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

An iteration-depth cross-scan runs the map to several depths and reads off the error at each, showing how the residual shrinks, stalls, or grows with more iterations. The arithmetic is exact only for the stated depths, map rule, and rows in the ideal discrete-map model. Real systems add continuous dynamics, noise, unmodeled feedback, finite sampling, parameter uncertainty, and model/source limits.

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}