"These simulations used 36-bit floating point arithmetic, which was adequate for trajectory simulations but marginal for implementing the Riccati equation solution for the measurement updates in the Kalman filter. Performance was not reliable in 36-bit floating point arithmetic, and the Apollo flight computer would have to implement the Kalman filter in 15-bit fixed-point arithmetic. Microprocessors were still a long way off. J.E. Potter was at that time a graduate student in mathematics at MIT, working part time at the Instrumentation Laboratory on the Apollo Project. He took the problem home with him one Friday afternoon and arrived back on Monday with a solution. The trick was to use a Cholesky factor of the covariance matrix as the dependent variable in an equivalent Riccati equation. The solution was brilliant, and the improvement was profound. The approach came to be called square-root filtering, and alternative implementations of square-root filters with better numerical stability were soon discovered."
excerpted from: https://www.researchgate.net/profile/Mohinder-Grewal/publica...