论文标题
使用Hermitian矩阵分辨率转移的二次形式的兰开斯方法
Shifted Lanczos method for quadratic forms with Hermitian matrix resolvents
论文作者
论文摘要
Hermitian矩阵分解的二次形式涉及转移的线性系统的解决方案。高效的迭代解决方案使用Krylov子空间的偏移不变属性。Hermitian兰开斯方法将给定的载体和矩阵还原为jacobi矩阵（实际对称的Tridiagonal矩阵，带有正面超级和亚比亚角条目），并使用Jacobi Matrix近似于Quadratic Matrix。这项研究开发了一种转移的兰开斯方法，该方法直接处理了Hermitian矩阵解决方案。我们得出了线性操作员的矩阵表示，该矩阵表示通过求解与移位的Lanczos方法相关的Vorobyev力矩问题来近似分解。我们表明，雅各比矩阵分解的条目可以近似二次形式，与矩匹配。我们提供了足够的条件，使该方法不会分解，错误绑定和错误估计。从现实世界应用中绘制的矩阵上的数值实验将所提出的方法与以前的方法进行了比较，并表明所提出的方法在解决某些问题方面优于公认的方法。
Quadratic forms of Hermitian matrix resolvents involve the solutions of shifted linear systems. Efficient iterative solutions use the shift-invariance property of Krylov subspaces The Hermitian Lanczos method reduces a given vector and matrix to a Jacobi matrix (real symmetric tridiagonal matrix with positive super and sub-diagonal entries) and approximates the quadratic form using the Jacobi matrix. This study develops a shifted Lanczos method that deals directly with the Hermitian matrix resolvent. We derive a matrix representation of a linear operator that approximates the resolvent by solving a Vorobyev moment problem associated with the shifted Lanczos method. We show that an entry of the Jacobi matrix resolvent can approximate the quadratic form, matching the moments. We give a sufficient condition such that the method does not break down, an error bound, and error estimates. Numerical experiments on matrices drawn from real-world applications compare the proposed method with previous methods and show that the proposed method outperforms well-established methods in solving some problems.
