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An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. (English) Zbl 0666.65033

An efficient method for computing a given number of leading eigenvalues (i.e., having largest real parts) and the corresponding eigenvectors of a large asymmetric matrix M is presented. The method consists of three main steps. The first is a filtering process in which the equation \(\dot x=Mx\) is solved for an arbitrary initial condition x(0) yielding: \(x(t)=e^{Mt}x(0)\). The second step is the construction of \((n+1)\) linearly independent vectors \(v_ m=M^ mx\), \(0\leq m\leq n\) or \(v_ m=e^{mM\tau}x\) (\(\tau\) being a “short” time interval). By construction, the vectors \(v_ m\) are combinations of only a small number of leading eigenvectors of M. The third step consists of an analysis of the vectors \(\{v_ m\}\) that yields the eigenvalues and eigenvectors.
The proposed method has been successfully tested on several systems. Here we present results pertaining to the Orr-Sommerfeld equation. The method should be useful for many computations in which present methods are too slow or necessitate excessive memory. In particular, we believe it is well suited for hydrodynamic and mechanical stability investigations.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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