multi_newton

multi_newton:
  Use Newton's method to seek a root of the multi function.
it = 2: alpha = 0.715909, log(alpha)= -0.334202
it = 3: alpha = 0.707069, log(alpha)= -0.346627, r = 1.03718
it = 4: alpha = 0.698484, log(alpha)= -0.358843, r = 1.03524
it = 5: alpha = 0.690791, log(alpha)= -0.369917, r = 1.03086
it = 6: alpha = 0.684371, log(alpha)= -0.379254, r = 1.02524
it = 7: alpha = 0.679321, log(alpha)= -0.386661, r = 1.01953
it = 8: alpha = 0.67553, log(alpha)= -0.392258, r = 1.01448
it = 9: alpha = 0.672782, log(alpha)= -0.396334, r = 1.01039
it = 10: alpha = 0.670841, log(alpha)= -0.399223, r = 1.00729
it = 11: alpha = 0.669494, log(alpha)= -0.401232, r = 1.00503
it = 12: alpha = 0.668572, log(alpha)= -0.402611, r = 1.00343
it = 13: alpha = 0.667946, log(alpha)= -0.403547, r = 1.00233
it = 14: alpha = 0.667524, log(alpha)= -0.40418, r = 1.00157
it = 15: alpha = 0.66724, log(alpha)= -0.404605, r = 1.00105
it = 16: alpha = 0.66705, log(alpha)= -0.404891, r = 1.00071
it = 17: alpha = 0.666922, log(alpha)= -0.405081, r = 1.00047
it = 18: alpha = 0.666837, log(alpha)= -0.405209, r = 1.00032
it = 19: alpha = 0.666781, log(alpha)= -0.405294, r = 1.00021
it = 20: alpha = 0.666743, log(alpha)= -0.405351, r = 1.00014
it = 21: alpha = 0.666717, log(alpha)= -0.40539, r = 1.0001
it = 22: alpha = 0.666701, log(alpha)= -0.405414, r = 1.00006
it = 23: alpha = 0.666692, log(alpha)= -0.405427, r = 1.00003
it = 24: alpha = 0.666683, log(alpha)= -0.405441, r = 1.00004
it = 25: alpha = 0.666657, log(alpha)= -0.40548, r = 1.0001
it = 26: alpha = 0.666817, log(alpha)= -0.40524, r = 0.999409
it = 27: alpha = 0.666458, log(alpha)= -0.405778, r = 1.00133
it = 28: alpha = 0.666992, log(alpha)= -0.404978, r = 0.998028
it = 29: alpha = 0.666947, log(alpha)= -0.405044, r = 1.00016

  Number of steps = 29
  f(1.00002) = 2.75335e-14
quit