class_docstr: Geodesic boundary value problem using an ODE solver.
  
  Parameters
  ----------
  n_nodes : Number of mesh nodes.
  integrator : ScipySolveBVP
      Instance of ScipySolveBVP.
  initialization : callable
      Function to provide initial solution. `f(space, point, base_point)`.
      Defaults to linear initialization.
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docstr: Jacobian of boundary value problem.
  
  Parameters
  ----------
  _ : float
      Unused.
  raveled_state : array-like, shape=[2*dim, n_nodes]
      Vector of state variables (position and speed).
  
  Returns
  -------
  jac : array-like, shape=[dim, dim, n_nodes]
code: def _jacobian(self, _, raveled_state, space):
      """Jacobian of boundary value problem.
  
      Parameters
      ----------
      _ : float
          Unused.
      raveled_state : array-like, shape=[2*dim, n_nodes]
          Vector of state variables (position and speed).
  
      Returns
      -------
      jac : array-like, shape=[dim, dim, n_nodes]
      """
      dim = space.dim
      n_nodes = raveled_state.shape[-1]
      position, velocity = raveled_state[:dim], raveled_state[dim:]
  
      dgamma = space.metric.jacobian_christoffels(
          gs.transpose(position)
      )
  
      df_dposition = -gs.einsum(
          "j...,...ijkl,k...->il...", velocity, dgamma, velocity
      )
  
      gamma = space.metric.christoffels(gs.transpose(position))
      df_dvelocity = -2 * gs.einsum(
          "...ijk,k...->ij...", gamma, velocity
      )
  
      jac_nw = gs.zeros((dim, dim, raveled_state.shape[1]))
      jac_ne = gs.squeeze(
          gs.transpose(gs.tile(gs.eye(dim), (n_nodes, 1, 1)))
      )
      jac_sw = df_dposition
      jac_se = df_dvelocity
      jac = gs.concatenate(
          (
              gs.concatenate((jac_nw, jac_ne), axis=1),
              gs.concatenate((jac_sw, jac_se), axis=1),
          ),
          axis=0,
      )
  
      return jac
  
imports: import math
  from abc import ABC, abstractmethod
  import geomstats.backend
  from geomstats.numerics.bvp import ScipySolveBVP
  from geomstats.numerics.ivp import GSIVPIntegrator
  from geomstats.numerics.optimizers import ScipyMinimize
  from geomstats.vectorization import get_batch_shape
label_desc: Use image node_id 6 for calling the LogODESolver obj's underlying member method code with example usage: obj._jacobian(_, raveled_state, space) and returns: jac