Newton's Method Step

Newton's method finds a root of f(x) = 0 by repeatedly replacing the curve with its tangent line. From a guess x₀, the tangent there crosses zero at x₁ = x₀ − f(x₀)/f′(x₀); that crossing is usually much closer to the root, so it becomes the next guess. Worked example: find √2 as the root of f(x) = x² − 2, starting at x₀ = 1.5. There f = 0.25 and f′ = 2x = 3, so x₁ = 1.5 − 0.25/3 = 1.416667. One more step gives 1.4142157, and a third gives 1.4142135624, correct to every digit shown. Near a simple root the number of correct digits roughly doubles each step.

The classic mistake is starting where f′ is close to zero. A nearly flat tangent crosses the axis a long way off: the same function from x₀ = 0.01 sends the next guess to x₁ = 100.005. Some functions never recover at all: f(x) = x³ − 2x + 2 started at 0 steps to 1 and back to 0 forever. Check that |f| is shrinking from step to step, and when it is not, start somewhere else. The method is far older than calculus in one case: the Babylonian square-root rule, average x and 2/x, is exactly Newton's step on x² − 2. It is still how equations with no algebraic solution get solved, such as Kepler's equation for where a planet is on its orbit, or the interest rate that makes a loan's payments add up.