The computational method of conjugate gradients for linear operator equations is given. This technique is applied to the solution of the integral equation that relates the aperture distribution and the far-field pattern. This problem is treated from the variational setting by minimizing an error functional. In searching for the solution, a sequence of expanding subspaces in a function space is generated via the Frechet differential of the error functional. The error functional is minimized in each of the subspaces and thus forms a sequence of approximating solutions. This iterative procedure may be conveniently implemented on a computer. Two synthesis problems are treated using this technique, and the results are discussed.