Lanczos-QR方法在大型非比例阻尼结构复模态计算中的应用

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振动与冲击 ›› 2011, Vol. 30 ›› Issue (5) : 222-225.

论文

Lanczos-QR方法在大型非比例阻尼结构复模态计算中的应用

张静1 刘明辉1 郑钢铁1,2

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摘要

摘　要：本文跟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，本文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。本文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。

Abstract

Abstract: In this paper, the Lanczos-QR algorithm was introduced into the computation of the large complex eigenvalue problem. This method, which combines the efficiency of the Lanczos algorithm and the stability of the QR algorithm, firstly reduces the matrix scale with the Lanczos algorithm, and then solves the complex eigenvalues and the complex eigenvectors of the reduced matrix with the QR algorithm. In order to improve the algorithm’s computing stability and precision, the partitioned matrix triangle decomposition was introduced to achieve the standard eigenvalue problem. A step-by step procedure of this method was summarized and programmed in FORTRAN language. Numerical examples show that the improved Lanczos-QR algorithm is precise and stable.