Numerical safeguarded use of the implicit restarted Lanczos algorithm for solving nonlinear eigenvalue problems and its monotonicity analysis
Numerical safeguarded use of the implicit restarted Lanczos algorithm for solving nonlinear eigenvalue problems and its monotonicity analysis
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隐式重启Lanczos算法求解非线性特征值问题的数值保障及其单调性分析
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发表时间：
1993
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通讯作者：
M. R. Abdel
中科院分区：
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作者：
M. R. Abdel
在本论文中，我们开发了一种高效精确的数值算法，用于评估大规模非线性特征值问题的一些最小特征值及其相应的特征向量。这些问题中的矩阵项是由有理函数很好地近似的超越函数。该算法基于隐式重新启动 Lanczos 方法，该方法用于解决线性特征值子问题，该问题与使用有理函数插值来近似广义特征值的新找零技术一起出现。我们已经在高性能计算机上测试了这项技术，并提出了一些数值实验来证明该过程的效率和准确性。
我们的单调性分析理论表明，参数化特征值曲线（单调递增）比具有不稳定行为的参数化行列式曲线表现得更好。
我们的数值和单调性分析足够普遍，适用于任何具有单调递增广义特征值的问题。此类问题与混合有限元公式相关，该公式涉及频率无关的刚度和频率相关的质量矩阵。
In this thesis, we develop an efficient accurate numerical algorithm for evaluating a few of the smallest eigenvalues and their corresponding eigenvectors for large scale nonlinear eigenproblems. The entries of the matrices in these problems are transcendental functions approximated well by rational functions. This algorithm is based upon the Implicit Restarted Lanczos method for solving the linear eigenvalue sub-problems that arise in conjunction with a new zero-finding technique that uses rational function interpolation to approximate the generalized eigenvalues. We have tested this technique on high performance computers and we present some numerical experiments that demonstrate the efficiency and the accuracy of this procedure.
Our monotonicity analysis theory shows that the parameterized eigenvalue curves (monotone increasing) are much better behaved than the parameterized determinant curves that have erratic behavior.
Our numerical and monotonicity analyses are sufficiently general that they hold for any problem having monotone increasing generalized eigenvalues. This type of problem is associated with the mixed finite element formulation that involves a frequency independent stiffness and frequency dependent mass matrices.