New Methods for Calculations of the Lowest Eigenvalues of the Real Symmetric Generalized Eigenvalue Problem

Lagrange‐type iterative methods for calculation of extreme eigenvalues of generalized eigenvalue problem with large symmetric matrices

“…(1), which can be considered as a block generalization of the algorithm of the Newton-Rayleigh method developed earlier [13]. The algorithm of the new method is presented below…”

“…The block generalization of the Newton-Rayleigh iterative method [13] is obtained from the Algorithm 1 by replacing Eqs. (8) and (9) for calculation of a new Lagrange correction vector at Step 4 by the following equation…”

“…It was also shown that the quadratic MITIN convergence of the Jacobi-Davidson method can be achieved by a proper choice of the projection operators [15]. Then, it was noted that the JacobiDavidson and the Newton-Rapson correction vectors have similar orthogonal properties [13]. These facts implicitly point out that the Jacobi-Davidson method is a Newton-Raphson type method.…”

“…The other four methods are the block generalization of the Newton-Rayleigh, the diagonal Newton-Rayleigh iterative methods proposed in Ref. [13], the Jacobi-Davidson [14], and the diagonal Jacobi-Davidson method [13]. A block generalization of the Jacobi-Davidson method is not unique defined.…”

“…There are a few approaches for the construction of correction vectors. Among them, the orthogonalization of the Krylov basis [11], the use of relaxation vectors [1], minimization of the residual vector [12] and the Rayleigh functional [13] can be mentioned.…”