Eigenvalue Problems
Eigenvalue Problems
复制标题
DOI：
10.1007/978-3-319-00401-3_9
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Philipp Scherer
中科院分区：
文献类型：
--
作者：
Philipp Scherer
特征值问题在物理学中无处不在。重要的例子是有限正交基上与时间无关的Schrödinger方程或分子围绕其平衡结构的调和运动。最重要的是普通特征值问题，它涉及具有厄米矩阵（或对称矩阵，如果是实数）的齐次线性方程组的解。小维矩阵可以通过确定特征多项式的根和求解齐次线性方程组直接对角化。雅可比方法使用连续旋转来对角化具有酉变换的矩阵。对于不太大的对称矩阵，一种非常流行的方法是将矩阵简化为三对角形式，这种形式可以用QL算法有效地对角化。一些特殊的三对角矩阵可以解析对角化。克雷洛夫空间算法是非常大维矩阵的选择。我们着重讨论了Arnoldi和Lanczos的方法。
Eigenvalue problems are omnipresent in physics. Important examples are the time independent Schrödinger equation in a finite orthogonal basis or the harmonic motion of a molecule around its equilibrium structure. Most important are ordinary eigenvalue problems, which involve the solution of a homogeneous system of linear equations with a Hermitian (or symmetric, if real) matrix. Matrices of small dimension can be diagonalized directly by determining the roots of the characteristic polynomial and solving a homogeneous system of linear equations. The Jacobi method uses successive rotations to diagonalize a matrix with a unitary transformation. A very popular method for not too large symmetric matrices reduces the matrix to tridiagonal form which can be diagonalized efficiently with the QL algorithm. Some special tridiagonal matrices can be diagonalized analytically. Krylov-space algorithms are the choice for matrices of very large dimension. We discuss especially the methods by Arnoldi and Lanczos.