/**
 * Open (non-bracketing) scalar root-finders.
 *
 * Complements the bracketing `findRoot` (bisection/Brent, `../typed/numeric.ts`)
 * with classic open methods that iterate from a starting point (or two)
 * without requiring a sign-change bracket: Newton–Raphson, secant, and
 * Halley's method (cubic convergence).
 *
 * @packageDocumentation
 */
type f64 = number;
type i32 = number;
/**
 * Options for Newton's method.
 */
export interface NewtonOptions {
    /** Analytic derivative f'(x). If omitted, a central-difference estimate is used. */
    fprime?: (x: f64) => f64;
    /** Absolute tolerance for convergence (default 1e-12) */
    tol?: f64;
    /** Maximum iterations (default 100) */
    maxIter?: i32;
}
/**
 * Find a root of f(x) = 0 via Newton–Raphson iteration:
 * `x_{k+1} = x_k - f(x_k) / f'(x_k)`.
 *
 * If `opts.fprime` is not supplied, the derivative is estimated by a
 * central difference with step `h = max(1, |x|) * cbrt(eps)`.
 *
 * @param f - Function whose root is sought
 * @param x0 - Initial guess
 * @param opts - Options (fprime, tol, maxIter)
 * @returns Approximate root
 *
 * @example
 * newton(x => x ** 2 - 2, 1) // => ~1.4142135623730951
 */
export declare function newton(f: (x: f64) => f64, x0: f64, opts?: NewtonOptions): f64;
/**
 * Options for the secant method.
 */
export interface SecantOptions {
    /** Absolute tolerance for convergence (default 1e-12) */
    tol?: f64;
    /** Maximum iterations (default 100) */
    maxIter?: i32;
}
/**
 * Find a root of f(x) = 0 via the secant method:
 * `x_{k+1} = x_k - f(x_k) (x_k - x_{k-1}) / (f(x_k) - f(x_{k-1}))`.
 *
 * Requires no derivative, but two initial estimates.
 *
 * @param f - Function whose root is sought
 * @param x0 - First initial estimate
 * @param x1 - Second initial estimate
 * @param opts - Options (tol, maxIter)
 * @returns Approximate root
 *
 * @example
 * secant(x => x ** 2 - 2, 1, 2) // => ~1.4142135623730951
 */
export declare function secant(f: (x: f64) => f64, x0: f64, x1: f64, opts?: SecantOptions): f64;
/**
 * Options for Halley's method.
 */
export interface HalleyOptions {
    /** Analytic first derivative f'(x). If omitted, estimated via central difference. */
    fprime?: (x: f64) => f64;
    /** Analytic second derivative f''(x). If omitted, estimated via central difference. */
    fprime2?: (x: f64) => f64;
    /** Absolute tolerance for convergence (default 1e-12) */
    tol?: f64;
    /** Maximum iterations (default 100) */
    maxIter?: i32;
}
/**
 * Find a root of f(x) = 0 via Halley's method (cubic convergence):
 * `x_{k+1} = x_k - 2 f f' / (2 f'^2 - f f'')`.
 *
 * If `opts.fprime` / `opts.fprime2` are not supplied, both derivatives are
 * estimated by central differences with step `h = max(1, |x|) * cbrt(eps)`.
 *
 * @param f - Function whose root is sought
 * @param x0 - Initial guess
 * @param opts - Options (fprime, fprime2, tol, maxIter)
 * @returns Approximate root
 *
 * @example
 * halley(x => x ** 3 - 2, 1) // => ~1.2599210498948732
 */
export declare function halley(f: (x: f64) => f64, x0: f64, opts?: HalleyOptions): f64;
export {};
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