import numpy as np
# Usage:
# x = np.linspace(-1.5, 1.2, 400)
# y = np.linspace(-0.2, 2.0, 400)
# X, Y = np.meshgrid(x, y)
# Z = muller_brown([X, Y])
[docs]
def muller_brown(x):
"""Evaluate the Muller-Brown potential at position *x*.
The Muller-Brown surface is a standard 2D test potential with three minima
and two saddle points, commonly used for benchmarking path-finding and
optimization algorithms.
Args:
x: Position as a 2-element sequence ``[x, y]``. Each element may be a
scalar or an array (for vectorized evaluation over a meshgrid).
Returns:
Potential energy value(s) at the given position(s).
.. versionadded:: 0.0.1
"""
A = [-200, -100, -170, 15]
a = [-1, -1, -6.5, 0.7]
b = [0, 0, 11, 0.6]
c = [-10, -10, -6.5, 0.7]
x0 = [1, 0, -0.5, -1]
y0 = [0, 0.5, 1.5, 1]
value = 0
for i in range(4):
value += A[i] * np.exp(
a[i] * (x[0] - x0[i]) ** 2
+ b[i] * (x[0] - x0[i]) * (x[1] - y0[i])
+ c[i] * (x[1] - y0[i]) ** 2
)
return value
[docs]
def muller_brown_gradient(x):
"""Compute the analytical gradient of the Muller-Brown potential at *x*.
Args:
x: Position as a 2-element array ``[x, y]``.
Returns:
numpy.ndarray: Gradient vector ``[dV/dx, dV/dy]``.
.. versionadded:: 0.0.1
"""
A = [-200, -100, -170, 15]
a = [-1, -1, -6.5, 0.7]
b = [0, 0, 11, 0.6]
c = [-10, -10, -6.5, 0.7]
x0 = [1, 0, -0.5, -1]
y0 = [0, 0.5, 1.5, 1]
dfdx = 0
dfdy = 0
for i in range(4):
dfdx += (
A[i]
* (2 * a[i] * (x[0] - x0[i]) + b[i] * (x[1] - y0[i]))
* np.exp(
a[i] * (x[0] - x0[i]) ** 2
+ b[i] * (x[0] - x0[i]) * (x[1] - y0[i])
+ c[i] * (x[1] - y0[i]) ** 2
)
)
dfdy += (
A[i]
* (b[i] * (x[0] - x0[i]) + 2 * c[i] * (x[1] - y0[i]))
* np.exp(
a[i] * (x[0] - x0[i]) ** 2
+ b[i] * (x[0] - x0[i]) * (x[1] - y0[i])
+ c[i] * (x[1] - y0[i]) ** 2
)
)
return np.array([dfdx, dfdy])
