Some new square-root algorithms for Bryson-Frazier smoothing formulas are suggested : square-root algorithms and a fast square-root (or so-called Chandrasekhar type) algorithm. The new square-root algorithms use square-root arrays composed of smoothed estimates and their error covariances. These algorithms provide many advantages over the conventional algorithms with respect to systolic array and parallel implementations as well as numerical stability and conditioning. For the case of constant-parameter systems, a fast square-root algorithm is suggested, which requires less computation than others.