The Fast Triangular Factorization Algorithms Of Special Matrices And Their Inversion

It is mainly to some simple matrices to the research of the fast triangular factorization algorithms of special matrices up to now. For example, to Toeplitz matrices, Loewner matrices, Vandermonde matrices, Hankel matrices, etc. We have gotten some effective fast triangular factorization algorithms to these matrices. They are all need of O(n2) operations. In this paper, we research some more general special matrices, for example, Teoplitz type matrices, Loewner type matrices, symmetrical Loewner matrices and Vandermonde type matrices, and so on. We respectively get their fast triangular factorization algorithms according to the character of these special matrices.In § 2, we give the theoretical base of all the algorithms in this paper.In § 3, we first give the definition of Toeplitz type matrices, then we give the triangular factorization algorithm of the inversion of Toeplitz type matrices. We give the triangular factorization algorithm of Toeplitz type matrices in the end.In §4, we first give the definition of Loewner type matrices, then we give the triangular factorization algorithm of the inversion of Loewner type matrices. We give the triangular factorization algorithm of Loewner type matrices in the end.In §5, we first give the definition of symmetric Loewner type matrices, then we give the triangular factorization algorithm of the inversion of symmetric Loewner type matrices. We give the triangular factorization algorithm of symmetric Loewner type matrices in the end.In §6, we first give the definition of Vandermonde type matrices, then we give the triangular factorization algorithm of the inversion of Vandermonde type matrices .In §7, we first give the definition of Hankel matrices, then we give the triangular factorization algorithm of the inversion of Hankel matrices.In §8, we give some numerical examples to check the validity of these algorithms.