# Root Finding > Use when a task asks for a root finder, zero finding, nonlinear equation solving, compressible flow Mach number solution, or implicit performance equation inversion in aerospace analysis. Determine the root of a nonlinear scalar equation f(x) = 0 numerically with the bisection method, Newton-Raphson, the secant method, or fixed-point iteration: bracket the root or supply an initial guess, apply the iteration, and produce the root to a specified tolerance under function tolerance, step tolerance, and max iteration convergence criteria. Produces the root, the method, and the iteration count. Trigger: root-finding, bisection, newton-raphson, secant-method, fixed-point iteration, mach number root, nonlinear equation, zero finding. - Skill: `ashfordeou/root-finding` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/root-finding` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/root-finding/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Productivity - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/root-finding --- # Root Finding (cross-cutting/numerics/root-finding) Use when the task is solving a nonlinear scalar equation f(x) = 0 numerically: choosing the bracketing or iteration method, supplying the bracket or initial guess, and reporting the root to a specified tolerance. ## Domain quick reference All worked numbers are for f(x) = x**2 - 2, whose positive root is sqrt(2) = 1.4142135623730951. - Bisection: with f(a) * f(b) < 0 (a sign change is bracketed), evaluate the midpoint c = (a + b) / 2 and halve the interval on the side that keeps the sign change. Linear convergence: one binary digit per step, error bounded by the interval half-width, so about 34 steps take [1, 2] to sqrt(2) at tol = 1e-10. Robust and derivative-free; needs only sign information. - Newton-Raphson: x_{k+1} = x_k - f(x_k) / f'(x_k). Quadratic convergence for a simple root when the initial guess is close enough; needs the derivative f'. From x0 = 1.5 on x**2 - 2 the iterate reaches 1.41421356237 in three steps. Fails when f'(x) is zero at a step (the update is undefined). - Secant method: x_{k+1} = x_k - f(x_k) * (x_k - x_{k-1}) / (f(x_k) - f(x_{k-1})). Approximates the derivative with the secant slope, so no analytic derivative is needed; superlinear convergence of order about 1.618 for a simple root. Fails when the two function values are equal (the slope is zero). - Fixed-point iteration: rewrite f(x) = 0 as x = g(x) and iterate x_{k+1} = g(x_k). Converges when g is a contraction at the fixed point, in practice when abs(g'(x*)) < 1; a repelling fixed point with abs(g'(x*)) > 1 diverges. - Bracketing and initial guess requirements: bisection needs a bracket whose endpoints straddle zero; Newton-Raphson and the secant method need initial guesses near the root; fixed-point iteration needs a contractive rewrite. There is no global method for arbitrary nonlinear equations. - Convergence criteria: the function tolerance abs(f(x)) < tol declares convergence for Newton-Raphson and the secant method; the step tolerance (interval half-width, or abs(x_{k+1} - x_k) for fixed-point iteration) declares convergence for bisection and fixed-point iteration; every method stops with an error after max_iter iterations. - Non-monotonic and ill-conditioned behavior: bisection still converges on a non-monotonic function as long as the endpoints straddle zero, but a double root (f touches zero without a sign change, e.g. x**2 = 0) has no straddle; Newton-Raphson can overshoot on non-convex functions or converge to a different root; a near-zero derivative amplifies function error and makes the problem ill-conditioned; the secant method can blow up when f(x_k) is close to f(x_{k-1}). - Aerospace application: the isentropic area-Mach relation A/A* = (1/M) * ((2/(gamma+1)) * (1 + (gamma-1)/2 * M**2)) ** ((gamma+1)/(2*(gamma-1))) with gamma = 1.4 is solved for M given A/A*; A/A* = 1.2 gives the subsonic root M = 0.59024876099 and the supersonic root M = 1.53414976720. Implicit performance equations (range, climb, thrust balance) are inverted the same way. - All functions are deterministic and stdlib-only; no network, no third-party numerical libraries. ## Workflow 1. Write the problem as f(x) = 0 and record the tolerance and the max iteration budget. 2. Bracket the root when possible: find a and b with f(a) * f(b) < 0 and call bisection(f, a, b, tol, max_iter); otherwise pick initial guesses for the derivative-based methods. 3. Pick the method: bisection for robustness and guaranteed convergence on a bracket, newton_raphson when the derivative is cheap, secant when it is not, fixed_point_iteration when a contractive rewrite x = g(x) is natural. 4. Call the method; a ValueError means the bracket did not straddle zero, the derivative or secant slope was zero, or the iteration did not converge within max_iter. 5. Verify the result: evaluate f(root) and confirm it is below the function tolerance, and report the root, the method, and the iteration count together. ## Pitfalls - Confusing root finding with ODE solving: ode-solvers marches a state y(t) forward in time from an initial condition; root finding solves f(x) = 0 for a single scalar. A task that says "solve the equation" without time marching is a root-finding problem and routes here, not to ode-solvers. - Confusing root finding with numerical integration: quadrature returns a definite integral, a single number, in one pass; root finding locates where a function vanishes. Integrals route to numerical-integration. - Confusing root finding with interpolation: interpolation fits a curve through given data points; root finding locates zeros of a given function. Fitting questions route to interpolation. - Confusing root finding with convergence verification: that leaf estimates discretization error with Richardson extrapolation, it does not locate zeros. Error estimation routes to convergence-verification. - Calling bisection without a straddle: f(a) * f(b) >= 0 raises ValueError, never silently searches. A double root (x**2 = 0) touches zero without changing sign and cannot be bracketed by bisection alone. - Calling newton_raphson where the derivative vanishes: f'(x) = 0 at a step raises ValueError; the update x - f/f' is undefined there. - Trusting a root without the residual check: always evaluate f(root) against the function tolerance; a non-converged or wrong-branch root is caught by the residual, not by the iteration count. - Using a loose max_iter with a tight tol: bisection needs about log2((b - a) / tol) steps; a max_iter below that raises ValueError on convergence failure, which is the contract, not a bug. - Expecting one method to work everywhere: no method is global. Newton-Raphson and the secant method converge locally and can diverge or land on a different root; fixed-point iteration needs a contractive rewrite. Report the bracket or initial guess so the result is reproducible. - Routing compressible-flow Mach table lookups here: closed-form property tables and airfoil data stay with their data leaves; the inversion of a compressible-flow relation for a Mach root is the root-finding task and routes here. ## Behavior contract (gate 3) The bisection, Newton-Raphson, secant, and fixed-point iteration logic, the bracket and initial guess requirements, the convergence criteria, and the ValueError contracts (non-straddling bracket, zero derivative or secant slope, convergence failure) are exercised by the gate 3 contract test: scripts/test_root_finding_logic.py against scripts/root_finding_logic.py (stdlib unittest, offline). Run: python3 scripts/test_root_finding_logic.py ## Compliance - Standards referenced, not reproduced: NACA TR-824 anchors the compressible-flow tables and isentropic area-Mach relations that these root-finding methods invert; the numerical methods themselves are classical numerical-analysis methodology, summary-only per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.