Read the equation
The average recurrence has fixed points at one and minus one half. Substituting either value into F(z) returns that value. The presence of two fixed points already rules out a single global destination for every allowed real starting state.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| z_fixed_1 | Positive fixed point 1 |
| z_fixed_2 | Negative fixed point −1/2 |
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
Read the surrounding source passage
7z=5z+1+1/z⇔2z2−z−1=0⇔2z+1z−1=0. Hence zfixed,1=1, zfixed,2=−12. The derivative is F′z=5−z−27, F′1=47, F′−1/2=17. Both fixed points are locally attracting. This statement is local for the negative fixed point; the proof below establishes global convergence only for positive starts.
Exact original notation
{z}_{fixed,1}=1, {z}_{fixed,2}=-\frac{1}{2}.Sources & provenance
{z}_{fixed,1}=1, {z}_{fixed,2}=-\frac{1}{2}.
{z}_{fixed,1}=1, {z}_{fixed,2}=-\frac{1}{2}.