We describe a quantum algorithm that generalizes the quantum linear system algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] to arbitrary problem specifications. We develop a new state preparation routine and show how simple ancilla measurements can efficiently estimate the overlap between the linear system solution and an arbitrary state vector. To demonstrate the algorithm's applicability, we show how it can be used to compute the electromagnetic scattering cross section of an arbitrary target, using the finite element method. This quantum algorithm can provide exponential speedup over the best classical algorithm, greatly improving the runtime and allowing for the modeling of far more complex objects than possible on a classical computer.