content: # Author: Travis Oliphant
  from __future__ import division, print_function, absolute_import
  
  __all__ = ['odeint']
  
  from . import _odepack
  from copy import copy
  import warnings
  
  class ODEintWarning(Warning):
      pass
  
  _msgs = {2: "Integration successful.",
           1: "Nothing was done; the integration time was 0.",
           -1: "Excess work done on this call (perhaps wrong Dfun type).",
           -2: "Excess accuracy requested (tolerances too small).",
           -3: "Illegal input detected (internal error).",
           -4: "Repeated error test failures (internal error).",
           -5: "Repeated convergence failures (perhaps bad Jacobian or tolerances).",
           -6: "Error weight became zero during problem.",
           -7: "Internal workspace insufficient to finish (internal error)."
           }
  
  
  def odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0,
             ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0,
             hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12,
             mxords=5, printmessg=0):
      """
      Integrate a system of ordinary differential equations.
  
      Solve a system of ordinary differential equations using lsoda from the
      FORTRAN library odepack.
  
      Solves the initial value problem for stiff or non-stiff systems
      of first order ode-s::
  
          dy/dt = func(y, t0, ...)
  
      where y can be a vector.
  
      *Note*: The first two arguments of ``func(y, t0, ...)`` are in the
      opposite order of the arguments in the system definition function used
      by the `scipy.integrate.ode` class.
  
      Parameters
      ----------
      func : callable(y, t0, ...)
          Computes the derivative of y at t0.
      y0 : array
          Initial condition on y (can be a vector).
      t : array
          A sequence of time points for which to solve for y.  The initial
          value point should be the first element of this sequence.
      args : tuple, optional
          Extra arguments to pass to function.
      Dfun : callable(y, t0, ...)
          Gradient (Jacobian) of `func`.
      col_deriv : bool, optional
          True if `Dfun` defines derivatives down columns (faster),
          otherwise `Dfun` should define derivatives across rows.
      full_output : bool, optional
          True if to return a dictionary of optional outputs as the second output
      printmessg : bool, optional
          Whether to print the convergence message
  
      Returns
      -------
      y : array, shape (len(t), len(y0))
          Array containing the value of y for each desired time in t,
          with the initial value `y0` in the first row.
      infodict : dict, only returned if full_output == True
          Dictionary containing additional output information
  
          =======  ============================================================
          key      meaning
          =======  ============================================================
          'hu'     vector of step sizes successfully used for each time step.
          'tcur'   vector with the value of t reached for each time step.
                   (will always be at least as large as the input times).
          'tolsf'  vector of tolerance scale factors, greater than 1.0,
                   computed when a request for too much accuracy was detected.
          'tsw'    value of t at the time of the last method switch
                   (given for each time step)
          'nst'    cumulative number of time steps
          'nfe'    cumulative number of function evaluations for each time step
          'nje'    cumulative number of jacobian evaluations for each time step
          'nqu'    a vector of method orders for each successful step.
          'imxer'  index of the component of largest magnitude in the
                   weighted local error vector (e / ewt) on an error return, -1
                   otherwise.
          'lenrw'  the length of the double work array required.
          'leniw'  the length of integer work array required.
          'mused'  a vector of method indicators for each successful time step:
                   1: adams (nonstiff), 2: bdf (stiff)
          =======  ============================================================
  
