Preconditioned conjugate gradient method for rank deficient least-squares problems

On generalized symmetric SOR method for augmented systems

“…There have been several studies for solving augmented system (1.1). Santos et al [8], and Santos and Yuan [9] studied preconditioned iterative method for solving system (1.1) with A = I . Yuan [12] presented preconditioned conjugate gradient methods for solving general augmented systems like (1.1) where A can be symmetric and positive semidefinite and B can be rank deficient.…”

On the generalized MSSOR method for augmented systems

“…where m m A R × ∈ is a symmetric positive definite matrix, and m n B R × ∈ is a matrix of full column rank. The augmented system like (1) appears in many different applications of scientific computing, such as the finite element approximation to solve the Navier-Stokes equation [1,2], the constrained least squares problems and generalized least squares problems [3,4,5,6,7,8] and constrained optimization [9].…”

“…There are several works for solving the augmented system (1). Santos et al [5], Santos and Yuan [6] studied preconditioned iterative method for solving the augmented system (1) with A I = . Yuan [6,8] presented preconditioned conjugate gradient methods for solving general systems like (1), where A is symmetric and positive semidefinite and B is rank deficient.…”

A modified symmetric successive overrelaxation method for augmented systems

“…System (1) can be found in many different applications of scientific computing, e.g., finite element discretization to solve partial differential equations including Stokes equations and Navier-Stokes equations, constrained least squares problems, and generalized least squares problems (see [1][2][3][4][5][6]). Such systems typically result from mixed or hybrid finite element approximations of second-order elliptic problems, elasticity problems or the Stokes equations [7], and from Lagrange multiplier methods [8].…”