Exercise 1: What Central Differences Fix, and What They Don't — Possible Solution ==================================================================== WHAT CENTRAL DIFFERENCING FIXES ------------------------------ Central differencing's advantage at moderate step sizes comes from its symmetric structure. The forward-difference formula's truncation error (the error from approximating a curve with a straight line, separate from any floating-point rounding) shrinks proportionally to h itself. The central-difference formula's truncation error shrinks proportionally to h^2, because the symmetric sampling on both sides of x causes the first-order error terms in the underlying Taylor expansion to cancel out algebraically, leaving only a smaller, higher-order term behind. This is a genuine mathematical improvement in how well the formula approximates the true derivative for a given step size - verified directly as nearly five orders of magnitude better accuracy at h=10^-4. WHAT CENTRAL DIFFERENCING DOES NOT FIX ------------------------------ Central differencing does nothing to change the fact that the formula still subtracts two function values, f(x+h) and f(x-h), that converge toward each other as h shrinks - exactly Chapter 4's cancellation pattern. Once h becomes small enough that f(x+h) and f(x-h) round to values close enough that their difference loses most of its significant digits, central differencing suffers the same kind of catastrophic collapse forward differencing does - just from a slightly better starting position, so it takes a smaller h to trigger the same failure. This chapter verified this directly: at h=10^-16, central differencing's error (0.0148) is much smaller than forward differencing's (0.54), but it is still a real, serious degradation compared to the ~10^-9 accuracy both methods achieved at their respective best step sizes. WHY BOTH FACTS ARE TRUE AT ONCE ------------------------------ Central differencing improves the TRUNCATION error (how good the approximation is in principle, for a given h, ignoring floating-point effects) without touching the CANCELLATION error (how much precision is lost to floating-point subtraction as h shrinks). These are two separate, independently-acting error sources, and central differencing only addresses one of them - which is exactly why it still has an optimal middle step size rather than getting arbitrarily accurate as h shrinks toward zero. WHY THIS WORKS AS AN ANSWER ------------------------------ The explanation separates the two distinct error sources (truncation error, which central differencing genuinely reduces, and cancellation error, which it does not) rather than treating the improvement as unconditional, and grounds both claims in this chapter's own verified numbers at both a moderate and an extreme step size.