content: """Lite version of scipy.linalg.
  
  Notes
  -----
  This module is a lite version of the linalg.py module in SciPy which
  contains high-level Python interface to the LAPACK library.  The lite
  version only accesses the following LAPACK functions: dgesv, zgesv,
  dgeev, zgeev, dgesdd, zgesdd, dgelsd, zgelsd, dsyevd, zheevd, dgetrf,
  zgetrf, dpotrf, zpotrf, dgeqrf, zgeqrf, zungqr, dorgqr.
  """
  
  __all__ = ['matrix_power', 'solve', 'tensorsolve', 'tensorinv', 'inv',
             'cholesky', 'eigvals', 'eigvalsh', 'pinv', 'slogdet', 'det',
             'svd', 'eig', 'eigh','lstsq', 'norm', 'qr', 'cond', 'matrix_rank',
             'LinAlgError']
  
  import sys
  from numpy.core import array, asarray, zeros, empty, transpose, \
          intc, single, double, csingle, cdouble, inexact, complexfloating, \
          newaxis, ravel, all, Inf, dot, add, multiply, identity, sqrt, \
          maximum, flatnonzero, diagonal, arange, fastCopyAndTranspose, sum, \
          isfinite, size, finfo, absolute, log, exp
  from numpy.lib import triu
  from numpy.linalg import lapack_lite
  from numpy.matrixlib.defmatrix import matrix_power
  from numpy.compat import asbytes
  
  # For Python2/3 compatibility
  _N = asbytes('N')
  _V = asbytes('V')
  _A = asbytes('A')
  _S = asbytes('S')
  _L = asbytes('L')
  
  fortran_int = intc
  
  # Error object
  class LinAlgError(Exception):
      """
      Generic Python-exception-derived object raised by linalg functions.
  
      General purpose exception class, derived from Python's exception.Exception
      class, programmatically raised in linalg functions when a Linear
      Algebra-related condition would prevent further correct execution of the
      function.
  
      Parameters
      ----------
      None
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> LA.inv(np.zeros((2,2)))
      Traceback (most recent call last):
        File "<stdin>", line 1, in <module>
        File "...linalg.py", line 350,
          in inv return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))
        File "...linalg.py", line 249,
          in solve
          raise LinAlgError, 'Singular matrix'
      numpy.linalg.linalg.LinAlgError: Singular matrix
  
      """
      pass
  
  def _makearray(a):
      new = asarray(a)
      wrap = getattr(a, "__array_prepare__", new.__array_wrap__)
      return new, wrap
  
  def isComplexType(t):
      return issubclass(t, complexfloating)
  
  _real_types_map = {single : single,
                     double : double,
                     csingle : single,
                     cdouble : double}
  
  _complex_types_map = {single : csingle,
                        double : cdouble,
                        csingle : csingle,
                        cdouble : cdouble}
  
  def _realType(t, default=double):
      return _real_types_map.get(t, default)
  
  def _complexType(t, default=cdouble):
      return _complex_types_map.get(t, default)
  
  def _linalgRealType(t):
      """Cast the type t to either double or cdouble."""
      return double
  
  _complex_types_map = {single : csingle,
                        double : cdouble,
                        csingle : csingle,
                        cdouble : cdouble}
  
  def _commonType(*arrays):
      # in lite version, use higher precision (always double or cdouble)
      result_type = single
      is_complex = False
      for a in arrays:
          if issubclass(a.dtype.type, inexact):
              if isComplexType(a.dtype.type):
                  is_complex = True
              rt = _realType(a.dtype.type, default=None)
              if rt is None:
                  # unsupported inexact scalar
                  raise TypeError("array type %s is unsupported in linalg" %
                          (a.dtype.name,))
          else:
              rt = double
          if rt is double:
              result_type = double
      if is_complex:
          t = cdouble
          result_type = _complex_types_map[result_type]
      else:
          t = double
      return t, result_type
  
  # _fastCopyAndTranpose assumes the input is 2D (as all the calls in here are).
  
  _fastCT = fastCopyAndTranspose
  
  def _to_native_byte_order(*arrays):
      ret = []
      for arr in arrays:
          if arr.dtype.byteorder not in ('=', '|'):
              ret.append(asarray(arr, dtype=arr.dtype.newbyteorder('=')))
          else:
              ret.append(arr)
      if len(ret) == 1:
          return ret[0]
      else:
          return ret
  
  def _fastCopyAndTranspose(type, *arrays):
      cast_arrays = ()
      for a in arrays:
          if a.dtype.type is type:
              cast_arrays = cast_arrays + (_fastCT(a),)
          else:
              cast_arrays = cast_arrays + (_fastCT(a.astype(type)),)
      if len(cast_arrays) == 1:
          return cast_arrays[0]
      else:
          return cast_arrays
  
  def _assertRank2(*arrays):
      for a in arrays:
          if len(a.shape) != 2:
              raise LinAlgError, '%d-dimensional array given. Array must be \
              two-dimensional' % len(a.shape)
  
  def _assertSquareness(*arrays):
      for a in arrays:
          if max(a.shape) != min(a.shape):
              raise LinAlgError, 'Array must be square'
  
  def _assertFinite(*arrays):
      for a in arrays:
          if not (isfinite(a).all()):
              raise LinAlgError, "Array must not contain infs or NaNs"
  
  def _assertNonEmpty(*arrays):
      for a in arrays:
          if size(a) == 0:
              raise LinAlgError("Arrays cannot be empty")
  
