Tridiagonal matrix algorithm

Abstract: For the further development of semiconductor heterostructures and devices based on it we have to know precisely the basic parameters such as energy band discontinuities, free carriers distribution, the magnitude of the accumulated charge, etc. Physical description of systems with reduced dimensionality gives the system of Schrodinger and Poisson equations that are too complicated for solving analytically and is usually solved by numerical methods. In this regard, in the paper questions of numerical calculation and energy band offsets determination on the example of quantum well by solving self-consistently Schrodinger and Poisson are considered. The determination of the Schrodinger equation is implemented by the “shooting” algorithm, and the determination of the Poisson equation using Thomas algorithm, in connection with the speed of operation and optimum accuracy of the solution.

Abstract: Let Am (m = 1, …, k) be n × n matrices, the joint numerical range is defined by $$JNR[{A_1}, \ldots ,{A_k}] = \left\{ {\left( {{x^*}{A_1}x, \ldots ,{A_k}x} \right):x \in {C^n},\parallel x\parallel = 1} \right\}.$$ In this paper, some geometric properties of JNR are presented, when the hermitian Am are bordered or (2μ–C1)–Cdiagonal matrices and the convexity of JNR is investigated.

Abstract: The k-tridiagonal matrices have received much attention in recent years. Many different algorithms have been proposed to improve the efficiency of k-tridiagonal matrix estimation. A novel method based on interval analysis has been identified to improve the efficiency of the calculation. This paper presents efficient and reliable computational algorithms for determining the determinant and inverse of general k-tridiagonal interval matrices built on generalized interval arithmetic. This study is based on the Doolittle LU factorization of the interval matrix. Finally, examples are presented to illustrate the algorithms.

Abstract: The following study introduces an algorithm for numerically solving a coupled system of differential equations that discretize to produce a banded pentadiagonal matrix with tridiagonal submatrices on either side. The article then proceeds to illustrate a step-by-step procedure to solve a two-equation system in a coupled manner. A generalization of this method to multiequation system follows. The proposed algorithm is highly efficient as it fully exploits the banded nature of the matrix. Finally, the solution of the two-equation k − ω and the four-equation turbulence models in plane channel flows by the current method serves as a demonstration exercise and wraps up the paper.