symmetric-diagonal reductions as preprocessing for symmetric positive definite generalized eigenvalue solvers

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 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.