|For the eigenvalues of a large and sparse unsym-metric coeecient matrix, we propose an improved version of the unsymmetric Lanczos process combining elements of numerical stability and parallel algorithm design. Stability is obtained by a coupled two-term recurrences procedure that generates Lanczos vectors scaled to unit length. The algorithm is derived such that all inner products and matrix...

In this paper we try to show the relations between the Lanczos algorithm and Pad e approximations as used e.g. in identiication and model reduction of dynamical systems. 1

Rational Krylov is an extension of the Lanczos or Arnoldi eigenvalue algorithm where several shifts (matrix factorizations) are used in one run. It corresponds to multipoint moment matching in model reduction. A variant applicable to nonlinear eigenproblems is described.

The most expensive part of all Electronic Structure Calculations based on Density Functional Theory, lies in the computation of an invariant subspace associated with some of the smallest eigenvalues of a discretized Hamiltonian operator. The dimension of this subspace typically depends on the total number of valence electrons in the system, and can easily reach hundreds or even thousands when l...

The most expensive part of all electronic structure calculations based on density functional theory lies in the computation of an invariant subspace associated with some of the smallest eigenvalues of a discretized Hamiltonian operator. The dimension of this subspace typically depends on the total number of valence electrons in the system, and can easily reach hundreds or even thousands when la...

Numerically exact investigations of interacting spin systems provide a major tool for an understanding of their magnetic properties. For medium size systems the approximate Lanczos diagonalization is the most common method. In this article we suggest two improvements: efficient basis coding in subspaces and simple restructuring for openMP parallelization.

The nonsymmetric Lanczos tridiagonalization algorithm is essentially the Gram-Schmidt biorthogonalization method for generating biorthogonal bases of a pair of Krylov subspaces. It suffers from breakdown and instability when a pivot at some step is zero or nearly zero, which is often the result of mismatch of the two Krylov subspaces. In this paper, we propose to modify one of the two Krylov su...

Its goal is to propose a new parallel algorithm to calculate the nullspace of a given huge matrix, in the field of GF(2). This parallel algorithm is mainly based on the Block Lanczos algorithm, but it presents a new way for distributing the matrix elements as well as minimizing the communication time. This thesis contains the necessary theoretical background and concludes with experimental resu...

An iterative algorithm is presented for solving the RPA equations of linear response. The method optimally computes the energy-weighted moments of the strength function, allowing one to match the computational effort to the intrinsic accuracy of the basic mean-field approximation, avoiding the problem of solving very large matrices. For local interactions, the computational effort for the metho...

Many problems in applied mathematics require the evaluation of matrix functionals of the form F(A) := u T f(A)u, where A is a large symmetric matrix and u is a vector. Golub and collaborators have described how approximations of such functionals can be computed inexpensively by using the Lanczos algorithm. The present note shows that error bounds for these approximations can be computed essenti...