A Closed-Boundary Convergence Theorem for the Golden-Ratio Contraction Map(LLM)

Abstract

Abstract We examine the affine contraction[ T(x)=\varphi+\varphi^{-7}(x-\varphi) ]on the real line, where (\varphi=(1+\sqrt{5})/2) is the golden ratio. The map is a linear contraction with unique fixed point (\varphi) and ratio (r=\varphi^{-7}\approx0.03444). The orbit satisfies the exact geometric relation[ |x_n-\varphi|=r^n|x_0-\varphi| ]and therefore converges to (\varphi) at a uniform linear rate. We derive closed-form expressions for the minimal iteration count (N_{\min}(x_0,\varepsilon)) needed to drive the error below a prescribed tolerance (\varepsilon>0). Replacing the strict residual test by the non-strict test[ r^n|x_0-\varphi|\le\varepsilon ]renders the corresponding sublevel sets closed and removes an off-by-one ambiguity that occurs precisely when (\varepsilon/|x_0-\varphi|) is an integer power of (r). The corrected count admits the simple formula[ N_{\min}^{(\mathrm{corr})}(x_0,\varepsilon)=\Bigl\lceil\frac{\ln(\varepsilon/|x_0-\varphi|)}{\ln r}\Bigr\rceil ](with the convention that the value is zero when the initial error already satisfies the tolerance). Because (r) is small, double-precision machine epsilon ((\approx2.22\times10^{-16})) is reached from typical starting values in at most eleven iterations. Numerical experiments confirm the predicted constant slope on a semi-log scale, and a short complexity discussion shows that direct iteration is competitive with evaluation of the closed-form expression at ordinary working precisions. All statements are elementary consequences of the classical theory of one-dimensional affine contractions; the concrete exponent (-7) is chosen only for numerical convenience. The note is presented as a self-contained, fully rigorous worked example.