How to find roots with SciPy

Root finding solves equations by locating the input where a function evaluates to zero. In SciPy, this is a common step when a model equation, calibration target, or numerical residual needs a concrete solution instead of a plotted crossing point.

scipy.optimize.root_scalar() handles one-dimensional functions and returns a RootResults object with the candidate root, convergence flag, and stop reason. A bracketing method such as brentq starts with two interval endpoints where the function values have opposite signs, so it does not need a derivative.

A small cos(x) - x calculation keeps the convergence fields easy to inspect. For a system of equations or a vector-valued residual, use scipy.optimize.root() rather than reshaping data for root_scalar().

1. Create a Python script named root_find.py.

   root_find.py

   import numpy as np
   from scipy.optimize import root_scalar
    
   def equation(x):
       return np.cos(x) - x
    
   solution = root_scalar(equation, bracket=[0, 1], method="brentq", xtol=1e-12)
    
   print(f"converged: {solution.converged}")
   print(f"flag: {solution.flag}")
   print(f"root: {solution.root:.12f}")
   print(f"residual: {equation(solution.root):.3e}")
   print(f"iterations: {solution.iterations}")
   print(f"function calls: {solution.function_calls}")

   brentq is a bracketing solver. The bracket endpoints must surround a sign change in the function.

2. Run the script.

   $ python3 root_find.py
   converged: True
   flag: converged
   root: 0.739085133215
   residual: 7.994e-15
   iterations: 7
   function calls: 8

3. Confirm that the solver converged and the residual is near zero.

   solution.root is the x-value returned by SciPy. Re-evaluating equation(solution.root) checks the original equation rather than only trusting the solver status.

4. Replace equation() with the function from your project.

   If the two bracket values do not produce opposite signs, root_scalar() raises ValueError instead of returning a false root.

5. Adjust xtol only when the root position needs a different absolute tolerance.

   A small x-step and a small function value are different checks. Keep the residual printout until the accepted root is close enough for the model being solved.

6. Remove the demo script when it was only created for the check.

   $ rm root_find.py