Newton Stepper - Root Finding Practice

Predict how many Newton iterations it takes to drive x² - a = 0 from x₀ = 1 to within 1e-4 of √a - then watch the step table and see how quadratic convergence behaves.

Each round draws a random integer a from 3 to 20 and asks you to solve x² - a = 0 - that is, find √a - by Newton's method starting from the fixed point x₀ = 1. Before seeing any step, you type your estimate of how many iterations it takes for the iterate to land within 1e-4 of the true root.

The update rule is Newton's method on f(x) = x² - a with f'(x) = 2x: xₙ₊₁ = xₙ - (xₙ² - a)/(2xₙ). The engine runs up to 20 steps, stopping at the first step whose absolute error |xₙ - √a| is at or below 1e-4, and counts how many steps that took.

After you lock in a guess, the reveal shows the true root, the actual iteration count, and the full step table - each iterate with its error - so you can watch the error collapse. Then a fresh a arrives.

Why quant interviews test this

Newton's method is a standard numerical-methods screen for quant-research and quant-dev roles: derive the update from a Taylor expansion, state the convergence order, and explain when it fails (bad starting points, vanishing derivative, non-smooth f). The digit-doubling intuition - and being able to estimate an iteration count without running anything - is what distinguishes fluency from recitation.

It also anchors applied follow-ups: solving for implied volatility or yield-to-maturity is Newton in one dimension, and Newton with a Hessian is the reference point that gradient descent and quasi-Newton methods (BFGS) are compared against. Expect the compare-and-contrast question: quadratic convergence per step versus the cost of derivatives per step.

The Newton Stepper guide covers how scoring works, the strategy that wins, a worked example and the mistakes most players make.

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• Every game guide