An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices

Abstract

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 matrixM is presented. The method consists of three main steps. The first is a filtering process in which the equationx = Mx is solved for an arbitrary initial conditionx(0) yielding:x(t)=e ^Mt x(0). The second step is the construction of (n+1) linearly independent vectorsv _m =M ^m x, 0⩽m⩽n orv _m =e ^mMt x (τ being a “short” time interval). By construction, the vectorsv _m are combinations of only a small number of leading eigenvectors ofM. 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.