What is Newton's method?

Newton's method (also called Newton-Raphson) is an iterative algorithm for finding roots of a function, that is, values of x where f(x) = 0. Starting from an initial guess x0, each step computes x1 = x0 minus f(x0) divided by f'(x0), where f'(x0) is the derivative. This is geometrically equivalent to following the tangent line at the current point to where it crosses the x-axis.

How fast does Newton's method converge?

Newton's method converges quadratically when it converges, meaning the number of correct decimal places roughly doubles with each iteration. Starting near a simple root, you might get 1 correct digit after one iteration, 2 after two, 4 after three, and 8 after four. This very rapid convergence makes it one of the most efficient root-finding methods.

When does Newton's method fail?

Newton's method can fail or diverge if the derivative f'(x) is zero or near zero at an iteration point, if the starting guess is too far from the root, or if the function has a repeated root (which slows convergence). It also converges to different roots depending on the starting point for functions with multiple roots.

Newton's Method Calculator

Newton's method, also known as the Newton-Raphson method, is one of the most powerful and widely used algorithms for finding where a function equals zero. Starting from an initial guess x0, it repeatedly improves the estimate by following the tangent line at the current point to where it intersects the x-axis. The update formula is x1 = x0 minus f(x0) divided by f prime(x0). Because the method is based on a linear approximation of the function at each step, it converges quadratically near a simple root: the number of correct decimal places approximately doubles with each iteration. This means a few iterations from a reasonable starting point are usually enough to get many decimal places of accuracy. The classic example is finding the square root of 2 by solving x squared minus 2 = 0. Starting at x0 = 1.5, Newton's method converges to sqrt(2) = 1.4142135... in just a few steps. This calculator lets you enter any function f(x) using JavaScript math notation, an initial guess x0, and a maximum number of iterations up to 20. It computes the derivative numerically using the central difference formula and shows a table of each iteration with the x value, f(x) value, derivative value and the change in x. The iteration stops early if the change is smaller than 1e-12, indicating convergence.

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f(x) expression

Use x for the variable. Examples: x*x-2, Math.sin(x)-0.5, x**3-x-1. Use Math.pow(x,n), Math.sqrt(x), Math.log(x) etc.

Initial guess (x0)

Max iterations

1.4142

root (where f(x) = 0)

f(root)0.0000

Iterations4

ConvergedYes

Convergence is declared when |x1 - x0| < 1e-12. The derivative is computed numerically by central difference.

n	x	f(x)	f'(x)	step

How it works

Newton's method iterates the formula x_n+1 = x_n - f(x_n) / f'(x_n). The derivative f'(x) is approximated numerically using the central difference formula: f'(x) ≈ (f(x+h) - f(x-h)) / (2h) with h = 1×10^-7. Iteration stops when |step| < 1×10^-12 or the maximum number of iterations is reached. If the derivative is zero or near zero at any step, the method cannot continue and a warning is shown.

Worked example

f(x) = x² - 2, x0 = 1.5, 10 iterations. Step 1: x1 = 1.5 - (1.5² - 2)/(2 × 1.5) = 1.5 - 0.25/3 = 1.4167. Step 2: x2 = 1.4167 - (1.4167² - 2)/(2 × 1.4167) = 1.4142. Step 3: the change is already less than 1e-12, so the method converges in about 3 to 4 steps to 1.4142 (√2), matching the defaults pre-filled above.

Related calculators

Implicit Differentiation Calculator: find dy/dx for implicit equations at a point.
Taylor Series Calculator: approximate functions with Taylor polynomial series.
Trig Identity Calculator: verify trig identities numerically at any angle.
Percent Error Calculator: compare your computed root to a known reference value.

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