Gram-Schmidt algorithms for covariance propagation†

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Abstract
This paper addresses the time propagation of triangular covariance factors. Attention is focused on the square-root free factorization, P =:UDUT , where U is unit upper triangular and D is diagonal. An efficient and reliable algorithm for U-D propagation is derived which employs Gram-Schmidt orthogonalization. Partitioning the state vector to distinguish bias and coloured process noise parameters increases mapping efficiency. Cost comparisons of the U-D, Schmidt square-root covariance and conventional covariance propagation methods are made using weighted arithmetic operation counts. The U-D time update is shown to be less costly than the Schmidt method ; and, except in unusual circumstances, it is within 20% of the cost of conventional propagation.

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