We propose a generalization of the conjugate gradient method that uses multiple preconditioners, combining them automatically in an optimal way. The derivation is described in detail, and analytical observations are made. A short recurrence relation does not hold in general for this new method, but in at least one case such a relation is satisfied: for two symmetric positive definite preconditi...

Sturmian eigenstates specified by stationary scattering boundary conditions are particularly useful in contexts such as forming simple separable two nu-cleon t matrices, and are determined via solution of generalised eigenvalue equation using real and symmetric matrices. In general, the spectrum of such an equation may contain complex eigenvalues. But to each complex eigenvalue there is a corre...

We study the behavior of the conjugate-gradient method for solving a set of linear equations, where the matrix is symmetric and positive definite with one set of eigenvalues that are large and the remaining are small. We characterize the behavior of the residuals associated with the large eigenvalues throughout the iterations, and also characterize the behavior of the residuals associated with ...

PREQN is a package of Fortran subroutines for automatically generating pre conditioners for the conjugate gradient method It is designed for solving a sequence of linear systems Aix bi i t where the coe cient matrices Ai are symmetric and positive de nite and vary slowly Problems of this type arise for example in non linear optimization The preconditioners are based on limited memory quasi Newt...

The Conjugate Gradient Squared (CGS) is a well-known and widely used iterative method for solving non-symmetric linear systems of equations. In practice the method converges fast, often twice as fast as the Bi-Conjugate Gradient (Bi-CG) method. However, during the iteration large residual norms may appear, which may lead to inaccurate approximate solutions or may even deteriorate the convergenc...

An important variation of preconditioned conjugate gradient algorithms is inexact precon-ditioner implemented with inner-outer iterations 5], where the preconditioner is solved by an inner iteration to a prescribed precision. In this paper, we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive deenite system and analyze its convergence property. We establi...

We simulate elastic light scattering with the coupled dipole method. The kernel of this method is a large set of linear equations. The n×n system matrix is complex, symmetric, full, and diagonally dominant. This application is a typical example of problems arising in computational electromagnetics. The matrix equations are usually solved with (preconditioned) conjugate gradient methods. For rea...

Beamspace Root-MUSIC (BM-RTMUSIC) algorithm cannot resolve coherent sources, and the use of spatial smoothing pre-process leads to array aperture loss and reduction in detectable source numbers, a new beamspace conjugate Root-MUSIC (BMC-RTMUSIC) algorithm is proposed to estimate the Direction-Of-Arrival (DOA) for non-circular signals. The proposed method employs the properties of noncircular si...

In this paper we discuss current activities at Cray Research to develop general-purpose, production-quality software for the eecient solution of sparse linear systems. In particular, we discuss our development of a package of iterative methods that includes Conjugate Gradient and related methods (GMRES, ORTHOMIN and others) along with several preconditioners (incomplete Cholesky and LU factoriz...

The standard formulation of the conjugate gradient algorithm involves two inner product computations. The results of these two inner products are needed to update the search direction and the computed solution. Since these inner products are mutually interdependent, in a distributed memory parallel environment their computation and subsequent distribution requires two separate communication and...