We consider on efficient and stable solution for linear systems by product-type iterative methods. By means of improvement on ordering of Lanczos and stability polynomials which construct algorithm of product-type BiCG method, we derive new algorithm based on BiCGSafe method with safety convergence. We will apply the proposed BiCGSafe method to solve linear systems which appeared in a variety o...

This paper analyzes the important steps of an O(n 2 log n) algorithm for nding the eigenvalues of a complex Hankel matrix. The three key steps are a Lanczos-type tridiagonalization algorithm, a fast FFT-based Hankel matrix-vector product procedure, and a QR eigenvalue method based on complex-orthogonal transformations. In this paper, we present an error analysis of the three steps, as well as r...

The Lanczos process constructs a sequence of orthonormal vectors vm spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary Lanczos process, starting at an arbitrary Lanczos vector vm and how to use this secondary process to efficiently obtain computable error estimates and error bounds...

A new method for computing transient electromagnetic wavefields in inhomogeneous and lossy media is presented. The method utilizes a modified Lanczos scheme, where a socalled reduced model is constructed. A discretization of the time variable is then superfluous. This reduced model represents the transient electromagnetic wavefield on a certain bounded interval in time. Some theoretical aspects...

We describe the application of the PVL algorithm to the small-signal analysis of circuits, including sensitivity computations. The PVL algorithm is based on the eecient computation of the Pad e approximation of the network transfer function via the Lanczos process. The numerical stability of the algorithm permits the computation of the Pad e approximation to any accuracy over a certain frequenc...

The Paige style Lanczos algorithm is an iterative method for finding a few eigenvalues of large sparse symmetric matrices. Some beautiful relationships among the elements of the eigenvectors of a symmetric tridiagonal matrix are used to derive a perverse starting vector which delays convergence as long as possible. Why i such slow convergence is never seen in practice is also examined.

Integer factorization is known to be one of the most important and useful methods in number theory and arithmetic. It also has a very close relationship to some algorithms in cryptography such as RSA algorithm. The RSA cryptosystem is one of the most popular and attractive public-key cryptosystems in theworld today. Its security is based on thedifficulty of integer factorization. Solving a larg...

This technical report introduces some software packages for partial SVD computation, including optimized PROPACK, modified PROPACK for computing singular values above a threshold and the corresponding singular vectors, and block Lanczos with warm start (BLWS). The current version is preliminary. The details will be enriched soon.

The nonsymmetric Lanczos method has recently received signiicant attention as a model reduction technique for large-scale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, inexpensive implicit restarts are developed which can be employed to stabilize the Lanczos generated model.

The CGS and BiCGstab algorithms were derived by their authors from the bi-conjugate gradient iteration. In this paper a derivation from the (unnormalized) Lanczos algorithm is given. Experimental results are given, showing that these methods offer no immediate advantage over their earlier counterparts.