hybrid.hybrid

hybrid.hybrid(
    f,
    dfdx,
    xmin,
    xmax,
    tol1,
    tol2,
    max_iter1=500,
    max_iter2=500,
    n=50,
)

Find multiple roots of a scalar function using a hybrid Bisection-Newton method.

This algorithm combines a robust bisection-based root searching stage with Newton-Raphson refinement to efficiently locate all detectable roots within a given interval.

Parameters

Name Type Description Default
f callable Function whose roots are sought. Must accept a single scalar argument. required
dfdx callable Derivative of f. Must accept a single scalar argument. required
xmin float Lower bound of the search interval. required
xmax float Upper bound of the search interval. required
tol1 float Absolute or relative convergence tolerance used by the bisection-based root search. This tolerance controls the accuracy of the initial guesses passed to Newton’s method and must be strictly positive. required
tol2 float Relative convergence tolerance used by the Newton-Raphson method. Must be strictly positive. required
max_iter1 int Maximum number of iteration for bisection method, Default = 500 500
max_iter2 int Maximum number of iteration for Newton’s method, Default = 500 500
n int number of initial guess if bisection fails for some reason (ex: when f(x) = x^2) 50

Returns

Name Type Description
roots Sequence[float] A sequence of estimated roots of the function f within the interval [xmin, xmax]. Only roots for which Newton’s method converges successfully are returned. The output may be empty if no convergent roots are found.

Notes

The hybrid algorithm proceeds in two stages:

  1. Bisection-based root search
    The interval [xmin, xmax] is subdivided and examined for sign changes of f. Each detected sign change (or exact zero) produces an initial root estimate using the bisection method via bisection_find_roots.

    This stage is robust but may:

    • Miss roots of even multiplicity
    • Miss roots if the subdivision is too coarse
    • Return duplicate or closely spaced root estimates

  2. Newton-Raphson refinement
    Each bisection-derived estimate is passed as an initial guess to newton1d, which rapidly refines the root when the derivative is well-behaved.

    Newton’s method may fail to converge if:

    • The derivative is zero or nearly zero
    • The initial guess is too far from a true root
    • The iterates diverge or become non-finite

Duplicate roots produced by either stage may be filtered using the Newton convergence tolerance tol2.

Convergence for Newton’s method is declared when:

|x_{n+1} - x_n| <= tol2 * max(1, |x_{n+1}|).

This hybrid approach improves robustness compared to Newton’s method alone while achieving faster convergence than pure bisection.

See Also

bisection_find_roots : Bisection-based root search for detecting multiple candidate roots. newton1d : Newton-Raphson method for fast local root refinement.

Examples

>>> f = lambda x: x**2 - 4
>>> df = lambda x: 2*x
>>> roots = hybrid(f, df, -3, 3, tol1=1e-6, tol2=1e-12)
>>> sorted(roots)
[-2.0, 2.0]
>>> f = lambda x: x**3 - x
>>> df = lambda x: 3*x**2 - 1
>>> roots = hybrid(f, df, -2, 2, tol1=1e-6, tol2=1e-12)
>>> sorted(roots)
[-1.0, 0.0, 1.0]