Preconditioned conjugate gradients for solving singular systems

AbstractIn this paper the preconditioned conjugate gradient method is used to solve the system of linear equations Ax = b, where A is a singular symmetric positive semi-definite matrix. The method diverges if b is not exactly in the range R(A) of A. If the null space N(A) of A is explicitly known, then this divergence can be avoided by subtracting from b its orthogonal projection onto N(A).As well as analysing this subtraction, conditions necessary for the existence of a nonsingular incomplete Cholesky decomposition are given. Finally, the theory is applied to the discretized semi-definite Neumann problem