The orthogonality requirement in either iterative diagonalization or conjugate gradient approaches to the single particle Schr6dinger equation/~/~u = E~u leads to an overall N 3 scaling of the effort with the number N of atoms. We show that the Lanczos method circumvents this problem even when applied to all occupied states. Our implementation shows that the method is stable, exact, scales as N...

Lanczos-hybrid regularization methods have been proposed as effective approaches for solving largescale ill-posed inverse problems. Lanczos methods restrict the solution to lie in a Krylov subspace, but they are hindered by semi-convergence behavior, in that the quality of the solution first increases and then decreases. Hybrid methods apply a standard regularization technique, such as Tikhonov...

Resolution matrices are useful in seismic tomography because they allow us to evaluate the information content of reconstructed images. Techniques based on the multiplicity of equivalent exact formulas that may be used to define the resolution matrices have been used previously by the author to design algorithms that avoid the need for any singular value decomposition of the ray-path matrix. An...

Lanczos bidiagonalization is a competitive method for computing a partial singular value decomposition of a large sparse matrix, that is, when only a subset of the singular values and corresponding singular vectors are required. However, a straightforward implementation of the algorithm has the problem of loss of orthogonality between computed Lanczos vectors, and some reorthogonalization techn...

The Lanczos algorithm for tridiagonalizing a given matrix A generates a sequence of approximating matrices A, that can naturally be obtained as restrictions to subspaces. The eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of A. In this paper we produce explicit formulas for the characteristic polynomials of the A,, in terms of the...

We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements, without restricting to variational ansatzes. The lattice of size N is partitioned into two subclusters. At each iteration the Lanczos vector is projected into two sets of n(svd) smaller subcluster vectors using singular value decomposition. For low entanglement entropy S(ee...

A closed form expression for the ground-state energy density of the general extensive many-body problem is given in terms of the Lanczos tridiagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, an(N) and bn(N), asymptotic forms a(z) and b(z) can be defined in terms of a parameter z[n/N (n is the Lanczos itera...

The Lancws procedure is widely used in model reduction of large-scale djnamical systems. In this note we introduce an exact expression for the Hz norm of the error system in the Lanczos procedure. To our knowledge, this is the first such global result fur the Lanczos procedure. Then we tnm our attention to the rational Krylov method, which has the disadvantage that the selection of interpolatio...