1D Root-Finding Solvers

Root finding is ubiquitous across quantitative finance, from inverting yields and implied volatilities to solving clearing equilibrium prices. SabiQuant includes an allocation-free numerical solver suite.

Implemented Solvers

1. Brent’s Method (BrentSolver)

Combines inverse quadratic interpolation, the secant method, and bisection. Guaranteed convergence for continuous functions where roots are bracketed $[a, b]$ with $f(a) \cdot f(b) < 0$.

2. Safeguarded Newton-Raphson (SafeguardedNewtonSolver)

Uses analytical derivative $f'(x)$ for quadratic convergence:

$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$

If $x_{k+1}$ jumps outside the valid bracket $[a, b]$ or does not decrease error fast enough, the solver falls back to a bisection step.

3. Bisection Solver (BisectionSolver)

Robust, unconditionally stable $O(\log_2(1/\varepsilon))$ linear convergence method.

---

Rust Code Example

use sabiquant::math::solvers::{BrentSolver, Solver1D};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Solve f(x) = x^3 - 2x - 5 = 0  (Known root ~ 2.09455148)
    let f = |x: f64| x.powi(3) - 2.0 * x - 5.0;

    let solver = BrentSolver::new()
        .max_iterations(100)
        .tolerance(1e-12);

    let bracket_low = 1.0;
    let bracket_high = 3.0;

    let root = solver.solve(&f, bracket_low, bracket_high)?;

    println!("Found Root: {:.12}", root);
    println!("Residual:   {:.2e}", f(root));

    assert!((f(root)).abs() < 1e-12);

    Ok(())
}