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A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems
SIAM J. MATRIX ANAL. APPL. c
1996 Society for Industrial and Applied MathematicsVol. 17, No. 2, pp. 401{425, April 1996 010A JACOBI{DAVIDSON ITERATION METHOD FOR LINEAREIGENVALUE PROBLEMSGERARD L.G. SLEIJPENy AND HENK A. VAN DER VORSTyAbstract. In this paper we propose a new method for the iterative computation of a few of theextremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is basedon an old and almost unknown method of Jacobi. Jacobis approach, combined with Davidsonsmethod, leads to a new method that has improved convergence properties and that may be usedfor general matrices. We also propose a variant of the new method that may be useful for thecomputation of nonextremal eigenvalues as well.Key words. eigenvalues and eigenvectors, Davidsons method, Jacobi iterations, harmonic RitzvaluesAMS subject classi cations. 65F15, 65N251. Introduction. Suppose we want to compute one or more eigenvalues andtheir corresponding eigenvectors of the n n matrix A. Several iterative methodsare available: Jacobis diagonalization method [9], [22], the power method [9], themethod of Lanczos [13], [22], Arnoldis method [1], [25], and Davidsons method [4],[25], [3], [14], [17]. The latter method has been reported to be quite successful, mostnotably in connection with certain symmetric problems in computational chemistry[4], [5], [31]. The success of the method seems to depend quite heavily on the (strong)diagonal dominance of A.The method of Davidson is commonly seen as an extension to Lanczoss method,but as Saad [25] points out, from the implementation point of view it is more relatedto Arnoldis method. In spite of these relations, the success of the method is notwell understood [25]. Some recent convergence results and improvements, as well asnumerical experiments, are reported in [3], [14], [15], [17], [16], [18], [27].Jacobi [12] proposed a method for eigenvalue approximation that essentially wasa combination of (1) Jacobi rotations, (2)
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