在笛卡尔坐标系中，使用离散透明边界条件对一维薛定谔方程进行数值模拟的 MATLAB 代码.zip

# Tridiagonal Matrix Algorithm [](https://www.mathworks.com/matlabcentral/fileexchange/85438-tridiagonal-matrix-algorithm) <br/><br/> # `tridiagonal_matrix` Solves the tridiagonal linear system <img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{A}\mathbf{x}=\mathbf{d}"/> for <img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{x}\in\mathbb{R}^{n}" title="\mathbf{x}\in\mathbb{R}^{n}" /> using the matrix implementation of the tridiagonal matrix algorithm. ## Syntax `x = tridiagonal_matrix(A,d)` ## Description `x = tridiagonal_matrix(A,d)` solves the tridiagonal linear system <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{A}\mathbf{x}=\mathbf{d}"/> for <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{x}\in\mathbb{R}^{n}" title="\mathbf{x}\in\mathbb{R}^{n}" />, where <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{A}\in&space;{\mathbb{R}}^{n\times&space;n}"/> is a tridiagonal matrix and <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{d}\in&space;{\mathbb{R}}^n"/>. <br/><br/> # `tridiagonal_vector` Solves the tridiagonal linear system <img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{A}\mathbf{x}=\mathbf{d}"/> for <img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{x}\in\mathbb{R}^{n}" title="\mathbf{x}\in\mathbb{R}^{n}" /> using the vector implementation of the tridiagonal matrix algorithm. ## Syntax `x = tridiagonal_vector(a,b,c,d)` ## Description `x = tridiagonal_vector(a,b,c,d)` solves the tridiagonal linear system <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{A}\mathbf{x}=\mathbf{d}"/> for <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{x}\in\mathbb{R}^{n}" title="\mathbf{x}\in\mathbb{R}^{n}" />, where <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{A}\in&space;{\mathbb{R}}^{n\times&space;n}"/> is a tridiagonal matrix defined using the tridiagonal vectors (<img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{a}\in\mathbb{R}^{n-1}"/>, <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{b}\in\mathbb{R}^{n}"/>, and <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{c}\in\mathbb{R}^{n-1}"/>) and where <img src="https://latex.codecogs.com/svg.image?\inline&space;\mathbf{d}\in&space;{\mathbb{R}}^n"/>. <br/><br/> # Tridiagonal Matrix Convention For these implementations, I use the following convention for denoting the elements of the tridiagonal matrix <img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{A}"/>: <img src="https://latex.codecogs.com/svg.latex?\mathbf{A}=\left\lbrack&space;\begin{array}{cccccc}&space;b_1&space;&space;&&space;c_1&space;&space;&&space;&space;&&space;&space;&&space;&space;&&space;\\&space;a_1&space;&space;&&space;b_2&space;&space;&&space;c_2&space;&space;&&space;&space;&&space;&space;&&space;\\&space;&space;&&space;a_2&space;&space;&&space;\ddots&space;&space;&&space;\ddots&space;&space;&&space;&space;&&space;\\&space;&space;&&space;&space;&&space;\ddots&space;&space;&&space;\ddots&space;&space;&&space;c_{n-2}&space;&space;&&space;\\&space;&space;&&space;&space;&&space;&space;&&space;a_{n-2}&space;&space;&&space;b_{n-1}&space;&space;&&space;c_{n-1}&space;\\&space;&space;&&space;&space;&&space;&space;&&space;&space;&&space;a_{n-1}&space;&space;&&space;b_n&space;&space;\end{array}\right\rbrack"/> Most other references have <img src="https://latex.codecogs.com/svg.latex?\inline&space;a_i"/>'s ranging from <img src="https://latex.codecogs.com/svg.latex?\inline&space;a_2"/> to <img src="https://latex.codecogs.com/svg.latex?\inline&space;a_n"/> both in the definition of the tridiagonal matrix and in the algorithm used to solve the corresponding linear system. In this implementation, I have the <img src="https://latex.codecogs.com/svg.latex?\inline&space;a_i"/>'s ranging from <img src="https://latex.codecogs.com/svg.latex?\inline&space;a_1"/> to <img src="https://latex.codecogs.com/svg.latex?\inline&space;a_{n-1}"/>; this makes the algorithm slightly more straightforward to implement. <br/><br/> # Examples and Additional Documentation - See "EXAMPLES.mlx" or the "Examples" tab on the File Exchange page for examples. - See ["Tridiagonal_Matrix_Algorithm.pdf"](https://tamaskis.github.io/files/Tridiagonal_Matrix_Algorithm.pdf) (also included with download) for the technical documentation.