An Algorithm for the Inversion of Finite Toeplitz Matrices
An Algorithm for the Inversion of Finite Toeplitz Matrices
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DOI：
10.1137/0112045
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发表时间：
1964-09
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影响因子：
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通讯作者：
W. F. Trench
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文献类型：
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作者：
W. F. Trench
具有特性（1.1），（1.2）和（1.3）。为了找到（yo，yi，yn）或任何n + 1连续变体的关节概率密度函数，有必要倒入矩阵TN。在本文中，我们得出了一个确切的递归程序，用于对有限顺序的任意正定确定的toeplitz矩阵的数值反转，该过程充分利用了（1.1），（1.3）和（1.3）对其元素放置的强限制的优势。 。使用此过程的第n阶toeplitz矩阵倒置所需的乘数数量与N2成比例，而不是与N'成正比，例如适用于任意遗传矩阵的方法。据作者所知，这种反转算法是第一个专门设计的，以利用一般Toeplitz矩阵的特殊简单性。此外，本文的截止部分专门用于形式（1.1）lnon-Hernmitian矩阵反转算法的陈述（1.1）。
possesses properties (1.1), (1.2), and (1.3). In order to find the joint probability density function of (yo, yi , , yn), or of any n + 1 successive variates, it is necessary to invert the matrix Tn . In this paper, we derive an exact recursive procedure for the numerical inversion of an arbitrary positive definite Toeplitz matrix of finite order, which takes full advantage of the strong restrictions placed on its elements by (1.1), (1.2), and (1.3). The number of multiplications required for the inversion of an nth order Toeplitz matrix, using this procedure, is proportional to n2, rather than to n', as in the case of methods which are suitable for arbitrary Hermitian matrices. To the author's knowledge, this inversion algorithm is the first to be specifically designed to take advantage of the peculiar simplicity of the general Toeplitz matrix. In addition, the closing section of the paper is devoted to a statement of an algorithm for the inversion of lnon-Hernmitian matrices of the form (1.1).