Methods of conjugate gradients for solving linear systems

Abstract

An iterative algorithm is given for solving a system Ax = k of n linear equations in n unknowns. The solution is given in n steps . It is shown that this m ethod is a special case of a very general m et hod which also includes Gaussian elimination . Th ese general algorithms are essentially algorithms for findin g an n dimensional ellipsoid . Connections a re m a de wit h the theo ry of orthogonal p olynomials and continued fractions.

Structural Transition in Layered As1–xPx Compounds: A Computational Study

“…We used a mesh cutoff energy of 180 Ry to determine the self-consistent charge density, which provided us with a precision in total energy of ≤ 2 meV/atom. All geometries have been optimized by SIESTA using the conjugate gradient method [27], until none of the residual Hellmann-Feynman forces exceeded 10 −2 eV/Å.…”

LSRN: A Parallel Iterative Solver for Strongly Over- or Underdetermined Systems

“…For example, the conjugate gradient (CG) method on the normal equation leads to the min-length solution (see Paige and Saunders [20]). In practice, CGLS [16] or LSQR [21] are preferable because they are equivalent to applying CG to the normal equation in exact arithmetic but they are numerically more stable. Other Krylov subspace methods such as the CS method [12] and LSMR [10] can solve (1.1) as well.…”

An Investigation on Numerical Characterization of Scattering From Target in a Dielectric Rough Soil Surface

“…Usually, when the number of sampling for the target, for the rough soil surface increases, the computational cost of solving the matrix equation using the conjugate gradient method (CGM) [33] or biconjugate gradient method (BCGM) or the direct LU inversion becomes prohibitive, in the following section, the efficient numerical method, i.e., the EPILE + FBM is adopted to speed up the scattering calculation.…”