      Other Parameters
      ----------------
      ml, mu : int, optional
          If either of these are not None or non-negative, then the
          Jacobian is assumed to be banded.  These give the number of
          lower and upper non-zero diagonals in this banded matrix.
          For the banded case, `Dfun` should return a matrix whose
          rows contain the non-zero bands (starting with the lowest diagonal).
          Thus, the return matrix `jac` from `Dfun` should have shape
          ``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``.
          The data in `jac` must be stored such that ``jac[i - j + mu, j]``
          holds the derivative of the `i`th equation with respect to the `j`th
          state variable.  If `col_deriv` is True, the transpose of this
          `jac` must be returned.
      rtol, atol : float, optional
          The input parameters `rtol` and `atol` determine the error
          control performed by the solver.  The solver will control the
          vector, e, of estimated local errors in y, according to an
          inequality of the form ``max-norm of (e / ewt) <= 1``,
          where ewt is a vector of positive error weights computed as
          ``ewt = rtol * abs(y) + atol``.
          rtol and atol can be either vectors the same length as y or scalars.
          Defaults to 1.49012e-8.
      tcrit : ndarray, optional
          Vector of critical points (e.g. singularities) where integration
          care should be taken.
      h0 : float, (0: solver-determined), optional
          The step size to be attempted on the first step.
      hmax : float, (0: solver-determined), optional
          The maximum absolute step size allowed.
      hmin : float, (0: solver-determined), optional
          The minimum absolute step size allowed.
      ixpr : bool, optional
          Whether to generate extra printing at method switches.
      mxstep : int, (0: solver-determined), optional
          Maximum number of (internally defined) steps allowed for each
          integration point in t.
      mxhnil : int, (0: solver-determined), optional
          Maximum number of messages printed.
      mxordn : int, (0: solver-determined), optional
          Maximum order to be allowed for the non-stiff (Adams) method.
      mxords : int, (0: solver-determined), optional
          Maximum order to be allowed for the stiff (BDF) method.
  
      See Also
      --------
      ode : a more object-oriented integrator based on VODE.
      quad : for finding the area under a curve.
  
      Examples
      --------
      The second order differential equation for the angle `theta` of a
      pendulum acted on by gravity with friction can be written::
  
          theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0
  
      where `b` and `c` are positive constants, and a prime (') denotes a
      derivative.  To solve this equation with `odeint`, we must first convert
      it to a system of first order equations.  By defining the angular
      velocity ``omega(t) = theta'(t)``, we obtain the system::
  
          theta'(t) = omega(t)
          omega'(t) = -b*omega(t) - c*sin(theta(t))
  
      Let `y` be the vector [`theta`, `omega`].  We implement this system
      in python as:
  
      >>> def pend(y, t, b, c):
      ...     theta, omega = y
      ...     dydt = [omega, -b*omega - c*np.sin(theta)]
      ...     return dydt
      ...
      
      We assume the constants are `b` = 0.25 and `c` = 5.0:
  
      >>> b = 0.25
      >>> c = 5.0
  
      For initial conditions, we assume the pendulum is nearly vertical
      with `theta(0)` = `pi` - 0.1, and it initially at rest, so
      `omega(0)` = 0.  Then the vector of initial conditions is
  
      >>> y0 = [np.pi - 0.1, 0.0]
  
      We generate a solution 101 evenly spaced samples in the interval
      0 <= `t` <= 10.  So our array of times is:
  
      >>> t = np.linspace(0, 10, 101)
  
      Call `odeint` to generate the solution.  To pass the parameters
      `b` and `c` to `pend`, we give them to `odeint` using the `args`
      argument.
  
      >>> from scipy.integrate import odeint
      >>> sol = odeint(pend, y0, t, args=(b, c))
  
      The solution is an array with shape (101, 2).  The first column
      is `theta(t)`, and the second is `omega(t)`.  The following code
      plots both components.
  
      >>> import matplotlib.pyplot as plt
      >>> plt.plot(t, sol[:, 0], 'b', label='theta(t)')
      >>> plt.plot(t, sol[:, 1], 'g', label='omega(t)')
      >>> plt.legend(loc='best')
      >>> plt.xlabel('t')
      >>> plt.grid()
      >>> plt.show()
      """
  
      if ml is None:
          ml = -1  # changed to zero inside function call
      if mu is None:
          mu = -1  # changed to zero inside function call
      t = copy(t)
      y0 = copy(y0)
      output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu,
                               full_output, rtol, atol, tcrit, h0, hmax, hmin,
                               ixpr, mxstep, mxhnil, mxordn, mxords)
      if output[-1] < 0:
          warning_msg = _msgs[output[-1]] + " Run with full_output = 1 to get quantitative information."
          warnings.warn(warning_msg, ODEintWarning)
      elif printmessg:
          warning_msg = _msgs[output[-1]]
          warnings.warn(warning_msg, ODEintWarning)
  
      if full_output:
          output[1]['message'] = _msgs[output[-1]]
  
      output = output[:-1]
      if len(output) == 1:
          return output[0]
      else:
          return output
  