code: def solve_ivp(
      fun,
      t_span,
      y0,
      method="RK45",
      t_eval=None,
      dense_output=False,
      events=None,
      vectorized=False,
      **options,
  ):
      """Solve an initial value problem for a system of ODEs.
  
      This function numerically integrates a system of ordinary differential
      equations given an initial value::
  
          dy / dt = f(t, y)
          y(t0) = y0
  
      Here t is a one-dimensional independent variable (time), y(t) is an
      n-dimensional vector-valued function (state), and an n-dimensional
      vector-valued function f(t, y) determines the differential equations.
      The goal is to find y(t) approximately satisfying the differential
      equations, given an initial value y(t0)=y0.
  
      Some of the solvers support integration in the complex domain, but note that
      for stiff ODE solvers, the right-hand side must be complex-differentiable
      (satisfy Cauchy-Riemann equations [11]_). To solve a problem in the complex
      domain, pass y0 with a complex data type. Another option is always to
      rewrite your problem for real and imaginary parts separately.
  
      Parameters
      ----------
      fun : callable
          Right-hand side of the system. The calling signature is ``fun(t, y)``.
          Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
          It can either have shape (n,); then ``fun`` must return array_like with
          shape (n,). Alternatively it can have shape (n, k); then ``fun``
          must return an array_like with shape (n, k), i.e. each column
          corresponds to a single column in ``y``. The choice between the two
          options is determined by `vectorized` argument (see below). The
          vectorized implementation allows a faster approximation of the Jacobian
          by finite differences (required for stiff solvers).
      t_span : 2-tuple of floats
          Interval of integration (t0, tf). The solver starts with t=t0 and
          integrates until it reaches t=tf.
      y0 : array_like, shape (n,)
          Initial state. For problems in the complex domain, pass `y0` with a
          complex data type (even if the initial guess is purely real).
      method : string or `OdeSolver`, optional
          Integration method to use:
  
              * 'RK45' (default): Explicit Runge-Kutta method of order 5(4) [1]_.
                The error is controlled assuming accuracy of the fourth-order
                method, but steps are taken using the fifth-order accurate formula
                (local extrapolation is done). A quartic interpolation polynomial
                is used for the dense output [2]_. Can be applied in the complex domain.
              * 'RK23': Explicit Runge-Kutta method of order 3(2) [3]_. The error
                is controlled assuming accuracy of the second-order method, but
                steps are taken using the third-order accurate formula (local
                extrapolation is done). A cubic Hermite polynomial is used for the
                dense output. Can be applied in the complex domain.
              * 'Radau': Implicit Runge-Kutta method of the Radau IIA family of
                order 5 [4]_. The error is controlled with a third-order accurate
                embedded formula. A cubic polynomial which satisfies the
                collocation conditions is used for the dense output.
              * 'BDF': Implicit multi-step variable-order (1 to 5) method based
                on a backward differentiation formula for the derivative
                approximation [5]_. The implementation follows the one described
                in [6]_. A quasi-constant step scheme is used and accuracy is
                enhanced using the NDF modification. Can be applied in the complex
                domain.
              * 'LSODA': Adams/BDF method with automatic stiffness detection and
                switching [7]_, [8]_. This is a wrapper of the Fortran solver
                from ODEPACK.
  
          You should use the 'RK45' or 'RK23' method for non-stiff problems and
          'Radau' or 'BDF' for stiff problems [9]_. If not sure, first try to run
          'RK45'. If needs unusually many iterations, diverges, or fails, your
          problem is likely to be stiff and you should use 'Radau' or 'BDF'.
          'LSODA' can also be a good universal choice, but it might be somewhat
          less convenient to work with as it wraps old Fortran code.
  
          You can also pass an arbitrary class derived from `OdeSolver` which
          implements the solver.
      dense_output : bool, optional
          Whether to compute a continuous solution. Default is False.
      t_eval : array_like or None, optional
          Times at which to store the computed solution, must be sorted and lie
          within `t_span`. If None (default), use points selected by the solver.
      events : callable, list of callables or None, optional
          Types of events to track. Each is defined by a continuous function of
          time and state that becomes zero value in case of an event. Each function
          must have the signature ``event(t, y)`` and return a float. The solver will
          find an accurate value of ``t`` at which ``event(t, y(t)) = 0`` using a
          root-finding algorithm. Additionally each ``event`` function might have
          the following attributes:
  
              * terminal: bool, whether to terminate integration if this
                event occurs. Implicitly False if not assigned.
              * direction: float, direction of a zero crossing. If `direction`
                is positive, `event` must go from negative to positive, and
                vice versa if `direction` is negative. If 0, then either direction
                will count. Implicitly 0 if not assigned.
  
