解大规模非对称矩阵特征问题Lanczos方法的两种精化版本

【作者】 吴钢；

【导师】 贾仲孝；

【作者基本信息】 大连理工大学 ， 计算数学， 2001， 硕士

【摘要】 在许多科学与工程计算中，经常必须数值求解大规模矩阵特征问题，理论分析和大量的数值实验已经表明了求解此类问题的经典正交投影、斜投影方法都存在着Ritz值收敛而Ritz向量可能不收敛的隐患。为此，贾对经典的正交投影和斜投影方法进行了重大革新，使用精化向量取代Ritz向量作为待求特征向量的近似，由此导出的精化投影方法不但克服了原有方法的隐患，使得算法的可靠性增强，而且收敛的速度也大大加快。 进一步将精化策略和求解大规模矩阵问题的许多其它重要技术或方法相结合，研究和开发出更多更高效的新算法，是一个十分重要而且迫切需要解决的课题。本文正是在这一思想的引导下，在下述几个方面进行了研究： 1．将精化投影方法的思想和拟精化思想相结合，提出了拟精化 双正交Lanczos方法，建立了拟精化近似特征对和精化近似特 征对对应的残量范数之间的关系，给出了拟精化双正交 Lanczos算法，并进行了数值实验，具体算例表明，该算法比 双正交Lanczos算法优越； 2．根据精化投影方法的思想，提出了半精化双正交Lanczos方 法，建立了半精化特征对和精化特征对对应的残量范数之间 的关系，给出了半精化双正交Lanczos算法，数值实验表明， 该算法比双正交Lanczos算法和拟精化双正交Lanczos算法 优越。更多还原

【Abstract】 Large unsymmetric eigenproblems arise in many applications in scientific-and engineering computing. Theoretical analysis and numerical experiments show that the classical .orthogonal and the oblique projection methods have potential danger in finding eigenvectors of the matrix involved. That is, Ritz vectors may not converge to the desired eigenvectors even if the corresponding Ritz values converge. In order to circumvent the flaw, Jia suggested to use certain refined vectors to approximate the desired eigenvectors. The resulting refined projection methods avoid the above danger and converge faster. It is important to combine the refined strategy with many other techniques or methods. More robust and reliable algorithms can be derived from the combination. This thesis consists of the following parts.1.The quasi-refined biorthogonalization Lanczos method is proposed according to the refined projection strategy and the quasi-refined idea. Moreover, we discuss the relationship between the residual norm of the quasi-refined approximate eigenpair and that of the refined eigenpair, and the quasi-refined biorthogonalization Lanczos algorithm is given. Some numerical experiments are carried on, and the results illustrate that the newalgorithm is superior to the classic one. 2.We propose the semi-refined biorthogonalization Lanczos method which is based on the refined projection idea. Furthermore, the relationship between the residual norm of the semi-refined approximate eigenpair and that of the approximate refined eigenpair is discussed. The semi-refined biorthogonalization Lanczos algorithm is given. We carry on some numerical experiments, and the results show that the new algorithm is more efficient than its conventional counterpart.更多还原