Find zeros of a function using the Newton-Raphson method

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Find zeros of a function using Newton's Method

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The Newton-Raphson method uses the tangent of a curve to iteratively approximate a zero of a function, f(x). This yields the update:

x_{i + 1} = x_i -\frac{f(x_i)}{f'(x_i)}.


Consider the zero of (x + 2) * (x - 1) at x = 1:
var nr = require('newton-raphson-method');

function f (x) { return (x - 1) * (x + 2); }
function fp (x) { return (x - 1) + (x + 2); }

// Using the derivative:
nr(f, fp, 2)
// => 1.0000000000000000 (6 iterations)

// Using a numerical derivative:
nr(f, 2)
// => 1.0000000000000000 (6 iterations)


$ npm install newton-raphson-method


require('newton-raphson-method')(f[, fp], x0[, options])

Given a real-valued function of one variable, iteratively improves and returns a guess of a zero.
  • f: The numerical function of one variable of which to compute the zero.
  • fp (optional): The first derivative of f. If not provided, is computed numerically using a fourth order central difference with step size h.
  • x0: A number representing the intial guess of the zero.
  • options (optional): An object permitting the following options:
- tolerance (default: 1e-7): The tolerance by which convergence is measured. Convergence is met if |x[n+1] - x[n]| <= tolerance * |x[n+1]|. - epsilon (default: 2.220446049250313e-16 (double-precision epsilon)): A threshold against which the first derivative is tested. Algorithm fails if |y'| < epsilon * |y|. - maxIterations (default: 20): Maximum permitted iterations. - h (default: 1e-4): Step size for numerical differentiation. - verbose (default: false): Output additional information about guesses, convergence, and failure.
Returns: If convergence is achieved, returns an approximation of the zero. If the algorithm fails, returns false.

See Also

  • modified-newton-raphson: A simple modification of Newton-Raphson that may exhibit improved convergence.
  • newton-raphson: A similar and lovely implementation that differs (only?) in requiring a first derivative.


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Ricky Reusser