          You can assign attributes like ``event.terminal = True`` to any
          function in Python. If None (default), events won't be tracked.
      vectorized : bool, optional
          Whether `fun` is implemented in a vectorized fashion. Default is False.
      options
          Options passed to a chosen solver. All options available for already
          implemented solvers are listed below.
      first_step : float or None, optional
          Initial step size. Default is ``None`` which means that the algorithm
          should choose.
      max_step : float, optional
          Maximum allowed step size. Default is np.inf, i.e. the step size is not
          bounded and determined solely by the solver.
      rtol, atol : float and array_like, optional
          Relative and absolute tolerances. The solver keeps the local error
          estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
          relative accuracy (number of correct digits). But if a component of `y`
          is approximately below `atol`, the error only needs to fall within
          the same `atol` threshold, and the number of correct digits is not
          guaranteed. If components of y have different scales, it might be
          beneficial to set different `atol` values for different components by
          passing array_like with shape (n,) for `atol`. Default values are
          1e-3 for `rtol` and 1e-6 for `atol`.
      jac : {None, array_like, sparse_matrix, callable}, optional
          Jacobian matrix of the right-hand side of the system with respect to
          y, required by the 'Radau', 'BDF' and 'LSODA' method. The Jacobian matrix
          has shape (n, n) and its element (i, j) is equal to ``d f_i / d y_j``.
          There are three ways to define the Jacobian:
  
              * If array_like or sparse_matrix, the Jacobian is assumed to
                be constant. Not supported by 'LSODA'.
              * If callable, the Jacobian is assumed to depend on both
                t and y; it will be called as ``jac(t, y)`` as necessary.
                For the 'Radau' and 'BDF' methods, the return value might be a
                sparse matrix.
              * If None (default), the Jacobian will be approximated by
                finite differences.
  
          It is generally recommended to provide the Jacobian rather than
          relying on a finite-difference approximation.
      jac_sparsity : {None, array_like, sparse matrix}, optional
          Defines a sparsity structure of the Jacobian matrix for a
          finite-difference approximation. Its shape must be (n, n). This argument
          is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
          elements in *each* row, providing the sparsity structure will greatly
          speed up the computations [10]_. A zero entry means that a corresponding
          element in the Jacobian is always zero. If None (default), the Jacobian
          is assumed to be dense.
          Not supported by 'LSODA', see `lband` and `uband` instead.
      lband, uband : int or None
          Parameters defining the bandwidth of the Jacobian for the 'LSODA' method,
          i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. Setting
          these requires your jac routine to return the Jacobian in the packed format:
          the returned array must have ``n`` columns and ``uband + lband + 1``
          rows in which Jacobian diagonals are written. Specifically
          ``jac_packed[uband + i - j , j] = jac[i, j]``. The same format is used
          in `scipy.linalg.solve_banded` (check for an illustration).
          These parameters can be also used with ``jac=None`` to reduce the
          number of Jacobian elements estimated by finite differences.
      min_step : float, optional
          The minimum allowed step size for 'LSODA' method.
          By default `min_step` is zero.
  
      Returns
      -------
      Bunch object with the following fields defined:
      t : ndarray, shape (n_points,)
          Time points.
      y : ndarray, shape (n, n_points)
          Values of the solution at `t`.
      sol : `OdeSolution` or None
          Found solution as `OdeSolution` instance; None if `dense_output` was
          set to False.
      t_events : list of ndarray or None
          Contains for each event type a list of arrays at which an event of
          that type event was detected. None if `events` was None.
      nfev : int
          Number of evaluations of the right-hand side.
      njev : int
          Number of evaluations of the Jacobian.
      nlu : int
          Number of LU decompositions.
      status : int
          Reason for algorithm termination:
  
              * -1: Integration step failed.
              *  0: The solver successfully reached the end of `tspan`.
              *  1: A termination event occurred.
  
      message : string
          Human-readable description of the termination reason.
      success : bool
          True if the solver reached the interval end or a termination event
          occurred (``status >= 0``).
  
