Definition:Stencil
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Definition
A quintuple $\tuple {X, S_X, \map S 0, \Delta, \delta}$ defines a stencil if and only if:
\((1)\) | $:$ | Index Domain: | $X$ is discrete: $X \subseteq \Z^{\size X}$ and of finite dimension: $\size X \in \N$ | ||||||
\((2)\) | $:$ | State Range: | $S_X$ is a well-defined set | ||||||
\((3)\) | $:$ | Initial State: | $\map S 0$ maps from index space to state space $\map S 0: X \to S_X$ | ||||||
\((4)\) | $:$ | Neighbourhood Delta: | $\Delta$ is a vector of index offsets: $\Delta \in \paren {\Z^{\size X} }^{\size \Delta}$ | ||||||
\((5)\) | $:$ | Transition Combinator: | $\delta$ is a mapping $\delta: S_X^{\size \Delta} \to S_X$ from neighbourhood states to states |
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By means of the update sweep $\forall \vec x \in X: \map {\map S {n + 1} } {\vec x} = \ds \map \delta {\prod_{i \mathop = 1}^{\size \Delta} \map {\map S n} {\vec x + \Delta_i} }$ this induces a stencil evolution $\N \ni n \mapsto \map S n \in \paren {X \to S_X}$ from the initial state $\map S 0$.