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scipy.optimize.

# bracket

scipy.optimize.bracket(func, xa=0.0, xb=1.0, args=(), grow_limit=110.0, maxiter=1000)

  Bracket the minimum of a function.

  Given a function and distinct initial points, search in the downhill direction (as defined by the initial points) and return three points that bracket the minimum of the function.

  Parameters:
    func callable f(x,*args)

      Objective function to minimize.

    xa, xb float, optional

      Initial points. Defaults xa to 0.0, and xb to 1.0. A local minimum need not be contained within this interval.

    args tuple, optional

      Additional arguments (if present), passed to func.

    grow_limit float, optional

      Maximum grow limit. Defaults to 110.0

    maxiter int, optional

      Maximum number of iterations to perform. Defaults to 1000.

  Returns:
    xa, xb, xc float

      Final points of the bracket.

    fa, fb, fc float

      Objective function values at the bracket points.

    funcalls int

      Number of function evaluations made.

  Raises:
    BracketError

      If no valid bracket is found before the algorithm terminates. See notes for conditions of a valid bracket.

  Notes

  The algorithm attempts to find three strictly ordered points (i.e. \(x_a < x_b < x_c\) or \(x_c < x_b < x_a\)) satisfying \(f(x_b) ≤ f(x_a)\) and \(f(x_b) ≤ f(x_c)\), where one of the inequalities must be satisfied strictly and all \(x_i\) must be finite.

  Examples

  This function can find a downward convex region of a function:

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> from scipy.optimize import bracket
      >>> def f(x):
      ...     return 10*x**2 + 3*x + 5
      >>> x = np.linspace(-2, 2)
      >>> y = f(x)
      >>> init_xa, init_xb = 0.1, 1
      >>> xa, xb, xc, fa, fb, fc, funcalls = bracket(f, xa=init_xa, xb=init_xb)
      >>> plt.axvline(x=init_xa, color="k", linestyle="--")
      >>> plt.axvline(x=init_xb, color="k", linestyle="--")
      >>> plt.plot(x, y, "-k")
      >>> plt.plot(xa, fa, "bx")
      >>> plt.plot(xb, fb, "rx")
      >>> plt.plot(xc, fc, "bx")
      >>> plt.show()

  Note that both initial points were to the right of the minimum, and the third point was found in the “downhill” direction: the direction in which the function appeared to be decreasing (to the left). The final points are strictly ordered, and the function value at the middle point is less than the function values at the endpoints; it follows that a minimum must lie within the bracket.