Recently, Lainiotia (1971 c), using the so-called ' partition theorem ', has obtained an optimal linear smoothing algorithm in explicit, closed-form expressions that are attractive, both from a computational and an analysis point of view. Lainiotis (1971c( 'partition smoothing' algorithm is re-examined herein, and its computational and analytical advantages studied. It is compared to the previously established two-filter smoothing algorithm of Mayne (1966), Fraser (1967), and Mehra (1968), as well as to the ' innovation smoothing ' algorithm of Kailath and Frost (1968) Subsequently, the so-called ' iterative ' or ' reprocessed ' smoothing scheme, used extensively as a data reduction process in the Apollo Space Programme, is studied using the ' partition smoothing ' algorithm. The resulting explicit and closed-form expressions are readily amenable to interpretation and optimization, and are, moreover, both theoretically interesting as well as practically useful. The statistical and limiting properties of the ' partition reprocessed smoothing ' algorithm are obtained and are thoroughly examined.
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