For MinRes and SymmLQ it is essential to compute a QR decomposition of a tridiagonal coefficient matrix gained in the Lanczos process. This QR decomposition is constructed by an update scheme applying in every step a single Givens rotation. Using complex Householder reflections we generalize this idea to block tridiagonal matrices that occur in generalizations of MinRes and SymmLQ to block meth...

For the solutions of large and sparse linear systems of equations with unsymmetric coeÆcient matrices, we propose an improved version of the BiConjugate Gradient method (IBiCG) method based on [5, 6] by using the Lanczos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a coupled two-term procedure...

It is well known that the Lanczos process suffers from loss of orthogonality in the case of finite-precision arithmetic. Several approaches have been proposed in order to address this issue, thus enabling the successful computation of approximate eigensolutions. However, these techniques have been studied mainly in the context of long Lanczos runs, but not for restarted Lanczos eigensolvers. Se...

Currently, RSA is a very popular, widely used and secure public key cryptosystem, but the security of the RSA cryptosystem is based on the difficulty of factoring large integers. The General Number Field Sieve (GNFS) algorithm is the best known method for factoring large integers over 110 digits. Our previous work on the parallel GNFS algorithm, which integrated the Montgomery’s block Lanczos a...