# Source code for quantecon.optimize.scalar_maximization

import numpy as np
from numba import jit, njit

[docs]@njit
def brent_max(func, a, b, args=(), xtol=1e-5, maxiter=500):
"""
Uses a jitted version of the maximization routine from SciPy's fminbound.
The algorithm is identical except that it's been switched to maximization
rather than minimization, and the tests for convergence have been stripped
out to allow for jit compilation.

Note that the input function func must be jitted or the call will fail.

Parameters
----------
func : jitted function
a : scalar
Lower bound for search
b : scalar
Upper bound for search
args : tuple, optional
Extra arguments passed to the objective function.
maxiter : int, optional
Maximum number of iterations to perform.
xtol : float, optional
Absolute error in solution xopt acceptable for convergence.

Returns
-------
xf : float
The maximizer
fval : float
The maximum value attained
info : tuple
A tuple of the form (status_flag, num_iter).  Here status_flag
indicates whether or not the maximum number of function calls was
attained.  A value of 0 implies that the maximum was not hit.
The value num_iter is the number of function calls.

Example
-------


@njit
def f(x):
return -(x + 2.0)**2 + 1.0

xf, fval, info = brent_max(f, -2, 2)


"""
if not np.isfinite(a):
raise ValueError("a must be finite.")

if not np.isfinite(b):
raise ValueError("b must be finite.")

if not a < b:
raise ValueError("a must be less than b.")

maxfun = maxiter
status_flag = 0

sqrt_eps = np.sqrt(2.2e-16)
golden_mean = 0.5 * (3.0 - np.sqrt(5.0))

fulc = a + golden_mean * (b - a)
nfc, xf = fulc, fulc
rat = e = 0.0
x = xf
fx = -func(x, *args)
num = 1

ffulc = fnfc = fx
xm = 0.5 * (a + b)
tol1 = sqrt_eps * np.abs(xf) + xtol / 3.0
tol2 = 2.0 * tol1

while (np.abs(xf - xm) > (tol2 - 0.5 * (b - a))):
golden = 1
# Check for parabolic fit
if np.abs(e) > tol1:
golden = 0
r = (xf - nfc) * (fx - ffulc)
q = (xf - fulc) * (fx - fnfc)
p = (xf - fulc) * q - (xf - nfc) * r
q = 2.0 * (q - r)
if q > 0.0:
p = -p
q = np.abs(q)
r = e
e = rat

# Check for acceptability of parabola
if ((np.abs(p) < np.abs(0.5*q*r)) and (p > q*(a - xf)) and
(p < q * (b - xf))):
rat = (p + 0.0) / q
x = xf + rat

if ((x - a) < tol2) or ((b - x) < tol2):
si = np.sign(xm - xf) + ((xm - xf) == 0)
rat = tol1 * si
else:      # do a golden section step
golden = 1

if golden:  # Do a golden-section step
if xf >= xm:
e = a - xf
else:
e = b - xf
rat = golden_mean*e

if rat == 0:
si = np.sign(rat) + 1
else:
si = np.sign(rat)

x = xf + si * np.maximum(np.abs(rat), tol1)
fu = -func(x, *args)
num += 1

if fu <= fx:
if x >= xf:
a = xf
else:
b = xf
fulc, ffulc = nfc, fnfc
nfc, fnfc = xf, fx
xf, fx = x, fu
else:
if x < xf:
a = x
else:
b = x
if (fu <= fnfc) or (nfc == xf):
fulc, ffulc = nfc, fnfc
nfc, fnfc = x, fu
elif (fu <= ffulc) or (fulc == xf) or (fulc == nfc):
fulc, ffulc = x, fu

xm = 0.5 * (a + b)
tol1 = sqrt_eps * np.abs(xf) + xtol / 3.0
tol2 = 2.0 * tol1

if num >= maxfun:
status_flag = 1
break

fval = -fx
info = status_flag, num

return xf, fval, info