We discuss some potential advantages of the orthogonal symmetric-diagonal reduction in two main versions of the Schur-QR method for symmetric positive definite generalized eigenvalue problems. We also advise and use the appropriate reductions as preprocessing on the solvers, mainly the Cholesky-QR method, of the considered problems. We study and compare performance of the considered methods. We discuss numerical stability of the methods via providing upper bound for backward error of the computed eigenpairs and via investigating two kinds of scaled residual errors. We also propose and apply two kinds of symmetrizing which improve the stability and the performance of the methods. Numerical experiments show that the implemented versions of the Schur-QR method and the preprocessed versions of the Cholesky-QR method are usually more stable than the Cholesky-QR method.
