Structure Matrices

Some special matrices that arise in science and engineering are researched in this paper. Such as block circulate tridiagonal matrix, block tridiagonal matrix, block pentandiagonal matrix, block quasi-tridiagonal matrix, Hankel-type matrix, Vandermonde-type matrix, Loewner-type matrix and symmetrical Loewner-type matrix. Algorithms of solving linear systems, triangular factorization and inverse are presented, among which some are new; some are improvements and extensions of existing ones.Firstly, new algorithms of solving block circulate tridiagonal systems, block tridiagonal systems and block pentandiagonal system are proposed, which are based on the special factorization of these block matrices. The new algorithms include selective parameter-matrices. Properly selected parameter-matrices can make new algorithms higher in precision than the famous Thomas algorithm. While the Thomas algorithm invalidate for some numerical examples, the new algorithms can solve.Secondly, three direct methods for solving linear system whose coefficient matrix is block circulate tridiagonal matrix are presented, which are Thomas method, three-parametric method and linear interpolation method. The stability and advantages of algorithms are also analyzed. Numerical examples illustrate that the problems which one of these algorithms can't solve, the others can. so they are supplement to each other. Beside, the PE_k method of solving block circulate tridiagonal is given, the posstiblity and convergence of PE_k method are discussed as well, and some numerical examples are given to illustrate that the method is better than some existing methods.