text: """
  Functions
  ---------
  .. autosummary::
     :toctree: generated/
  
      fmin_l_bfgs_b
  
  """
  
  ## License for the Python wrapper
  ## ==============================
  
  ## Copyright (c) 2004 David M. Cooke <cookedm@physics.mcmaster.ca>
  
  ## Permission is hereby granted, free of charge, to any person obtaining a
  ## copy of this software and associated documentation files (the "Software"),
  ## to deal in the Software without restriction, including without limitation
  ## the rights to use, copy, modify, merge, publish, distribute, sublicense,
  ## and/or sell copies of the Software, and to permit persons to whom the
  ## Software is furnished to do so, subject to the following conditions:
  
  ## The above copyright notice and this permission notice shall be included in
  ## all copies or substantial portions of the Software.
  
  ## THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  ## IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  ## FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  ## AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  ## LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
  ## FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
  ## DEALINGS IN THE SOFTWARE.
  
  ## Modifications by Travis Oliphant and Enthought, Inc. for inclusion in SciPy
  
  from __future__ import division, print_function, absolute_import
  
  import numpy as np
  from numpy import array, asarray, float64, int32, zeros
  from . import _lbfgsb
  from .optimize import (MemoizeJac, OptimizeResult,
                         _check_unknown_options, wrap_function,
                         _approx_fprime_helper)
  from scipy.sparse.linalg import LinearOperator
  
  __all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct']
  
  
  def fmin_l_bfgs_b(func, x0, fprime=None, args=(),
                    approx_grad=0,
                    bounds=None, m=10, factr=1e7, pgtol=1e-5,
                    epsilon=1e-8,
                    iprint=-1, maxfun=15000, maxiter=15000, disp=None,
                    callback=None, maxls=20):
      """
      Minimize a function func using the L-BFGS-B algorithm.
  
      Parameters
      ----------
      func : callable f(x,*args)
          Function to minimize.
      x0 : ndarray
          Initial guess.
      fprime : callable fprime(x,*args), optional
          The gradient of `func`. If None, then `func` returns the function
          value and the gradient (``f, g = func(x, *args)``), unless
          `approx_grad` is True in which case `func` returns only ``f``.
      args : sequence, optional
          Arguments to pass to `func` and `fprime`.
      approx_grad : bool, optional
          Whether to approximate the gradient numerically (in which case
          `func` returns only the function value).
      bounds : list, optional
          ``(min, max)`` pairs for each element in ``x``, defining
          the bounds on that parameter. Use None or +-inf for one of ``min`` or
          ``max`` when there is no bound in that direction.
      m : int, optional
          The maximum number of variable metric corrections
          used to define the limited memory matrix. (The limited memory BFGS
          method does not store the full hessian but uses this many terms in an
          approximation to it.)
      factr : float, optional
          The iteration stops when
          ``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
          where ``eps`` is the machine precision, which is automatically
          generated by the code. Typical values for `factr` are: 1e12 for
          low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
          high accuracy. See Notes for relationship to `ftol`, which is exposed
          (instead of `factr`) by the `scipy.optimize.minimize` interface to
          L-BFGS-B.
      pgtol : float, optional
          The iteration will stop when
          ``max{|proj g_i | i = 1, ..., n} <= pgtol``
          where ``pg_i`` is the i-th component of the projected gradient.
      epsilon : float, optional
          Step size used when `approx_grad` is True, for numerically
          calculating the gradient
      iprint : int, optional
          Controls the frequency of output. ``iprint < 0`` means no output;
          ``iprint = 0``    print only one line at the last iteration;
          ``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
          ``iprint = 99``   print details of every iteration except n-vectors;
          ``iprint = 100``  print also the changes of active set and final x;
          ``iprint > 100``  print details of every iteration including x and g.
      disp : int, optional
          If zero, then no output. If a positive number, then this over-rides
          `iprint` (i.e., `iprint` gets the value of `disp`).
      maxfun : int, optional
          Maximum number of function evaluations.
      maxiter : int, optional
          Maximum number of iterations.
      callback : callable, optional
          Called after each iteration, as ``callback(xk)``, where ``xk`` is the
          current parameter vector.
      maxls : int, optional
          Maximum number of line search steps (per iteration). Default is 20.
  
