Newton's Method

To find where a curve hits zero, slide down its tangent line to the axis and repeat. Each step roughly doubles the number of correct digits. Drag the starting guess.

How it works

Tangent lines and linear approximation

In one sentence

Newton's method improves a root guess by following the tangent line to the axis, $x_{n+1} = x_n - f(x_n)/f'(x_n)$ , and converges quadratically near a simple root.

General formula

{\htmlData{sym=2}{x_{n+1}}} = {\htmlData{sym=3}{x_n}} - \frac{{\htmlData{sym=1}{f(x_n)}}}{{\htmlData{sym=0}{f'(x_n)}}}

The tangent line is the best straight-line stand-in for $f$ near $x_n$ ; where it hits zero is the next guess.

What the symbols mean

$f'(x_n)$
Slope at the guess
How steep the tangent line is at the current guess.
$f(x_n)$
Miss distance
How far the function is from zero at the current guess.
$x_{n+1}$
Next guess
Where the tangent line crosses zero: usually much closer to the root.
$x_n$
Current guess
The estimate of the root after n steps.

The tangent line at $x_n$ is the best straight-line stand-in for $f$ near that point.
Where that line crosses zero is the next guess.
A starting point near a flat spot can fling the next guess far away.

Learn the math

Finding Zeroes of PolynomialsAlgebra · Lesson 37, Polynomial Functions Linear ApproximationsCalc I · Lesson 36, Applications of Derivatives Newton's MethodCalc I · Lesson 38, Applications of Derivatives SequencesCalc II · Lesson 24, Sequences and Series Monotonic and Bounded SequencesCalc II · Lesson 25, Sequences and Series

Learn the math

Finding Zeroes of PolynomialsAlgebra · Lesson 37, Polynomial Functions

Linear ApproximationsCalc I · Lesson 36, Applications of Derivatives

Newton's MethodCalc I · Lesson 38, Applications of Derivatives

SequencesCalc II · Lesson 24, Sequences and Series

Monotonic and Bounded SequencesCalc II · Lesson 25, Sequences and Series