This paper discusses algorithms for attitude determination using GPS differential phase measurements, assuming that the cycle integer ambiguities are known. The problem of attitude determination is posed as a parameter optimization problem where a new quaternion-based cost function is used. Since the new cost function is not a simple quadratic form and therefore Davenport's q-Method is not applicable in this case. Three algorithms for finding the optimal quaternion are derived, two of which are discrete. The third one is a continuous version of the Newton-Raphson algorithm. This continuous version is new and has a guaranteed exponential convergence to the closest local minimum located on the gradient direction in regions where the associated Hessian matrix is positive definite. The algorithms presented in this paper can handle cases of planar antenna arrays and thus cover a deficiency in earlier algorithms. The efficiency of the new algorithms is demonstrated through numerical examples.