A uniform approach to gradient methods for linear operator equations

Let T be a bounded linear operator from one Hilbert space to another. A class of gradient methods for minimizing ∥Tx ? f ∥² is analyzed and characterized by the step-size used in the iteration $\text{x}{}_{\mathrm{n+1}}\text{= x}{}_{\mathrm{n}}\text{? s(x}{}_{\mathrm{n}}\text{) T}{}^1\text{(Tx}{}_{\mathrm{n}}\text{? f)}$. A general convergence theorem is proved under the simple assumption that the least-squares problem exhibits a solution. Specific convergence rates are established for operators with closed and nonclosed ranges.