Complex shift and invert strategies for real matrices
Complex shift and invert strategies for real matrices
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DOI：
10.1016/0024-3795(87)90126-1
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发表时间：
1987-04
影响因子：
1.1
通讯作者：
B. Parlett;Y. Saad
中科院分区：
文献类型：
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作者：
B. Parlett;Y. Saad
当使用迭代法求解形式为Fu =λMu的广义非对称特征值问题时，其中FandM是真实的带状矩阵，通常需要使用移位和逆算子B =（F−σM）−1Min来增强特征值分离并提高效率。不幸的是，移位σ通常是复杂的，矩阵B也是如此。那么问题是是否有可能避免复杂的算术，同时保持对（F，M）的带状性的任何优点。对于经典的问题，其中M = I和F是带状的，复杂的算术可以通过使用双移位来避免，即，通过使用真实的矩阵BB，其带宽是F的两倍。这个令人满意的解决方案扩展到的情况下，M是对角线。在一般情况下，对上述问题的回答是否定的，在这个意义上，复杂的算术只能以失去带状性的优势为代价来避免。一种解决方案是将移位矩阵F − σMin分解为复数运算，但随后在迭代过程中使用真实的运算。本文研究了几种方法，并讨论了它们在不同情况下各自的优点。
When using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the formFu=λMu, whereFandMare real banded matrices, it is often desirable to work with the shifted and inverted operatorB= (F−σM)−1Min order to enhance the eigenvalue separation and improve efficiency. Unfortunately, the shift σ is generally complex, and so is the matrixB. The question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair (F, M). For the classical problem whereM=IandFis banded, complex arithmetic can be avoided by using double shifts, i.e., by working with the real matrixBB̄, whose bandwidth is double that ofF. This satisfactory solution extends to the case whereMis diagonal as well. In the generalized case the answer to the above question is negative, in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness. One solution is to factor the shifted matrixF− σMin complex arithmetic but employ real arithmetic subsequently in the iterative procedure. This paper examines several approaches and discusses their respective merits under various circumstances.