The block Lanczos algorithm proposed by Peter Montgomery is an efficient means to tackle the sparse linear algebra problem which arises in the context of the number field sieve factoring algorithm and its predecessors. We present here a modified version of the algorithm, which incorporates several improvements: we discuss how to efficiently handle homogeneous systems and how to reduce the numbe...

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. G...

Krylov subspace based moment matching algorithms, such as PVL (Padé approximation Via the Lanczos process), have emerged as popular tools for efficient analyses of the impulse response in a large linear circuit. In this work, a new derivation of the PVL algorithm is presented from the matrix point of view. This approach simplifies the mathematical theory and derivation of the algorithm. Moreove...

In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and so-called look-ahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczos-based iterative schemes for solving nonsymmetric linear systems have been p...

This work presents an adaptive block Lanczos method for large-scale non-Hermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the non-Hermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria ...

Lanczos vectors computed in nite precision arithmetic by the three-term recurrence tend to lose their mutual biorthogonality. One either accepts this loss and takes more steps or re-biorthogonalizes the Lanczos vectors at each step. For the symmetric case, there is a compromise approach. This compromise, known as maintaining semi-orthogonality, minimizes the cost of re-orthogonalization. This p...

The Rivest-Shamir-Adleman (RSA) algorithm is a very popular and secure public key cryptosystem, but its security relies on the difficulty of factoring large integers. The General Number Field Sieve (GNFS) algorithm is currently 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 met...

A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in th...

T h e B R algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. I t is a bulge-chasingalgorithm like the Q R algorithm, but , unlike the QR algorithm, it is well adapted to computing the eigenvaluesof the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the B R algorithm and gives numerical...

Solutions to very large sparse systems of linear equations over small finite fields are essential in the best known algorithms for factoring integers and computing discrete logarithms. Due to the sizes of such systems, elimination-based approaches are impractical, so attention has shifted to iterative or black-box methods. This thesis examines one brand of such algorithms: those based on the La...