Exercise 3: What |g'(x)| Very Close to 1 Actually Predicts — Possible Solution ==================================================================== WHAT THE VERIFIED g(x)=2/x EXAMPLE SHOWS ------------------------------ This chapter verified that g(x)=2/x, whose derivative at the true root has magnitude |g'(sqrt(2))| approximately equal to 1 (measured at -0.9999999999999998, essentially exactly -1), does not converge to sqrt(2) and does not diverge to infinity either. Instead, it locks into a permanent, exact 2-cycle: 1.4 -> 1.42857142857... -> 1.4 -> 1.42857142857... forever, neither getting closer to the root nor moving further away from it over time. WHY THIS IS A GENUINELY THIRD OUTCOME ------------------------------ The convergence rule |g'(x)|<1 predicts shrinking error (the distance from the current iterate to the root gets smaller each step) and |g'(x)|>1 predicts growing error (the distance gets larger each step, diverging). At exactly |g'(x)|=1, neither description is accurate: the error near the fixed point is (to a first-order approximation) neither shrinking nor growing per step - it's being roughly preserved in magnitude, which allows the iterate to bounce back and forth around the fixed point without ever settling into it or escaping it entirely. This is a genuinely different qualitative behavior from both convergence and divergence, not just "very slow convergence" or "very slow divergence." WHY THIS MATTERS PRACTICALLY ------------------------------ A condition value very close to 1 is a real warning sign distinct from the clearly-safe (well below 1) and clearly-dangerous (well above 1) cases: a fixed-point iteration in this regime may appear to be "doing something" - the iterate keeps changing, it doesn't immediately blow up - while never actually making real progress toward a usable answer. This is arguably more dangerous in practice than outright divergence, because divergence is usually obvious quickly (values grow rapidly, as seen in this chapter's own g3 example), while a marginal cycle can look deceptively like slow but genuine progress if only the first couple of iterations are examined without checking whether the values are actually converging or just oscillating. WHY THIS WORKS AS AN ANSWER ------------------------------ The explanation identifies the marginal |g'(x)|=1 case as a distinct third outcome (oscillation/cycling) rather than folding it into "very slow convergence" or "very slow divergence," grounds the claim in this chapter's own verified g(x)=2/x behavior, and explains why this specific case is a practically important warning sign in its own right.