  In this paper we examine iterative methods for solving the forward ($A{\bf
x}={\bf b}$) and adjoint ($A^{T}{\bf y}={\bf g}$) systems of linear equations
used to approximate the scattering amplitude, defined by ${\bf g}^{T}{\bf
x}={\bf y}^{T}{\bf b}$. Based on an idea first proposed by Gene Golub, we use a
conjugate gradient-like iteration for a nonsymmetric saddle point matrix that
is constructed so as to have a real positive spectrum. Numerical experiments
show that this method is more consistent than known methods for computing the
scattering amplitude such as GLSQR or QMR. We then demonstrate that when
combined with known preconditioning techniques, the proposed method exhibits
more rapid convergence than state-of-the-art iterative methods for nonsymmetric
systems.
