newton1d.newton1d

newton1d.newton1d(f, df, x0, tol1, max_iter=1000)

Find roots of a scalar function using Newton–Raphson.

Parameters

Name	Type	Description	Default
f	callable	Function whose root is sought. Must accept a single scalar argument.	required
df	callable	Derivative of f. Must accept a single scalar argument.	required
x0	float or Sequence[float]	Initial guesses for the root.	required
tol1	float	Relative convergence tolerance for the Newton–Raphson method. Must be strictly positive.	required
max_iter	int	Maximum number of iterations before declaring non-convergence.	100

Returns

Name	Type	Description
x_arr	Sequence[float]	Estimated roots of the function f.

Notes

The Newton–Raphson method updates the current estimate using the local linear approximation of f at the current point:

x_{n+1} = x_n - f(x_n) / df(x_n).

Convergence is declared when the step size is sufficiently small in a relative sense:

|x_{n+1} - x_n| <= tol1 * max(1, |x_{n+1}|).

This is numerically safer than |(x_{n+1}-x_n)/x_{n+1}| because it avoids division by zero when x_{n+1} = 0.

The method often converges rapidly when x0 is sufficiently close to the true root and df(x) is well-behaved, but it may fail when df(x_n) = 0 (or very close to zero) or when the iterates diverge.

Raises

Name	Type	Description
	TypeError	If f or df is not callable.
	ValueError	If tol1 <= 0, max_iter <= 0, x0 is not finite, or df(x) is too close to zero during iteration.
	RuntimeError	If the method does not converge within max_iter iterations.

Examples

>>> f = lambda x: x**2 - 2
>>> df = lambda x: 2*x
>>> root = newton1d(f, df, 1.0, 1e-12)
>>> abs(root - 2**0.5) < 1e-10
True