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The /api/v1/newton/solve endpoint finds the optimum of a single-variable polynomial by applying the Newton-Raphson method to the first derivative f′(x). Starting from an initial guess x₀, each iteration refines the estimate using xₙ₊₁ = xₙ − f′(xₙ) / f″(xₙ) until convergence. The nature of the located critical point is verified using the second derivative f″(x). Compared to the Bisection method, Newton-Raphson typically converges much faster (quadratically) but requires a good initial guess and a non-zero second derivative at the solution.

​
Request

POST /api/v1/newton/solve 
Content-Type: application/json
​
coefficients
list[float]
required
Polynomial coefficients listed from highest to lowest degree. Must contain between 1 and 11 elements (degree 0 through 10). The array must not be all-zero.Examples:
  • [-1, 4, 0] → -x² + 4x
  • [1, -6, 9, 0] → x³ - 6x² + 9x
​
goal
string
required
Optimisation direction. Accepted values: "max" or "min".
​
x0
float
required
Initial guess for the location of the critical point. The algorithm converges to the nearest root of f′(x) from this starting value.
​
tolerance
float
default:"1e-6"
Convergence tolerance on the step size |xₙ₊₁ − xₙ|. Must be greater than 0 and at most 1.0.
​
max_iterations
integer
default:"100"
Maximum number of Newton-Raphson steps. Accepted range: 1–500.

​
Response

A successful call returns a NewtonResponse object.
​
status
string
Outcome of the solve. One of:
  • "optimal" — a critical point was found within tolerance.
  • "no_convergence" — max_iterations was reached without satisfying the tolerance.
​
optimal_x
float | null
The x-coordinate of the located critical point. null if the method did not converge.
​
optimal_value
float | null
The function value f(optimal_x). null if the method did not converge.
​
nature
string | null
Classification of the critical point: "maximum", "minimum", or "inflection point". null if the method did not converge.
​
goal_satisfied
boolean
true when the nature of the found point matches the requested goal.
​
iterations_count
integer
Actual number of Newton-Raphson iterations performed.
​
iterations
list[NewtonIteration]
Step-by-step iteration table, one entry per Newton-Raphson step.
​
function_str
string
Human-readable string representation of f(x) (e.g. "-x**2 + 4*x").
​
derivative_str
string
Human-readable string representation of f′(x) (e.g. "-2*x + 4").
​
second_derivative_str
string
Human-readable string representation of f″(x) (e.g. "-2").
​
message
string
Human-readable summary of the result.

​
Example

Maximize f(x) = −x² + 4x starting from x₀ = 0. Coefficients from highest to lowest degree: [-1, 4, 0]. 
curl -X POST https://api.example.com/api/v1/newton/solve \
  -H "Content-Type: application/json" \
  -d '{
    "coefficients": [-1, 4, 0],
    "goal": "max",
    "x0": 0,
    "tolerance": 1e-6,
    "max_iterations": 100
  }'

{
  "status": "optimal",
  "optimal_x": 2.0,
  "optimal_value": 4.0,
  "nature": "maximum",
  "goal_satisfied": true,
  "iterations_count": 1,
  "iterations": [
    {
      "iteration": 1,
      "x_n": 0.0,
      "f_xn": 0.0,
      "df_xn": 4.0,
      "d2f_xn": -2.0,
      "x_next": 2.0,
      "step": 2.0
    }
  ],
  "function_str": "-x**2 + 4*x",
  "derivative_str": "-2*x + 4",
  "second_derivative_str": "-2",
  "message": "Maximum found at x = 2.0 with f(x) = 4.0"
}
For a quadratic polynomial Newton-Raphson converges in a single iteration regardless of the starting point x₀, because f′(x) is linear. For higher-degree polynomials, choose x₀ close to the expected critical point for reliable convergence.
The Newton-Raphson step divides by f″(xₙ). If f″(xₙ) ≈ 0 at any iteration (e.g. near an inflection point), the step becomes very large and the method may diverge. The endpoint returns HTTP 422 Unprocessable Entity if a division-by-zero occurs.

​
Health Check

GET /api/v1/newton/health Returns a liveness confirmation for the Newton-Raphson Optimization service. 
curl https://api.example.com/api/v1/newton/health

{
  "status": "ok",
  "service": "newton-optimizer-api"
}

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