The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes . To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm . We desc...

We implement the Lanczos algorithm on an Intel Xeon Phi coprocessor and compare its performance to a multi-core Intel Xeon CPU and an NVIDIA graphics processor. The Xeon and the Xeon Phi are parallelized with OpenMP and the graphics processor is programmed with CUDA. The performance is evaluated by measuring the execution time of a single step in the Lanczos algorithm. We study two quantum latt...

We present a new variant of the block Lanczos algorithm for finding vectors in the kernel of a symmetric matrix over F2. Our algorithm is at least as efficient as that of Peter Montgomery [6], while the sequence of matrices Wi constructed here have different algebraic properties that may be useful in eventually providing a provable upper bound on the time required to solve this problem. Namely,...

Two different Lanczos-type methods for the linear response eigenvalue problem are analyzed. The first one is a natural extension of the classical Lanczos method for the symmetric eigenvalue problem while the second one was recently proposed by Tsiper specially for the linear response eigenvalue problem. Our analysis leads to bounds on errors for eigenvalue and eigenvector approximations by the ...

Parallel solution of irregular problems require solving the graph partitioning problem. The extended eigenproblem appears as the solution of some relaxed formulations of the graph partitioning problem. In this paper, a new subspace algorithm for the solving the extended eigenproblem is presented. The structure of this subspace method allows the incorporation of multigrid preconditioners. We num...

The spectral transformation Lanczos method is very popular for solving large scale Her-mitian generalized eigenvalue problems. The method uses a special inner product so that the symmetric Lanczos method can be used. Sometimes, a semi-inner product must be used. This may lead to instabilities and breakdown. In this paper, we suggest a cure for breakdown by use of an implicit restart in the Lanc...

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos–type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U , i.e., of the matrix U +UH . By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is ran...

We present a distributed scheme for initialization from atomic wavefunctions in ab initio molecular dynamics simulations. Good initial guesses for approximate wavefunctions are very important in order to enable practical simulations with thousands of atoms. The new scheme is based on a distributed implementation of the Lanczos algorithm for very large dense eigenproblems. We show that the massi...