''' a,b = bracket(f,xStart,h)
  Finds the brackets (a,b) of a minimum point of the
  user-supplied scalar function f(x).
  The search starts downhill from xStart with a step
  length h.
  x,fMin = search(f,a,b,tol=1.0e-6)
  Golden section method for determining x that minimizes
  the user-supplied scalar function f(x).
  The minimum must be bracketed in (a,b).
'''    
from math import log, ceil
def bracket(f,x1,h):
  c = 1.618033989 
  f1 = f(x1)
  x2 = x1 + h; f2 = f(x2)
 # Determine downhill direction and change sign of h if needed
  if f2 > f1:
    h = -h
    x2 = x1 + h; f2 = f(x2)
   # Check if minimum between x1 - h and x1 + h
    if f2 > f1: return x2,x1 - h 
 # Search loop
  for i in range (100):  
    h = c*h
    x3 = x2 + h; f3 = f(x3)
    if f3 > f2: return x1,x3
    x1 = x2; x2 = x3
    f1 = f2; f2 = f3
  print "Bracket did not find a mimimum"    
def search(f,a,b,tol=1.0e-9):
  nIter = int(ceil(-2.078087*log(tol/abs(b-a)))) # Eq. (10.4)
  R = 0.618033989
  C = 1.0 - R
 # First telescoping
  x1 = R*a + C*b; x2 = C*a + R*b
  f1 = f(x1); f2 = f(x2)
 # Main loop
  for i in range(nIter):
    if f1 > f2:
      a = x1
      x1 = x2; f1 = f2
      x2 = C*a + R*b; f2 = f(x2)
    else:
      b = x2
      x2 = x1; f2 = f1
      x1 = R*a + C*b; f1 = f(x1)
  if f1 < f2: return x1,f1
  else: return x2,f2