Bisection Search and the Perils of Floats turned a linear-cost search into a logarithmic one by using one bit of information per guess (is the midpoint too high or too low?). Newton–Raphson goes further: it uses the function's slope at the current guess to jump directly toward the root, rather than merely halving an interval. When it applies, it converges dramatically faster than bisection — but it needs a derivative, and it can fail outright where bisection cannot. This note derives the update rule for polynomial roots, implements it for square roots, and states precisely when to prefer it over bisection and when its extra assumptions make it the wrong tool.
Learning objectives. By the end of this note you should be able to:
- derive the Newton–Raphson update rule from a first-order Taylor approximation of a function around the current guess;
- implement Newton–Raphson to find an approximate square root and count its iterations against bisection's;
- explain why Newton–Raphson needs a derivative and an initial guess, and why bisection needs neither;
- identify at least one failure mode of Newton–Raphson (non-convergence or divergence) that bisection's halving guarantee rules out.
In this lesson
- Deriving the update rule
- When Newton–Raphson is the wrong tool
Before you start
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