content: """
  Tsitouras's 5/4 order Runge-Kutta method
  
  Runge–Kutta pairs of order 5(4) satisfying only the first column
  simplifying assumption,
  Ch. Tsitouras,
  Computers & Mathematics with Applications, Vol 62, Issue 2, 770-775
  Jan 2011
  """
  
  __all__ = ["tsit5_coeff"]
  import numpy as np
  order = 5
  rk_step = 7
  
  a = np.zeros((rk_step, rk_step), dtype=np.float64)
  b = np.zeros(rk_step, dtype=np.float64)
  c = np.zeros(rk_step, dtype=np.float64)
  e = np.zeros(rk_step, dtype=np.float64)
  bi = np.zeros((rk_step, rk_step), dtype=np.float64)
  
  c[0] = 0.
  c[1] = .161
  c[2] = .327
  c[3] = .9
  c[4] = .9800255409045097
  c[5] = 1.
  c[6] = 1.
  
  b[0] =  .09646076681806523
  b[1] =  .01
  b[2] =  .4798896504144996
  b[3] =  1.379008574103742
  b[4] = -3.290069515436081
  b[5] =  2.324710524099774
  b[6] = 0.
  
  e[0] =  .17800110522257773e-2
  e[1] =  .8164344596567463e-3
  e[2] = -.7880878010261994e-2
  e[3] =  .1447110071732629
  e[4] = -.5823571654525552
  e[5] =  .45808210592918686
  e[6] = -1/66
  
  a[1, 0] = c[1]
  
  a[2, 1] = 0.3354806554923570
  a[2, 0] = c[2] - a[2, 1]
  
  a[3, 1] = -6.359448489975075
  a[3, 2] = 4.362295432869581
  a[3, 0] = c[3] - a[3, 2] - a[3, 1]
  
  a[4, 1] = -11.74888356406283
  a[4, 2] = 7.495539342889836
  a[4, 3] = -0.09249506636175525
  a[4, 0] = c[4] - a[4, 3] - a[4, 2] - a[4, 1]
  
  a[5, 1] = -12.92096931784711
  a[5, 2] = 8.159367898576159
  a[5, 3] = -0.071584973281401
  a[5, 4] = -0.02826905039406838
  a[5, 0] = c[5] - a[5, 4] - a[5, 3] - a[5, 2] - a[5, 1]
  
  a[6, 0] =  .09646076681806523
  a[6, 1] =  .01
  a[6, 2] =  .4798896504144996
  a[6, 3] =  1.379008574103742
  a[6, 4] = -3.290069515436081
  a[6, 5] =  2.324710524099774
  
  bi[0, 0] = 1.
  bi[0, 1] = -2.763706197274826
  bi[0, 2] = 2.9132554618219126
  bi[0, 3] = -1.0530884977290216
  
  bi[1, 1] = 0.13169999999999998
  bi[1, 2] = -0.2234
  bi[1, 3] = 0.1017
  
  bi[2, 1] = 3.930296236894751
  bi[2, 2] = -5.941033872131505
  bi[2, 3] = 2.490627285651253
  
  bi[3, 1] = -12.411077166933676
  bi[3, 2] = 30.338188630282318
  bi[3, 3] = -16.548102889244902
  
  bi[4, 1] = 37.50931341651104
  bi[4, 2] = -88.1789048947664
  bi[4, 3] = 47.37952196281928
  
  bi[5, 1] = -27.896526289197286
  bi[5, 2] = 65.09189467479368
  bi[5, 3] = -34.87065786149661
  
  bi[6, 1] = 1.5
  bi[6, 2] = -4.0
  bi[6, 3] = 2.5
  
  
  tsit5_coeff = {'order': order, 'a': a, 'b': b, 'c': c, 'e': e, 'bi': bi}
  