      Returns
      -------
      x : array_like
          Estimated position of the minimum.
      f : float
          Value of `func` at the minimum.
      d : dict
          Information dictionary.
  
          * d['warnflag'] is
  
            - 0 if converged,
            - 1 if too many function evaluations or too many iterations,
            - 2 if stopped for another reason, given in d['task']
  
          * d['grad'] is the gradient at the minimum (should be 0 ish)
          * d['funcalls'] is the number of function calls made.
          * d['nit'] is the number of iterations.
  
      See also
      --------
      minimize: Interface to minimization algorithms for multivariate
          functions. See the 'L-BFGS-B' `method` in particular. Note that the
          `ftol` option is made available via that interface, while `factr` is
          provided via this interface, where `factr` is the factor multiplying
          the default machine floating-point precision to arrive at `ftol`:
          ``ftol = factr * numpy.finfo(float).eps``.
  
      Notes
      -----
      License of L-BFGS-B (FORTRAN code):
  
      The version included here (in fortran code) is 3.0
      (released April 25, 2011). It was written by Ciyou Zhu, Richard Byrd,
      and Jorge Nocedal <nocedal@ece.nwu.edu>. It carries the following
      condition for use:
  
      This software is freely available, but we expect that all publications
      describing work using this software, or all commercial products using it,
      quote at least one of the references given below. This software is released
      under the BSD License.
  
      References
      ----------
      * R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
        Constrained Optimization, (1995), SIAM Journal on Scientific and
        Statistical Computing, 16, 5, pp. 1190-1208.
      * C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
        FORTRAN routines for large scale bound constrained optimization (1997),
        ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.
      * J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B,
        FORTRAN routines for large scale bound constrained optimization (2011),
        ACM Transactions on Mathematical Software, 38, 1.
  
      """
      # handle fprime/approx_grad
      if approx_grad:
          fun = func
          jac = None
      elif fprime is None:
          fun = MemoizeJac(func)
          jac = fun.derivative
      else:
          fun = func
          jac = fprime
  
      # build options
      if disp is None:
          disp = iprint
      opts = {'disp': disp,
              'iprint': iprint,
              'maxcor': m,
              'ftol': factr * np.finfo(float).eps,
              'gtol': pgtol,
              'eps': epsilon,
              'maxfun': maxfun,
              'maxiter': maxiter,
              'callback': callback,
              'maxls': maxls}
  
      res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds,
                             **opts)
      d = {'grad': res['jac'],
           'task': res['message'],
           'funcalls': res['nfev'],
           'nit': res['nit'],
           'warnflag': res['status']}
      f = res['fun']
      x = res['x']
  
      return x, f, d
  
  
  def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None,
                       disp=None, maxcor=10, ftol=2.2204460492503131e-09,
                       gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000,
                       iprint=-1, callback=None, maxls=20, **unknown_options):
      """
      Minimize a scalar function of one or more variables using the L-BFGS-B
      algorithm.
  
