raygeo.geo.algo.rootfind

Bisection, secant, and Illinois on $x^3 - 2x - 5$. 1D root-finding methods: bisection, secant, Illinois.

Functions

bisect()

bisect(
    f: Callable[[float], float],
    lo: float,
    hi: float,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> tuple[float, str, int]

Bisection root-finding.

Parameter	Type	Description
f	Callable[[float], float]	Function to find the root of (takes float, returns float).
lo	float	Lower bound of the search interval.
hi	float	Upper bound of the search interval.
tol	float = 1e-10	Convergence tolerance (default 1e-10).
max_iter	int = 100	Maximum iterations (default 100).
Returns	tuple[float, str, int]	(root, status_string, iteration_count).

Error vs iteration count: secant fastest, bisection slowest.

Iterations to reach a given tolerance for sqrt(2): secant needs far fewer than bisection.

bisect_tracked()

bisect_tracked(
    f: Any,
    lo: float,
    hi: float,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> tuple[float, str, int, list[float]]

Tracked bisection root-finding.

Parameter	Type	Description
f	Any	Function to find the root of.
lo	float	Lower bound of the search interval.
hi	float	Upper bound of the search interval.
tol	float = 1e-10	Convergence tolerance.
max_iter	int = 100	Maximum number of iterations.
Returns	tuple[float, str, int, list[float]]	(root, status_string, iteration_count, history).

bracket_grid()

bracket_grid(
    f: Callable[[float], float],
    heading: float,
    max_deflection: float,
) -> tuple[float, str, int]

7-sample angular grid search with linear interpolation.

Samples f at heading + max_deflection * ratio for 7 ratios evenly spaced across [-1, -0.6, -0.2, 0, 0.2, 0.6, 1.0]. When a sign change is found between adjacent samples the root is linearly interpolated. Falls back to the sample with smallest absolute error.

Parameter	Type	Description
f	Callable[[float], float]	Error function f(angle) -> error.
heading	float	Centre angle in radians.
max_deflection	float	Maximum angular spread in radians.
Returns	tuple[float, str, int]	(root, status_string, sample_count).

7-sample angular grid search with linear interpolation: samples f(x) on a fan around heading and interpolates across adjacent sign changes.

illinois()

illinois(
    f: Callable[[float], float],
    lo: float,
    hi: float,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> tuple[float, str, int]

Illinois (safeguarded false-position) root-finding.

Parameter	Type	Description
f	Callable[[float], float]	Function to find the root of (takes float, returns float).
lo	float	Lower bound of the search interval.
hi	float	Upper bound of the search interval.
tol	float = 1e-10	Convergence tolerance (default 1e-10).
max_iter	int = 100	Maximum iterations (default 100).
Returns	tuple[float, str, int]	(root, status_string, iteration_count).

illinois_tracked()

illinois_tracked(
    f: Any,
    lo: float,
    hi: float,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> tuple[float, str, int, list[float]]

Tracked Illinois method root-finding.

Parameter	Type	Description
f	Any	Function to find the root of.
lo	float	Lower bound of the search interval.
hi	float	Upper bound of the search interval.
tol	float = 1e-10	Convergence tolerance.
max_iter	int = 100	Maximum number of iterations.
Returns	tuple[float, str, int, list[float]]	(root, status_string, iteration_count, history).

secant()

secant(
    f: Callable[[float], float],
    x0: float,
    x1: float,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> tuple[float, str, int]

Secant root-finding.

Parameter	Type	Description
f	Callable[[float], float]	Function to find the root of (takes float, returns float).
x0	float	First initial guess.
x1	float	Second initial guess.
tol	float = 1e-10	Convergence tolerance (default 1e-10).
max_iter	int = 100	Maximum iterations (default 100).
Returns	tuple[float, str, int]	(root, status_string, iteration_count).

secant_tracked()

secant_tracked(
    f: Any,
    x0: float,
    x1: float,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> tuple[float, str, int, list[float]]

Tracked secant method root-finding.

Parameter	Type	Description
f	Any	Function to find the root of.
x0	float	First initial guess.
x1	float	Second initial guess.
tol	float = 1e-10	Convergence tolerance.
max_iter	int = 100	Maximum number of iterations.
Returns	tuple[float, str, int, list[float]]	(root, status_string, iteration_count, history).