Consider a database containing not only base relations but also stored derived relations (also called materialized or concrete views). When a base relation is updated, it may also be necessary to update some of the derived relations. This paper gives sufficient and necessary conditions for detecting when an update of a base relation cannot affect a derived relation (an irrelevant update), and for detecting when a derived relation can be correctly updated using no data other than the derived relation itself and the given update operation (an autonomously computable update). The class of derived relations considered is restricted to those defined by PSJ-expressions, that is, any relational algebra expressions constructed from an arbitrary number of project, select and join operations (but containing no self-joins). The class of update operations consists of insertions, deletions, and modifications, where the set of tuples to be deleted or modified is specified by a selection condition on attributes of the relation being updated.