The core computational step in spectral learning – finding the projection of a function onto the eigenspace of a symmetric operator, such as a graph Laplacian – generally incurs a cubic computational complexity O(N). This paper describes the use of Lanczos eigenspace projections for accelerating spectral projections, which reduces the complexity to O(nTop + nN) operations, where n is the number...

The one-hole spectral weight for two chains and two dimensional lattices is studied numerically using a new method of analysis of the spectral function within the Lanczos iteration scheme: the Lanczos spectra decoding method. This technique is applied to the t−Jz model for Jz → 0, directly in the infinite size lattice. By a careful investigation of the first 13 Lanczos steps and the first 26 on...

We present a variant of the popular BiCGSTAB method for solving nonsymmetric linear systems. The method, which we denote by ML(k)BiCGSTAB, is derived from a variant of the BiCG method based on a Lanczos process using multiple (k > 1) starting left Lanczos vectors. Compared with the original BiCGSTAB method, our new method produces a residual polynomial which is of lower degree after the same nu...

We increase the efficiency of a recently proposed time integration scheme for time dependent quantum transport by using the Lanczos method for time evolution. We illustrate our modified scheme in terms of a simple one dimensional model. Our results show that the Lanczos-adapted scheme gives a large increase in numerical efficiency, and is an advantageous route for numerical time integration in ...

Recently, there has been a revival of interest in the Lanczos potential of the Weyl conformal tensor. Previous work by Novello and Neto has been done with the linearized Lanczos potential as a model of a spin-2 field, which depends on a massless limit of the field. In this paper, we look at an action based on a massless potential, and show that it is classically equivalent to the linearized reg...

The solution of linear systems with a parameter is an important problem in engineering applications, including structural dynamics, acoustics, and electronic circuit simulations, and in related model order reduction methods such as Padé via Lanczos. In this paper, we present a Lanczos-based method for solving parameterized symmetric linear systems with multiple right-hand sides. We show that fo...

We study the Lanczos method for solving symmetric linear systems with multiple right-hand sides. First, we propose a numerical method of implementing the Lanczos method, which can provide all approximations to the solution vectors of the remaining linear systems. We also seek possible application of this algorithm for solving the linear systems that occur in continuation problems. Sample numeri...

The matrix-form LSQR method is presented in this paper for solving the least squares problem of the matrix equation AXB = C with tridiagonal matrix constraint. Based on a matrix-form bidiagonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB = C reduces to a unconstrained least squares problem of linear system, which can be solved by u...

Two new algorithms for one-sided bidiagonalization are presented. The first is a block version which improves execution time by improving cache utilization from the use of BLAS 2.5 operations and more BLAS 3 operations. The second is adapted to parallel computation. When incorporated into singular value decomposition software, the second algorithm is faster than the corresponding ScaLAPACK rout...

Though the implicitly restarted Arnoldi/Lanczos method in ARPACK is a reliable method for computing a few eigenvalues of large-scale matrices, it can be inefficient because it only checks for convergence at restarts. Significant savings in runtime can be obtained by checking convergence at each Lanczos iteration. We describe a new convergence test for the maximum eigenvalue that is numerically ...