【作者】 徐猛；

【导师】 徐仲；

【作者基本信息】 西北工业大学 ， 计算数学， 2003， 硕士

【摘要】 对于特殊矩阵的快速三角分解算法的研究，目前主要是对一些较简单的矩阵进行的。如对Toeplitz矩阵、Loewner矩阵、Vandermonde矩阵、Hankel矩阵等，都得到了一些有效的三角分解算法，其计算量均为O(n~2)。本文研究更广类型的一些特殊矩阵，如Toeplitz型矩阵、Loewner型矩阵、对称Loewner型矩阵以及Vandermonde型矩阵等，根据这些特殊矩阵的结构特点，给出了相应的快速三角分解算法。 §2给出了本文所有算法的理论基础。 在§3中，首先给出Toeplitz型矩阵的定义，然后推导Toeplitz型矩阵的逆矩阵的快速三角分解算法。继而推导Toeplitz型矩阵的快速三角分解算法。 在§4中，首先给出Loewner型矩阵的定义，然后推导Loewner型矩阵的逆矩阵的快速三角分解算法。继而推导Loewner型矩阵的快速三角分解的算法。 在§5中，首先给出对称Loewner型矩阵的定义，然后推导对称Loewner型矩阵的逆矩阵的快速三角分解算法。继而推导对称Loewner型矩阵的快速三角分解算法。 在§6中，首先给出Vandermonde型矩阵的定义，然后推导Vandermonde型矩阵的逆矩阵的快速三角分解算法。 在§7中，首先给出Hankel矩阵的定义，然后推导Hankel矩阵的逆矩阵的快速三角分解算法。 在§8中，给出了本文算法的一些数值算例，说明了算法的有效性。更多还原

【Abstract】 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.更多还原