      References
      ----------
      .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
             formulae", Journal of Computational and Applied Mathematics, Vol. 6,
             No. 1, pp. 19-26, 1980.
      .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
             of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
      .. [3] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
             Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
      .. [4] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
             Stiff and Differential-Algebraic Problems", Sec. IV.8.
      .. [5] `Backward Differentiation Formula
              <https://en.wikipedia.org/wiki/Backward_differentiation_formula>`_
              on Wikipedia.
      .. [6] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
             COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
      .. [7] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
             Solvers," IMACS Transactions on Scientific Computation, Vol 1.,
             pp. 55-64, 1983.
      .. [8] L. Petzold, "Automatic selection of methods for solving stiff and
             nonstiff systems of ordinary differential equations", SIAM Journal
             on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148,
             1983.
      .. [9] `Stiff equation <https://en.wikipedia.org/wiki/Stiff_equation>`_ on
             Wikipedia.
      .. [10] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
              sparse Jacobian matrices", Journal of the Institute of Mathematics
              and its Applications, 13, pp. 117-120, 1974.
      .. [11] `Cauchy-Riemann equations
               <https://en.wikipedia.org/wiki/Cauchy-Riemann_equations>`_ on
               Wikipedia.
  
      Examples
      --------
      Basic exponential decay showing automatically chosen time points.
  
      >>> from scipy.integrate import solve_ivp
      >>> def exponential_decay(t, y): return -0.5 * y
      >>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8])
      >>> print(sol.t)
      [  0.           0.11487653   1.26364188   3.06061781   4.85759374
         6.65456967   8.4515456   10.        ]
      >>> print(sol.y)
      [[2.         1.88836035 1.06327177 0.43319312 0.17648948 0.0719045
        0.02929499 0.01350938]
       [4.         3.7767207  2.12654355 0.86638624 0.35297895 0.143809
        0.05858998 0.02701876]
       [8.         7.5534414  4.25308709 1.73277247 0.7059579  0.287618
        0.11717996 0.05403753]]
  
      Specifying points where the solution is desired.
  
      >>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8],
      ...                 t_eval=[0, 1, 2, 4, 10])
      >>> print(sol.t)
      [ 0  1  2  4 10]
      >>> print(sol.y)
      [[2.         1.21305369 0.73534021 0.27066736 0.01350938]
       [4.         2.42610739 1.47068043 0.54133472 0.02701876]
       [8.         4.85221478 2.94136085 1.08266944 0.05403753]]
  
      Cannon fired upward with terminal event upon impact. The ``terminal`` and
      ``direction`` fields of an event are applied by monkey patching a function.
      Here ``y[0]`` is position and ``y[1]`` is velocity. The projectile starts at
      position 0 with velocity +10. Note that the integration never reaches t=100
      because the event is terminal.
  
      >>> def upward_cannon(t, y): return [y[1], -0.5]
      >>> def hit_ground(t, y): return y[1]
      >>> hit_ground.terminal = True
      >>> hit_ground.direction = -1
      >>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground)
      >>> print(sol.t_events)
      [array([ 20.])]
      >>> print(sol.t)
      [0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
       1.11088891e-01 1.11098890e+00 1.11099890e+01 2.00000000e+01]
      """
      if method not in METHODS and not (
          inspect.isclass(method) and issubclass(method, OdeSolver)
      ):
          raise ValueError(
              "`method` must be one of {} or OdeSolver class.".format(
                  METHODS
              )
          )
  
      t0, tf = float(t_span[0]), float(t_span[1])
  
      if t_eval is not None:
          t_eval = np.asarray(t_eval)
          if t_eval.ndim != 1:
              raise ValueError("`t_eval` must be 1-dimensional.")
  
