''' root = ridder(f,a,b,tol=1.0e-9).
  Finds a root of f(x) = 0 with Ridder's method.
  The root must be bracketed in (a,b).
'''
import error
from math import sqrt
def ridder(f,a,b,tol=1.0e-9):  
  fa = f(a)
  if fa == 0.0: return a
  fb = f(b)
  if fb == 0.0: return b
  if fa*fb > 0.0: error.err('Root is not bracketed')
  for i in range(30):
   # Compute the improved root x from Ridder's formula
    c = 0.5*(a + b); fc = f(c)
    s = sqrt(fc**2 - fa*fb)
    if s == 0.0: return None
    dx = (c - a)*fc/s
    if (fa - fb) < 0.0: dx = -dx
    x = c + dx; fx = f(x)
   # Test for convergence
    if i > 0:
      if abs(x - xOld) < tol*max(abs(x),1.0): return x
    xOld = x
   # Re-bracket the root as tightly as possible
    if fc*fx > 0.0:
      if fa*fx < 0.0: b = x; fb = fx
      else:      a = x; fa = fx
    else:
      a = c; b = x; fa = fc; fb = fx
  return None
  print 'Too many iterations'