Root-finding is the process of solving equations of the form f(x) = 0, where the goal is to find values of x (called roots or zeros) that satisfy the equation. This fundamental problem appears throughout scientific computing, engineering, and quantitative analysis—from solving nonlinear equilibrium conditions in physics to finding break-even points in economics, calibrating financial models, and analyzing circuit behavior in electrical engineering. Root-finding is distinct from optimization: it seeks points where the function crosses zero rather than where it reaches a minimum or maximum. However, the two are intimately connected: finding a root of the derivative f'(x) = 0 is equivalent to finding a critical point of f(x).
Implementation: These tools leverage numerical root-finding methods from SciPy, symbolic approaches through CasADi, and classical iterative algorithms. SciPy provides robust, production-grade solvers for both scalar and systems of equations, CasADi enables symbolic manipulation and automatic Jacobian computation, while classical methods form the theoretical foundation underlying these libraries.
Scalar Root-Finding: When solving a single equation f(x) = 0 for a univariate function, specialized scalar methods are highly efficient. The ROOT_SCALAR tool implements SciPy’s scalar root solver, which automatically selects among Brent’s method, secant method, and bisection based on available information—whether the function is bracketed, whether the derivative is supplied, and the smoothness of the function. Scalar root-finding excels in applications like finding equilibrium prices, calibrating yield curves in finance, or determining operating points in circuit design.
Systems of Nonlinear Equations: Many applications require simultaneously solving multiple equations in multiple unknowns. The ROOT tool addresses square systems \mathbf{f}(\mathbf{x}) = \mathbf{0} using SciPy’s robust Newton-Krylov and hybrid methods, which combine Newton’s method with global damping strategies to handle poor initial guesses. For problems where Jacobian information is critical or where automatic differentiation is desired, the CA_ROOT tool leverages CasADi to solve systems with automatically computed Jacobians, providing symbolic precision and enabling better convergence in ill-conditioned problems.
Fixed-Point Iteration: Some root-finding problems naturally formulate as fixed-point iterations: finding x such that x = g(x). This is mathematically equivalent to solving f(x) = g(x) - x = 0, but fixed-point formulations sometimes offer computational advantages. The FIXED_POINT tool implements fixed-point iteration for scalar functions, useful in iterative refinement, dynamical systems analysis, and fractional equation solving.
The difficulty of root-finding depends heavily on problem characteristics: whether the function is scalar or multidimensional, smooth or discontinuous, whether derivatives are available, and whether roots are isolated or part of a continuum. Figure 1 illustrates the contrast between different problem classes and demonstrates how algorithm behavior varies with function topology.
C:\Users\brent\AppData\Local\Temp\ipykernel_2016\1746005838.py:40: UserWarning: The following kwargs were not used by contour: 'label'
ax2.contour(X_sys, Y_sys, f1, levels=[0], colors='blue', linewidths=2, label='x² + y² = 1')
C:\Users\brent\AppData\Local\Temp\ipykernel_2016\1746005838.py:41: UserWarning: The following kwargs were not used by contour: 'label'
ax2.contour(X_sys, Y_sys, f2, levels=[0], colors='orange', linewidths=2, label='x - y = 0.5')
| Tool | Description |
|---|---|
| CA_ROOT | Solve a system of nonlinear equations using CasADi with automatic Jacobian. |
| FIXED_POINT | Find a fixed point x such that f(x) = x for a scalar function expression. |
| ROOT | Solve a square nonlinear system using SciPy’s root solver. |
| ROOT_SCALAR | Find a real root of a scalar function using SciPy’s root_scalar. |