Abstract: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 $$m\times n$$
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globe network. Chebyshev polynomials are introduced to represent new potenti… Show more
“…Recently, the shift-and-invert Arnoldi or Lanczos method has been used in designing fast algorithms for the generalized Toeplitz eigenproblem [9], and Toeplitz matrix exponential [10,11]. Toeplitz matrices have various applications [12][13][14][15], due to the special structure of Toeplitz matrices, there are many fast algorithms for solving Toeplitz matrix problems [16][17][18][19] and various formula for the inversion of Toeplitz matrix [20][21][22], the products of the inverse of a Toeplitz matrix and a vector can be implemented using several FFTs [10,11,21]. For a Hankel matrix, the inverse can be obtained by solving two large Hankel linear systems, and the matrix-vector products in the shift-and-invert Arnoldi method can also be realized efficiently by using FFTs.…”
Krylov subspace method is an effective method for large-scale eigenproblems. The shift-and-invert Arnoldi method is employed to compute a few eigenpairs of a large Hankel matrix pencil. However, a crucial step in the process is computing products between the inversion of a Hankel matrix and vectors. The inversion of the Hankel matrix can be obtained by solving two Hankel systems. By establishing a relationship between the errors of systems and the residuals of the Hankel eigenproblem, we provide a practical stopping criterion for solving Hankel systems and propose an inexact shift-and-invert Arnoldi method for the generalized Hankel eigenproblem. Numerical experiments are presented to demonstrate the efficiency of the new algorithm and our theoretical results.
“…Recently, the shift-and-invert Arnoldi or Lanczos method has been used in designing fast algorithms for the generalized Toeplitz eigenproblem [9], and Toeplitz matrix exponential [10,11]. Toeplitz matrices have various applications [12][13][14][15], due to the special structure of Toeplitz matrices, there are many fast algorithms for solving Toeplitz matrix problems [16][17][18][19] and various formula for the inversion of Toeplitz matrix [20][21][22], the products of the inverse of a Toeplitz matrix and a vector can be implemented using several FFTs [10,11,21]. For a Hankel matrix, the inverse can be obtained by solving two large Hankel linear systems, and the matrix-vector products in the shift-and-invert Arnoldi method can also be realized efficiently by using FFTs.…”
Krylov subspace method is an effective method for large-scale eigenproblems. The shift-and-invert Arnoldi method is employed to compute a few eigenpairs of a large Hankel matrix pencil. However, a crucial step in the process is computing products between the inversion of a Hankel matrix and vectors. The inversion of the Hankel matrix can be obtained by solving two Hankel systems. By establishing a relationship between the errors of systems and the residuals of the Hankel eigenproblem, we provide a practical stopping criterion for solving Hankel systems and propose an inexact shift-and-invert Arnoldi method for the generalized Hankel eigenproblem. Numerical experiments are presented to demonstrate the efficiency of the new algorithm and our theoretical results.
“…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 $$m\times n$$
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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.
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