Conjugate gradient methods for the Rayleigh quotient minimization of generalized eigenvalue problems (Q1308505)

For the computation of extreme eigenvalues and corresponding eigenvectors of the generalized eigenvalue problem \(Ax= \lambda Bx\), where \(A\) is real symmetric and \(B\) is positive definite, the conjugate gradient method for minimization of the Rayleigh quotient by \textit{W. W. Bradbury} and \textit{R. Fletcher} [New iterative methods for solution of the eigenproblem. Numer. Math. 9, 259-267 (1966; Zbl 0202.435)] is discussed and modified. The author considers proper step sizes in the one-dimensional minimization of the Rayleigh quotient and discusses properties, in particular, the convergence of the modified conjugate gradient methods for the generalized eigenvalue problem.