56 research outputs found
A fast elementary algorithm for computing the determinant of toeplitz matrices
In recent years, a number of fast algorithms for computing the determinant of
a Toeplitz matrix were developed. The fastest algorithm we know so far is of
order , where is the number of rows of the Toeplitz matrix
and is the bandwidth size. This is possible because such a determinant can
be expressed as the determinant of certain parts of -th power of a related
companion matrix. In this paper, we give a new elementary proof of
this fact, and provide various examples. We give symbolic formulas for the
determinants of Toeplitz matrices in terms of the eigenvalues of the
corresponding companion matrices when is small.Comment: 12 pages. The article is rewritten completely. There are major
changes in the title, abstract and references. The results are generalized to
any Toeplitz matrix, but the formulas for Pentadiagonal case are still
include
Inverse properties of a class of seven-diagonal (near) Toeplitz matrices
This paper presents the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method
New algorithm for solving pentadiagonal CUPL-Toeplitz linear systems
In this paper, based on the structure of pentadiagonal CUPL-Toeplitz matrix and Sherman–Morrison–Woodbury formula, we develop a new algorithm for solving nonsingular pentadiagonal CUPL-Toeplitz linear system. Some numerical examples are given in order to illustrate the effectiveness of the proposed algorithms
Determinants of some pentadiagonal matrices
In this paper we consider pentadiagonal ((n+1)times(n+1)) matrices with two subdiagonals and two superdiagonals at distances (k) and (2k) from the main diagonal where (1le k < 2kle n). We give an explicit formula for their determinants and also consider the Toeplitz and “imperfect” Toeplitz versions of such matrices. Imperfectness means that the first and last (k) elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerváry and Szász we also show how these determinants can be factorized
Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic
It was shown in a series of recent publications that the eigenvalues of
Toeplitz matrices generated by so-called simple-loop symbols admit
certain regular asymptotic expansions into negative powers of . On the
other hand, recently two of the authors considered the pentadiagonal Toeplitz
matrices generated by the symbol , which does not satisfy
the simple-loop conditions, and derived asymptotic expansions of a more
complicated form. We here use these results to show that the eigenvalues of the
pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic
expansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m,
Garoni, and Serra-Capizzano and reveals that the simple-loop condition is
essential for the existence of the regular asymptotic expansion.Comment: 28 pages, 7 figure
ウィッテイカー・ヘンダーソン平滑化法に関する諸論
広島大学(Hiroshima University)博士(経済学)Doctor of Economicsdoctora
Spectral properties for a type of heptadiagonal symmetric matrices
In this paper we expressed the eigenvalues of a sort of heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From the prescribed eigenvalues, we computed eigenvectors for these types of matrices, giving also a formula not dependent on any unknown parameter for its determinant and inverse. Potential applications of the results are still provided
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