Lanczos-QR方法在大型非比例阻尼结构复模态计算中的应用

摘要

根据大型非比例阻尼结构的动力学模型特点,发展了适用于大型复特征值问题求解的Lanczos-QR方法.针对大规模非对称实矩阵的标准特征值问题,此方法首先采用Lanczos迭代方法对矩阵进行降阶;然后采用带移位的双重步位移QR方法求解降阶后的实三对角矩阵的全部特征值和特征向量,最后经过相应的变换得到原矩阵的特征解.另外,提出了分块矩阵三角分解方法将广义特征值进行标准化,该方既提高了计算效率,又避免了范数误差.还对奇异问题的处理进行了讨论.最后在FORTRAN语言中进行了程序实现,完成初步软件化,并且通过算例对算法及其程序的精度和收敛速度进行了验证.%Here, Lanczos-QR algorithm was introduced into the computation of a large complex eigenvalue problem. Combining efficiency of Lanczos algorithm and stability of QR algorithm, this method firstly reduced matrix scale with Lanczos algorithm, and then solved complex eigenvalues and complex eigenvectors of the reduced matrix with QR algorithm. In order to improve the algorithm's computing stability and precision, the partitioned matrix triangle decomposition was introduced to get a standard eigenvalue problem. A step-by-step procedure of this method was summarized and programmed with FORTRAN language. Numerical examples showed that the improved Lanczos-QR algorithm is precise and stable.