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The Bisection method finds an optimal point of a single-variable polynomial by locating a root of its first derivative f′(x) = 0 within a user-supplied interval [a, b]. At each iteration the interval is halved: the midpoint m = (a + b) / 2 is evaluated, and the sub-interval whose endpoints bracket the zero-crossing of f′ is retained. Iteration continues until f′(m) falls within the requested tolerance or the interval width itself becomes negligibly small. Once the stationary point x* is found, the sign of f″(x*) determines whether it is a minimum, maximum, or saddle/inflection point.

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Polynomial Representation

The function is modelled as a polynomial of arbitrary degree. Coefficients are supplied from highest to lowest degree, following NumPy convention: 
coefficients = [c_n, c_{n-1}, ..., c_1, c_0]
f(x) = c_n·xⁿ + c_{n-1}·xⁿ⁻¹ + ... + c_1·x + c_0
Examples:
coefficientsf(x)
[-1, 4, 0]−x² + 4x
[1, -3, 2]x² − 3x + 2
[1, 0, -4, 0, 3]x⁴ − 4x² + 3
The engine computes f′(x) and f″(x) analytically using NumPy’s polyder function.
f′(a) and f′(b) must have opposite signs for bisection to guarantee a stationary point inside [a, b]. If they share the same sign the endpoint returns a 422 error — adjust the interval so that the derivative changes sign within it.

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Endpoint

POST /bisection/solve

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Request

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coefficients
float[]
required
Polynomial coefficients listed from highest to lowest degree. Must contain between 1 and 11 elements; the list must not be all zeros.Example: [-1, 4, 0] encodes f(x) = −x² + 4x.
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goal
string
required
Optimization direction: "max" to find a maximum or "min" to find a minimum. The engine uses f″(x*) to classify the stationary point and reports whether it matches the requested goal.
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a
float
required
Left endpoint of the search interval. Must satisfy a < b.
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b
float
required
Right endpoint of the search interval. Must satisfy b > a.
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tolerance
float
default:"1e-6"
Convergence threshold. Iteration stops when |f′(midpoint)| < tolerance or the interval half-width < tolerance. Must be in the range (0, 1].
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max_iterations
integer
default:"100"
Maximum number of bisection steps. Must be between 1 and 500. The solver halts at this limit even if tolerance has not been reached.

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Response

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status
string
Always "optimal" when the solver completes without error. Error conditions are returned as HTTP 422 responses.
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optimal_x
float | null
The x-coordinate of the stationary point found by bisection.
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optimal_value
float | null
The function value f(optimal_x) at the stationary point.
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nature
string | null
Classification of the stationary point based on f″(x*):
  • "mínimo" — local minimum (f″ > 0).
  • "máximo" — local maximum (f″ < 0).
  • "punto de inflexión" — saddle or inflection point (f″ ≈ 0).
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goal_satisfied
boolean
true when the nature of the found point matches the requested goal. For example, goal = "max" with nature = "máximo" yields true.
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iterations_count
integer
Total number of bisection iterations executed before convergence.
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iterations
BisectionIteration[]
Step-by-step trace of the bisection process, one entry per iteration.
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function_str
string
Human-readable string representation of f(x), e.g. "-x² + 4x".
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derivative_str
string
Human-readable string representation of f′(x), e.g. "-2x + 4".
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message
string
Narrative summary of the result, including the stationary point’s nature and whether the goal was satisfied. Also warns when a saddle/inflection point was found.

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Example

Find the maximum of f(x) = −x² + 4x on [0, 4]: 
curl -X POST http://localhost:8000/bisection/solve \
  -H "Content-Type: application/json" \
  -d '{
    "coefficients": [-1, 4, 0],
    "goal": "max",
    "a": 0,
    "b": 4,
    "tolerance": 1e-6,
    "max_iterations": 100
  }'

{
  "status": "optimal",
  "optimal_x": 2.0,
  "optimal_value": 4.0,
  "nature": "máximo",
  "goal_satisfied": true,
  "iterations_count": 1,
  "iterations": [
    {
      "iteration": 1,
      "a": 0.0,
      "b": 4.0,
      "midpoint": 2.0,
      "f_mid": 4.0,
      "df_mid": 0.0,
      "interval_width": 2.0
    }
  ],
  "function_str": "-x² + 4x",
  "derivative_str": "-2x + 4",
  "message": "Óptimo encontrado: el punto x = 2 es un máximo de la función, coincidiendo con la meta solicitada (máximo)."
}
When goal_satisfied is false, the solver still converges to the nearest stationary point — it simply doesn’t match the requested direction. This happens, for example, when the only stationary point inside [a, b] is a minimum but goal = "max" was requested. Check the interval bounds or switch the goal.

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