          if np.any(t_eval < min(t0, tf)) or np.any(
              t_eval > max(t0, tf)
          ):
              raise ValueError(
                  "Values in `t_eval` are not within `t_span`."
              )
  
          d = np.diff(t_eval)
          if tf > t0 and np.any(d <= 0) or tf < t0 and np.any(d >= 0):
              raise ValueError(
                  "Values in `t_eval` are not properly sorted."
              )
  
          if tf > t0:
              t_eval_i = 0
          else:
              # Make order of t_eval decreasing to use np.searchsorted.
              t_eval = t_eval[::-1]
              # This will be an upper bound for slices.
              t_eval_i = t_eval.shape[0]
  
      if method in METHODS:
          method = METHODS[method]
  
      solver = method(fun, t0, y0, tf, vectorized=vectorized, **options)
  
      if t_eval is None:
          ts = [t0]
          ys = [y0]
      elif t_eval is not None and dense_output:
          ts = []
          ti = [t0]
          ys = []
      else:
          ts = []
          ys = []
  
      interpolants = []
  
      events, is_terminal, event_dir = prepare_events(events)
  
      if events is not None:
          g = [event(t0, y0) for event in events]
          t_events = [[] for _ in range(len(events))]
      else:
          t_events = None
  
      status = None
      while status is None:
          message = solver.step()
  
          if solver.status == "finished":
              status = 0
          elif solver.status == "failed":
              status = -1
              break
  
          t_old = solver.t_old
          t = solver.t
          y = solver.y
  
          if dense_output:
              sol = solver.dense_output()
              interpolants.append(sol)
          else:
              sol = None
  
          if events is not None:
              g_new = [event(t, y) for event in events]
              active_events = find_active_events(g, g_new, event_dir)
              if active_events.size > 0:
                  if sol is None:
                      sol = solver.dense_output()
  
                  root_indices, roots, terminate = handle_events(
                      sol, events, active_events, is_terminal, t_old, t
                  )
  
                  for e, te in zip(root_indices, roots):
                      t_events[e].append(te)
  
                  if terminate:
                      status = 1
                      t = roots[-1]
                      y = sol(t)
  
              g = g_new
  
          if t_eval is None:
              ts.append(t)
              ys.append(y)
          else:
              # The value in t_eval equal to t will be included.
              if solver.direction > 0:
                  t_eval_i_new = np.searchsorted(
                      t_eval, t, side="right"
                  )
                  t_eval_step = t_eval[t_eval_i:t_eval_i_new]
              else:
                  t_eval_i_new = np.searchsorted(t_eval, t, side="left")
                  # It has to be done with two slice operations, because
                  # you can't slice to 0-th element inclusive using backward
                  # slicing.
                  t_eval_step = t_eval[t_eval_i_new:t_eval_i][::-1]
  
              if t_eval_step.size > 0:
                  if sol is None:
                      sol = solver.dense_output()
                  ts.append(t_eval_step)
                  ys.append(sol(t_eval_step))
                  t_eval_i = t_eval_i_new
  
          if t_eval is not None and dense_output:
              ti.append(t)
  
      message = MESSAGES.get(status, message)
  
      if t_events is not None:
          t_events = [np.asarray(te) for te in t_events]
  
      if t_eval is None:
          ts = np.array(ts)
          ys = np.vstack(ys).T
      else:
          ts = np.hstack(ts)
          ys = np.hstack(ys)
  
      if dense_output:
          if t_eval is None:
              sol = OdeSolution(ts, interpolants)
          else:
              sol = OdeSolution(ti, interpolants)
      else:
          sol = None
  
      return OdeResult(
          t=ts,
          y=ys,
          sol=sol,
          t_events=t_events,
          nfev=solver.nfev,
          njev=solver.njev,
          nlu=solver.nlu,
          status=status,
          message=message,
          success=status >= 0,
      )
  
file_path: catboost/contrib/python/scipy/py2/scipy/integrate/_ivp/ivp.py
imports: from __future__ import division, print_function, absolute_import
  import inspect
  import numpy
  from .bdf import BDF
  from .radau import Radau
  from .rk import RK23, RK45
  from .lsoda import LSODA
  from scipy.optimize import OptimizeResult
  from .common import EPS, OdeSolution
  from .base import OdeSolver
label_desc: Use image node_id 5 for calling a global function with example usage: solve_ivp(fun, t_span, y0, method, t_eval, dense_output, events, vectorized) and returns: OdeResult