A priori state vector and error covariance computation for the Unscented Kalman Filter (UKF) is described. The original UKF propagates multiple sigma points to compute the a priori mean state vector and the error covariance, resulting in a higher computational time compared to the Extended Kalman Filter (EKF). In the proposed method, the posterior mean state vector is propagated and then the sigma points at the current time step are calculated using the first-order Taylor Series approximation. This reduces the computation time significantly, as demonstrated using two example applications which show improvements of 90.5% and 92.6%. This method shows the estimated state vector and the error covariance are accurate to the first-order Taylor series terms. A second method using Richardson Extrapolation improves prediction accuracy to the second-order Taylor series terms. This is implemented on the two examples, improving efficiency by 85.5% and 86.8%.