Average of the seven operators as a recurrence

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Averaging all seven operators at the same input gives (5z+1+1/z)/7. This parallel average differs from applying the operators sequentially. The recurrence is undefined at zero and has two real fixed points; its behavior depends on the starting domain.

Terms & definitions

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Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

• Do not substitute the constant-one result of sequential composition for this averaged recurrence.

Original context

6.2 Proposition 2: an averaged recurrence has different dynamics

The equal-average recurrence introduced in the discussion was

zn+1=Fzn=5zn+1+1/zn7,  zn≠0.

Averaging is an additional coupling rule. It is not implied by drawing seven operators around a center. It should not be called the unique consequence of the proposed geometry.

The fixed-point equation is

7z=5z+1+1/z⇔2z2−z−1=0⇔2z+1z−1=0.

{z}_{n+1}=F\left({z}_{n}\right)=\frac{5{z}_{n}+1+1/{z}_{n}}{7},  {z}_{n}\ne 0.