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...

We study the problem of complex-valued independent subspace analysis (ISA). We introduce complex flag manifolds to tackle this problem, and, based on Riemannian geometry, propose the natural conjugate gradient method on this class of manifolds. Numerical experiments demonstrate that the natural conjugate gradient method yields better convergence compared to the natural gradient geodesic search ...

We evaluate the High-Performance Fortran (HPF) language for the compact expression and eecient implementation of conjugate gradient iterative matrix-solvers on High Performance Computing and Communi-cations(HPCC) platforms. We discuss the use of intrinsic functions, data distribution directives and explicitly parallel constructs to optimize performance by minimizing communications requirements ...

We consider symmetric positive definite systems of linear equations with multiple right-hand sides. The seed conjugate gradient method solves one right-hand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other right-hand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the conjugate ...

Conjugate gradient methods are important for large-scale unconstrained optimization. This paper proposes an acceleration of these methods using a modification of steplength. The idea is to modify in a multiplicative manner the steplength k α , computed by Wolfe line search conditions, by means of a positive parameter k η , in such a way to improve the behavior of the classical conjugate gradien...

The method of conjugate gradients provides a very effective way to optimize large, deterministic systems by gradient descent. In its standard form, however, it is not amenable to stochastic approximation of the gradient. Here we explore a number of ways to adopt ideas from conjugate gradient in the stochastic setting, using fast Hessian-vector products to obtain curvature information cheaply. I...

The discretization of the semiconductor equations in three-dimensional device simulators leads to very large sparse linear systems of equations. While the solution of the Poisson equation e.g. by the conjugate gradient methodis straightforward, the iterative solution of the carrier continuity equations is nontrivial due to both the non-symmetry and the poor conditioning of the coefficient matri...