Algebraic multigrid for general inconsistent linear systems: The correction step

AbstractThis paper is a continuation of the previous Technical Report [2]. We are now focusingon the properties of the correction step of our AMG approach. We included also numericalexperiments for the 2D case to support our theoretical results.2000 MS Classification: 65F10, 65F20, 65N55Key words and phrases: algebraic multigrid, inconsistent least squares problems, cor-rection step 1 Problem formulation Let A be an m × n matrix and b ∈ IR m . We shall denote by (A) i ,(A) j ,(A) ij ,A T ,N(A),R(A),A + the i-th row, j-th column, (i,j)-th element, transpose, null space, range and Moore-Penrose pseu-doinverse of A, respectively. For a given vector subspace E ⊂ IR q , P E (x) will be the orthogonalprojection onto E of an element x ∈ IR q and E ⊥ will denote its orthogonal complement, with re-spect to the euclidean scalar product and norm, denoted by h·,·i and k · k. The following propertiesare known (see e.g. [1])AA + A = A, A + AA + = A + , (AA + ) T = AA + , (A + A) T = A + A, (A T ) + = (A