We present an iterative method based on the Hermitian and skew-Hermitian splitting for solving the continuous Sylvester equation. By using the Hermitian and skew-Hermitian splitting of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a CGNR-like method as inner iteration to approximate each outer iterate, while each outer iteratio...

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present general convergence results, independent of iterative solvers for inner linear systems. We prove that the method converges quadratically at least under a new condition, called the uniform positiveness condition. This condition can be much weaker than the commonly used one that at outer iteration k, requires the relative res...

Convergence results are provided for inexact inverse subspace iteration applied to the problem of finding the invariant subspace associated with a small number of eigenvalues of a large sparse matrix. These results are illustrated by the use of block-GMRES as the iterative solver. The costs of the inexact solves are measured by the number of inner iterations needed by the iterative solver at ea...

Newton's Method constitutes a nested iteration scheme with the Newton step as the outer iteration and a linear solver of the Jacobian system as the inner iteration. We examine the interaction between these two schemes and derive solution techniques for the linear system from the properties of the outer Newton iteration. Contrary to inexact Newton methods, our techniques do not rely on relaxed t...

We analyze the Chebyshev iteration in which the linear system involving the splitting matrix is solved inexactly by an inner iteration. We assume that the tolerance for the inner iteration may change from one outer iteration to the other. When the tolerance converges to zero, the asymptotic convergence rate is unaaected. Motivated by this result, we seek the sequence of tolerance values that yi...

We propose an extension to the double iteration method, the Multipower Defect Correction method, which is used to refine approximate eigenelements of integral operators. It consists of an inner/outer iteration where, inside a defect correction iteration, p steps of a power iteration are used. The approximate inverse used in the defect correction is built with an approximation to the reduced res...

This paper presents an eecient compilation technique to generate the local memory access sequences for block-cyclically distributed array references with aane subscripts in data-parallel programs. For the memory accesses of an array reference with aane subscript within a two-nested loop, there exist repetitive patterns both at the outer and inner loops. We use tables to record the memory access...

Analytical target cascading is a method for design optimization of hierarchical, multilevel systems. A quadratic penalty relaxation of the system consistency constraints is used to ensure subproblem feasibility. A typical nested solution strategy consists of inner and outer loops. In the inner loop, the coupled subproblems are solved iteratively with fixed penalty weights. After convergence of ...

We study an inexact implicitly restarted Arnoldi (IRA) method for computing a few eigenpairs of generalized non-Hermitian eigenvalue problems with spectral transformation, where in each Arnoldi step (outer iteration) the matrix-vector product involving the transformed operator is performed by iterative solution (inner iteration) of the corresponding linear system of equations. We provide new pe...

The recently developed perturbation iteration method is applied to boundary layer type singular problems for the first time. As a preliminary work on the topic, the simplest algorithm of PIA(1,1) is employed in the calculations. Linear and nonlinear problems are solved to outline the basic ideas of the new solution technique. The inner and outer solutions are determined with the iteration algor...