      Options
      -------
      disp : None or int
          If `disp is None` (the default), then the supplied version of `iprint`
          is used. If `disp is not None`, then it overrides the supplied version
          of `iprint` with the behaviour you outlined.
      maxcor : int
          The maximum number of variable metric corrections used to
          define the limited memory matrix. (The limited memory BFGS
          method does not store the full hessian but uses this many terms
          in an approximation to it.)
      ftol : float
          The iteration stops when ``(f^k -
          f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``.
      gtol : float
          The iteration will stop when ``max{|proj g_i | i = 1, ..., n}
          <= gtol`` where ``pg_i`` is the i-th component of the
          projected gradient.
      eps : float
          Step size used for numerical approximation of the Jacobian.
      maxfun : int
          Maximum number of function evaluations.
      maxiter : int
          Maximum number of iterations.
      iprint : int, optional
          Controls the frequency of output. ``iprint < 0`` means no output;
          ``iprint = 0``    print only one line at the last iteration;
          ``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
          ``iprint = 99``   print details of every iteration except n-vectors;
          ``iprint = 100``  print also the changes of active set and final x;
          ``iprint > 100``  print details of every iteration including x and g.
      callback : callable, optional
          Called after each iteration, as ``callback(xk)``, where ``xk`` is the
          current parameter vector.
      maxls : int, optional
          Maximum number of line search steps (per iteration). Default is 20.
  
      Notes
      -----
      The option `ftol` is exposed via the `scipy.optimize.minimize` interface,
      but calling `scipy.optimize.fmin_l_bfgs_b` directly exposes `factr`. The
      relationship between the two is ``ftol = factr * numpy.finfo(float).eps``.
      I.e., `factr` multiplies the default machine floating-point precision to
      arrive at `ftol`.
  
      """
      _check_unknown_options(unknown_options)
      m = maxcor
      epsilon = eps
      pgtol = gtol
      factr = ftol / np.finfo(float).eps
  
      x0 = asarray(x0).ravel()
      n, = x0.shape
  
      if bounds is None:
          bounds = [(None, None)] * n
      if len(bounds) != n:
          raise ValueError('length of x0 != length of bounds')
      # unbounded variables must use None, not +-inf, for optimizer to work properly
      bounds = [(None if l == -np.inf else l, None if u == np.inf else u) for l, u in bounds]
  
      if disp is not None:
          if disp == 0:
              iprint = -1
          else:
              iprint = disp
  
      n_function_evals, fun = wrap_function(fun, ())
      if jac is None:
          def func_and_grad(x):
              f = fun(x, *args)
              g = _approx_fprime_helper(x, fun, epsilon, args=args, f0=f)
              return f, g
      else:
          def func_and_grad(x):
              f = fun(x, *args)
              g = jac(x, *args)
              return f, g
  
      nbd = zeros(n, int32)
      low_bnd = zeros(n, float64)
      upper_bnd = zeros(n, float64)
      bounds_map = {(None, None): 0,
                    (1, None): 1,
                    (1, 1): 2,
                    (None, 1): 3}
      for i in range(0, n):
          l, u = bounds[i]
          if l is not None:
              low_bnd[i] = l
              l = 1
          if u is not None:
              upper_bnd[i] = u
              u = 1
          nbd[i] = bounds_map[l, u]
  
      if not maxls > 0:
          raise ValueError('maxls must be positive.')
  
      x = array(x0, float64)
      f = array(0.0, float64)
      g = zeros((n,), float64)
      wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64)
      iwa = zeros(3*n, int32)
      task = zeros(1, 'S60')
      csave = zeros(1, 'S60')
      lsave = zeros(4, int32)
      isave = zeros(44, int32)
      dsave = zeros(29, float64)
  
      task[:] = 'START'
  
      n_iterations = 0
  
      while 1:
          # x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \
          _lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr,
                         pgtol, wa, iwa, task, iprint, csave, lsave,
                         isave, dsave, maxls)
          task_str = task.tostring()
          if task_str.startswith(b'FG'):
              # The minimization routine wants f and g at the current x.
              # Note that interruptions due to maxfun are postponed
              # until the completion of the current minimization iteration.
              # Overwrite f and g:
              f, g = func_and_grad(x)
          elif task_str.startswith(b'NEW_X'):
              # new iteration
              n_iterations += 1
              if callback is not None:
                  callback(np.copy(x))
  
              if n_iterations >= maxiter:
                  task[:] = 'STOP: TOTAL NO. of ITERATIONS REACHED LIMIT'
              elif n_function_evals[0] > maxfun:
                  task[:] = ('STOP: TOTAL NO. of f AND g EVALUATIONS '
                             'EXCEEDS LIMIT')
          else:
              break
  
      task_str = task.tostring().strip(b'\x00').strip()
      if task_str.startswith(b'CONV'):
          warnflag = 0
      elif n_function_evals[0] > maxfun or n_iterations >= maxiter:
          warnflag = 1
      else:
          warnflag = 2
  
