Iteration Depth Error Cross-Scan - Nonlinear Dynamics and Stability Step by Step

Finite iteration depth is scanned against the remaining fixed-point gap. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Iteration depth is a finite ledger

The source ledger starts at zero and stops at the displayed depth.

0\to\frac{1}{4}\to\frac{3}{8}\to\frac{7}{16}

Each extra step halves the remaining gap here

The fixed point is one half. For this affine contraction, the displayed gap shrinks from one quarter to one sixteenth.

\begin{array}{c|c|c}n&x_n&x^*-x_n\\1&\frac{1}{4}&\frac{1}{4}\\2&\frac{3}{8}&\frac{1}{8}\\3&\frac{7}{16}&\frac{1}{16}\\\end{array}