Fixed-point polynomial of the average recurrence

Read the equation

At a nonzero fixed point, multiplying z=F(z) by 7z yields 2z²−z−1=0. Factoring gives (2z+1)(z−1)=0. This algebra identifies all finite real fixed points of the displayed recurrence.

Terms & definitions

How it works

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

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Original context

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.

Hence

zfixed,1=1,  zfixed,2=−12.

The derivative is

7z=5z+1+1/z⇔2{z}^{2}-z-1=0⇔\left(2z+1\right)\left(z-1\right)=0.