      # These two portions of the workspace are described in the mainlb
      # subroutine in lbfgsb.f. See line 363.
      s = wa[0: m*n].reshape(m, n)
      y = wa[m*n: 2*m*n].reshape(m, n)
  
      # See lbfgsb.f line 160 for this portion of the workspace.
      # isave(31) = the total number of BFGS updates prior the current iteration;
      n_bfgs_updates = isave[30]
  
      n_corrs = min(n_bfgs_updates, maxcor)
      hess_inv = LbfgsInvHessProduct(s[:n_corrs], y[:n_corrs])
  
      return OptimizeResult(fun=f, jac=g, nfev=n_function_evals[0],
                            nit=n_iterations, status=warnflag, message=task_str,
                            x=x, success=(warnflag == 0), hess_inv=hess_inv)
  
  
  class LbfgsInvHessProduct(LinearOperator):
      """Linear operator for the L-BFGS approximate inverse Hessian.
  
      This operator computes the product of a vector with the approximate inverse
      of the Hessian of the objective function, using the L-BFGS limited
      memory approximation to the inverse Hessian, accumulated during the
      optimization.
  
      Objects of this class implement the ``scipy.sparse.linalg.LinearOperator``
      interface.
  
      Parameters
      ----------
      sk : array_like, shape=(n_corr, n)
          Array of `n_corr` most recent updates to the solution vector.
          (See [1]).
      yk : array_like, shape=(n_corr, n)
          Array of `n_corr` most recent updates to the gradient. (See [1]).
  
      References
      ----------
      .. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited
         storage." Mathematics of computation 35.151 (1980): 773-782.
  
      """
      def __init__(self, sk, yk):
          """Construct the operator."""
          if sk.shape != yk.shape or sk.ndim != 2:
              raise ValueError('sk and yk must have matching shape, (n_corrs, n)')
          n_corrs, n = sk.shape
  
          super(LbfgsInvHessProduct, self).__init__(
              dtype=np.float64, shape=(n, n))
  
          self.sk = sk
          self.yk = yk
          self.n_corrs = n_corrs
          self.rho = 1 / np.einsum('ij,ij->i', sk, yk)
  
      def _matvec(self, x):
          """Efficient matrix-vector multiply with the BFGS matrices.
  
          This calculation is described in Section (4) of [1].
  
          Parameters
          ----------
          x : ndarray
              An array with shape (n,) or (n,1).
  
          Returns
          -------
          y : ndarray
              The matrix-vector product
  
          """
          s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
          q = np.array(x, dtype=self.dtype, copy=True)
          if q.ndim == 2 and q.shape[1] == 1:
              q = q.reshape(-1)
  
          alpha = np.zeros(n_corrs)
  
          for i in range(n_corrs-1, -1, -1):
              alpha[i] = rho[i] * np.dot(s[i], q)
              q = q - alpha[i]*y[i]
  
          r = q
          for i in range(n_corrs):
              beta = rho[i] * np.dot(y[i], r)
              r = r + s[i] * (alpha[i] - beta)
  
          return r
  
      def todense(self):
          """Return a dense array representation of this operator.
  
          Returns
          -------
          arr : ndarray, shape=(n, n)
              An array with the same shape and containing
              the same data represented by this `LinearOperator`.
  
          """
          s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
          I = np.eye(*self.shape, dtype=self.dtype)
          Hk = I
  
          for i in range(n_corrs):
              A1 = I - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i]
              A2 = I - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i]
  
              Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] *
                                                          s[i][np.newaxis, :])
          return Hk
  