The study focuses primarily on Vandermonde-like matrix systems. The idea is to express Vandermonde and Vandermonde-like matrix systems as the problems related to Krylov Matrices. The connection provides a different angle to view the Vandermondelike systems. Krylov subspace methods are strongly related to polynomial spaces, hence a nice connection can be established using LU factorization as pro...

Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A ≈ RTR up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method compresses off-diagonal blocks with rank-revealing orthogonal decompositions. In the meantime, positive semidefinite te...

We have recently developed a new program analysis strat egy called fractal symbolic analysis that addresses some of limitations of techniques such as dependence analysis In this paper we show how fractal symbolic analysis can be used to convert between left looking and right looking versions of three kernels of central importance in compu tational science Cholesky factorization LU factorization...

For a fast simulation of interconnect structures we consider preconditioned iterative solution methods for large complex valued linear systems. In many applications the discretized equations result in ill-conditioned matrices, and efficient preconditioners are indispensable to solve the linear systems accurately. We apply the dual threshold incomplete LU (ILUT) factorization as preconditioners ...

For a one-dimensional diffusion problem on an refined computational grid we present preconditioners based on the standard approximate inverse technique. Next, we determine its spectral condition number κ2 and perform numerical calculations which corroborate the result. Then we perform numerical calculations which show that the standard approximate inverse preconditioners and our modified versio...

This thesis presents two new linear equation solvers, and investigates their applications in VLSI design automation. Both solvers are derived in the context of a special class of large-scale sparse left-hand-side matrices that are commonly encountered in engineering applications, and techniques are presented that can potentially extend the theory to more general cases. The first is a stochastic...

We present an approximate structured factorization method which is efficient, robust, and also relatively insensitive to ill conditioning, high frequencies, or wavenumbers for some discretized PDEs. Given a sparse symmetric positive definite discretized matrix A, we compute a structured approximate factorization A ≈ LLT with a desired accuracy, where L is lower triangular and data sparse. This ...

In this paper, a method via sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix is proposed. The resulting factorized sparse approximate inverse is used as a preconditioner for solving symmetric positive definite linear systems of equations by using the preconditioned conjugate gradient algorithm. Some numerical experime...

This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The rst subset which can be termed \coarse" is obtained by using \block" independent sets, or \aggregates". Two aggregates have no coupling between them, but nodes in the same aggregate may be ...

Recently, we released two LAPACK subroutines that implement Aasen’s algorithms for solving a symmetric indefinite linear system of equations. The first implementation is based on a partitioned right-looking variant of Aasen’s algorithm (the column-wise left-looking panel factorization, followed by the right-looking trailing submatrix update using the panel). The second implements the two-stage ...