On the Convergence of the Conjugate Gradient Method for Singular Linear Operator Equations

Let T:X → Y be a bounded linear operator between two Hilbert spaces X and Y, and let T^ denote the generalized inverse of T, and (T) the range of T. If (T) is closed, then starting with x₀ ∈ X, the conjugate gradient method which minimizes the functional f(x) = ||Tx - y||² ,\, y ∈ Y at each step, converges to a least squares solution of $Tx = y$; namely T^ y + (I - p)x₀. Here $I - P$ denotes the orthogonal projection of X onto the null space of X. If (T) is not closed with x₀ ∈ (T^ * T) and Qy ∈ (TT^ * T), where Q is the orthogonal projection of Y onto the null space of the adjoint operator T^ *, then the conjugate gradient method converges to the least squares solution T^ y of minimal norm. Bounds on the rate of convergence are given in both cases.