F eb 1 99 8 A Lanczos algorithm for linear response

F eb 1 99 8 A canonical ensemble approach to graded - response perceptrons

Perceptrons with graded input-output relations and a limited output precision are studied within the Gardner–Derrida canonical ensemble approach. Soft non-negative error measures are introduced allowing for extended retrieval properties. In particular, the performance of these systems for a linear and quadratic error measure, corresponding to the perceptron respectively the adaline learning alg...

A Lanczos-type method for multiple starting vectors

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...

F eb 1 99 8 Riemann - Roch for Tensor Powers

Stability of the Lanczos Method for Matrix Function Approximation

Theoretically elegant and ubiquitous in practice, the Lanczos method can approximate f(A)x for any symmetric matrix A ∈ R, vector x ∈ R, and function f . In exact arithmetic, the method’s error after k iterations is bounded by the error of the best degree-k polynomial uniformly approximating the scalar function f(x) on the range [λmin(A), λmax(A)]. However, despite decades of work, it has been ...

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT.

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenval...