Exercise 1: Why "Always Use Newton's Method" Is Bad General Advice — Possible Solution ==================================================================== WHAT MAKES NEWTON'S METHOD ATTRACTIVE ------------------------------ This chapter verified a dramatic speed advantage: 4 iterations for Newton's method versus 39 for bisection, on the identical equation and tolerance. If speed were the only consideration, Newton's method would win every time, and "always use Newton's method" would be reasonable advice. WHY SPEED ISN'T THE ONLY CONSIDERATION ------------------------------ This chapter verified two genuinely different ways Newton's method can fail outright, from starting points that look entirely reasonable: 1. The permanent cycle: for f(x) = x^3 - 2x + 2 starting at x0=0, Newton's method doesn't converge slowly or converge to the wrong answer - it produces an exact, endless 0 -> 1 -> 0 -> 1 cycle and never approaches the real root at all. A caller waiting for this to converge would wait forever. 2. Degraded convergence at a double root: for f(x)=(x-1)^2, Newton's method still converges, but only linearly (error ratio exactly 0.5 every step) - the same speed as bisection, with none of Newton's usual advantage, because f'(x) approaches zero right along with f(x). WHY BISECTION DOESN'T HAVE THESE FAILURE MODES ------------------------------ Bisection's guarantee comes directly from the Intermediate Value Theorem: as long as the two starting points bracket a genuine sign change, the root is mathematically guaranteed to exist inside that bracket, and each step is guaranteed to halve the bracket size no matter what the function looks like in between. Bisection never "jumps" based on the function's slope the way Newton's method does, so it has nothing analogous to a bad slope, a zero slope, or a starting point that happens to land in a cycle. THE ACTUAL TRADEOFF ------------------------------ Newton's method is the better choice when it's known (or can be verified) that the function is well-behaved near the root and the starting point is reasonably close to it - situations where its speed advantage is real and its failure modes are unlikely. Bisection is the safer default when reliability matters more than speed, or when nothing is known in advance about the function's behavior, because its guarantee doesn't depend on any of the conditions that can break Newton's method. A well-designed root finder often combines both: try Newton's method for speed, but fall back to bisection (or detect non-convergence and bracket a sign change instead) if Newton's method fails to make progress. WHY THIS WORKS AS AN ANSWER ------------------------------ The explanation acknowledges the genuine, verified speed advantage rather than dismissing it, cites both specific verified failure modes by name rather than vaguely gesturing at "Newton's method can fail," explains why bisection's guarantee is structurally immune to those same failure modes, and proposes the real-world combined approach rather than declaring one method universally superior.