We describe the MRS solver, a Minimal Residual method based on the Lanczos algorithm that solves problems from the important class of linear systems with a shifted skew-symmetric coefficient matrix using short vector recurrences. The MRS solver is theoretically compared with other Krylov solvers and illustrated by some numerical experiments.

We report on experiments conducted with the implicitly restarted Lanczos algorithm for computing a few of the lowest frequencies of standing electromagnetic waves in resonant cavities with the nite element method. The linear systems that are caused by the shift and invert spectral transformation are solved by means of two-level hierarchical basis preconditioners.

Light-cone coordinates and supersymmetric discrete light-cone quantization are used to analyze the thermodynamics of two-dimensional supersymmetric quantum chromodynamics with a Chern–Simons term in the large-Nc approximation. This requires estimation of the entire spectrum of the theory, which is done with a new algorithm based on Lanczos iterations. Although this work is still in progress, so...

We provide the secret Diffie-Hellman-Key which is requested by Kevin McCurley’s challenge of 1989. The DH-protocol in question has been carried out in (ZZ/pZZ)∗ where p is a 129-digit prime of special form. Our method employed the Number Field Sieve. The linear algebra computation was done by the Lanczos algorithm.

For solving linear system of equations is known several algorithms. Iteration algorithms are recommended for the large linear systems with sparse matrix. But in the case of general n x m matrices the classic iterative algorithms are not applicable with a few exceptions. For example in some cases the Lanczos type algorithms are adequate. The algorithm presented here based on the minimization of ...

The Lanczos iteration for finding the largest eigenvalues has been implemented in C++. This implementation has been tested against ARPACK which is created for this purpose, but written in Fortran. The test were with regards to both speed and precision.

Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only few Lanczos vectors, using the BiConjugate Gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation ...

We present two expansions for the gradient of a random field. In the first approach, we differentiate its truncated KarhunenLoève expansion. In the second approach, the Karhunen-Loève expansion of the random field gradient is computed directly. Both strategies require the solution of dense, symmetric matrix eigenvalue problems which can be handled efficiently by combining hierachical matrix tec...

This paper describes a graphics processing unit (GPU) implementation of the Filtered Lanczos Procedure for the solution of large, sparse, symmetric eigenvalue problems. The Filtered Lanczos Procedure uses a carefully chosen polynomial spectral transformation to accelerate the convergence of the Lanczos method when computing eigenvalues within a desired interval. This method has proven particula...