Abstract:In this paper, for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, some properties, including the determinant, the inverse matrix, the eigenvalues and the eigenvectors, are studied in detail. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the well-known Fibonacci numbers; the inverse of the PTPT matrix can also be explicitly expressed using the Lucas number and only four elements in the PTPT matrix. Eigenvalues and eigenvectors can be obtained under certa… Show more
“…Recent research continues to highlight the importance of studying Topelitz matrices, such as [3,12,13,19].…”
Section: Introductionmentioning
confidence: 99%
“…Yaru Fu et al [12], in 2020, presented in their paper some properties for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, including the determinant, and the inverse matrix. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the wellknown Fibonacci numbers.…”
The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.
“…Recent research continues to highlight the importance of studying Topelitz matrices, such as [3,12,13,19].…”
Section: Introductionmentioning
confidence: 99%
“…Yaru Fu et al [12], in 2020, presented in their paper some properties for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, including the determinant, and the inverse matrix. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the wellknown Fibonacci numbers.…”
The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.
“…So the exact eigenvalues of the tridiagonal matrix need to be found. Tridiagonal matrices are used in many areas of science and engineering, and there are many good conclusions about it 54 – 61 . Figure 1 A cowbeb resistor network containing nodes and a zero potential point O .…”
The research of resistive network will become the basis of many fields. At present, many exact potential formulas of some complex resistor networks have been obtained. Computer numerical simulation is the trend of computing, but written calculation will limit the time and scale. In this paper, the potential formulas of a $$m\times n$$
m
×
n
scale cobweb resistor network and fan resistor network are optimized. Chebyshev polynomial of the second class and the absolute value function are used to express the novel potential formulas of the resistor network, and described in detail the derivation process of the explicit formula. Considering the influence of parameters on the potential formulas, several idiosyncratic potential formulas are proposed, and the corresponding three-dimensional dynamic images are drawn. Two numerical algorithms of the computing potential are presented by using the mathematical model and DST-VI. Finally, the efficiency of calculating potential by different methods are compared. The advantages of new potential formulas and numerical algorithms by the calculation efficiency of the three methods are shown. The optimized potential formulas and the presented numerical algorithms provide a powerful tool for the field of science and engineering.
“…RT method needs eigenvalues of a tridiagonal matrix to represent the potential formula. At present, there have been many results on tridiagonal matrices [45][46][47][48][49][50][51] , which are also widely used. It can be said that it is a powerful tool to solve the resistor network [33][34][35][36][37][38][39][40][41][42][43] .…”
Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an $$m\times n$$
m
×
n
globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.
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