Existence test for asynchronous interval iterations

In the search for regions that contain fixed points ofa real function of several variables, tests based on interval calculationscan be used to establish existence ornon-existence of fixed points in regions that are examined in the course ofthe search. The search can e.g. be performed as a synchronous (sequential) interval iteration:In each iteration step all components of the iterate are calculatedbased on the previous iterate. In this case it is straight forward to base simple interval existence and non-existencetests on the calculations done in each step of the iteration. The search can also be performed as an asynchronous (parallel) iteration: Only a few components are changed in each stepand this calculation is in general based on components from differentprevious iterates. For the asynchronous iteration it turns out thatsimple tests of existence and non-existence can be based on thecomponentwise calculations done in the course of the iteration. These componentwisetests are useful for parallel implementation of the search, sincethe tests can then be performed local to each processor and only when a test issuccessful do a processor communicate this result to other processors.