An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation is presented of a look-ahead version of the Lanczos algorithm that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead

A nested Krylov subspace method to compute the sign function of large complex matrices

We present an acceleration of the well-established Krylov-Ritz methods to compute the sign function of large complex matrices, as needed in lattice QCD simulations involving the overlap Dirac operator at both zero and nonzero baryon density. Krylov-Ritz methods approximate the sign function using a projection on a Krylov subspace. To achieve a high accuracy this subspace must be taken quite large, which makes the method too costly. The new idea is to make a further projection on an even smaller, nested Krylov subspace. If additionally an intermediate preconditioning step is applied, this projection can be performed without affecting the accuracy of the approximation, and a substantial gain in efficiency is achieved for both Hermitian and non-Hermitian matrices. The numerical efficiency of the method is demonstrated on lattice configurations of sizes ranging from 4^4 to 10^4, and the new results are compared with those obtained with rational approximation methods.Comment: 17 pages, 12 figures, minor corrections, extended analysis of the preconditioning ste

The cryptohermitian smeared-coordinate representation of wave functions

The one-dimensional real line of coordinates is replaced, for simplification or approximation purposes, by an N-plet of the so called Gauss-Hermite grid points. These grid points are interpreted as the eigenvalues of a tridiagonal matrix $\mathfrak{q}_0$ which proves rather complicated. Via the "zeroth" Dyson-map $\Omega_0$ the "operator of position" $\mathfrak{q}_0$ is then further simplified into an isospectral matrix $Q_0$ which is found optimal for the purpose. As long as the latter matrix appears non-Hermitian it is not an observable in the manifestly "false" Hilbert space ${\cal H}^{(F)}:=\mathbb{R}^N$. For this reason the optimal operator $Q_0$ is assigned the family of its isospectral avatars $\mathfrak{h}_\alpha$, $\alpha=(0,)\,1,2,...$. They are, by construction, selfadjoint in the respective $\alpha-$dependent image Hilbert spaces ${\cal H}^{(P)}_\alpha$ obtained from ${\cal H}^{(F)}$ by the respective "new" Dyson maps $\Omega_\alpha$. In the ultimate step of simplification, the inner product in the F-superscripted space is redefined in an {\it ad hoc}, $\alpha-$dependent manner. The resulting "simplest", S-superscripted representations ${\cal H}^{(S)}_\alpha$ of the eligible physical Hilbert spaces of states (offering different dynamics) then emerge as, by construction, unitary equivalent to the (i.e., indistinguishable from the) respective awkward, P-superscripted and $\alpha-$subscripted physical Hilbert spaces.Comment: 13. pp, 3 fig

Gegenbauer-solvable quantum chain model

In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, ...$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer's ultraspherical polynomials $f_n(x)=C_n^\alpha(x)$ are chosen for the detailed illustration of technicalities.Comment: 29 pp., 1 fi