An algorithm for characterizing attosecond extreme ultraviolet pulses that is not bandwidth-limited, requires no interpolation of the experimental data, and makes no approximations beyond the strong-field approximation is introduced. This approach fully incorporates the dipole transition matrix element into the retrieval process. Unlike attosecond retrieval methods such as phase retrieval by omega oscillation filtering (PROOF), or improved PROOF, it simultaneously retrieves both the attosecond and infrared (IR) pulses, without placing fundamental restrictions on the IR pulse duration, intensity or bandwidth. The new algorithm is validated both numerically and experimentally, and is also found to have practical advantages. These include an increased robustness to noise, and relaxed requirements for the size of the experimental dataset and the intensity of the streaking pulse.