  
  # Linear equations
  
  def tensorsolve(a, b, axes=None):
      """
      Solve the tensor equation ``a x = b`` for x.
  
      It is assumed that all indices of `x` are summed over in the product,
      together with the rightmost indices of `a`, as is done in, for example,
      ``tensordot(a, x, axes=len(b.shape))``.
  
      Parameters
      ----------
      a : array_like
          Coefficient tensor, of shape ``b.shape + Q``. `Q`, a tuple, equals
          the shape of that sub-tensor of `a` consisting of the appropriate
          number of its rightmost indices, and must be such that
         ``prod(Q) == prod(b.shape)`` (in which sense `a` is said to be
         'square').
      b : array_like
          Right-hand tensor, which can be of any shape.
      axes : tuple of ints, optional
          Axes in `a` to reorder to the right, before inversion.
          If None (default), no reordering is done.
  
      Returns
      -------
      x : ndarray, shape Q
  
      Raises
      ------
      LinAlgError
          If `a` is singular or not 'square' (in the above sense).
  
      See Also
      --------
      tensordot, tensorinv
  
      Examples
      --------
      >>> a = np.eye(2*3*4)
      >>> a.shape = (2*3, 4, 2, 3, 4)
      >>> b = np.random.randn(2*3, 4)
      >>> x = np.linalg.tensorsolve(a, b)
      >>> x.shape
      (2, 3, 4)
      >>> np.allclose(np.tensordot(a, x, axes=3), b)
      True
  
      """
      a,wrap = _makearray(a)
      b = asarray(b)
      an = a.ndim
  
      if axes is not None:
          allaxes = range(0, an)
          for k in axes:
              allaxes.remove(k)
              allaxes.insert(an, k)
          a = a.transpose(allaxes)
  
      oldshape = a.shape[-(an-b.ndim):]
      prod = 1
      for k in oldshape:
          prod *= k
  
      a = a.reshape(-1, prod)
      b = b.ravel()
      res = wrap(solve(a, b))
      res.shape = oldshape
      return res
  
  def solve(a, b):
      """
      Solve a linear matrix equation, or system of linear scalar equations.
  
      Computes the "exact" solution, `x`, of the well-determined, i.e., full
      rank, linear matrix equation `ax = b`.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          Coefficient matrix.
      b : array_like, shape (M,) or (M, N)
          Ordinate or "dependent variable" values.
  
      Returns
      -------
      x : ndarray, shape (M,) or (M, N) depending on b
          Solution to the system a x = b
  
      Raises
      ------
      LinAlgError
          If `a` is singular or not square.
  
      Notes
      -----
      `solve` is a wrapper for the LAPACK routines `dgesv`_ and
      `zgesv`_, the former being used if `a` is real-valued, the latter if
      it is complex-valued.  The solution to the system of linear equations
      is computed using an LU decomposition [1]_ with partial pivoting and
      row interchanges.
  
      .. _dgesv: http://www.netlib.org/lapack/double/dgesv.f
  
      .. _zgesv: http://www.netlib.org/lapack/complex16/zgesv.f
  
      `a` must be square and of full-rank, i.e., all rows (or, equivalently,
      columns) must be linearly independent; if either is not true, use
      `lstsq` for the least-squares best "solution" of the
      system/equation.
  
      References
      ----------
      .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
             FL, Academic Press, Inc., 1980, pg. 22.
  
      Examples
      --------
      Solve the system of equations ``3 * x0 + x1 = 9`` and ``x0 + 2 * x1 = 8``:
  
      >>> a = np.array([[3,1], [1,2]])
      >>> b = np.array([9,8])
      >>> x = np.linalg.solve(a, b)
      >>> x
      array([ 2.,  3.])
  
      Check that the solution is correct:
  
      >>> (np.dot(a, x) == b).all()
      True
  
      """
      a, _ = _makearray(a)
      b, wrap = _makearray(b)
      one_eq = len(b.shape) == 1
      if one_eq:
          b = b[:, newaxis]
      _assertRank2(a, b)
      _assertSquareness(a)
      n_eq = a.shape[0]
      n_rhs = b.shape[1]
      if n_eq != b.shape[0]:
          raise LinAlgError, 'Incompatible dimensions'
      t, result_t = _commonType(a, b)
  #    lapack_routine = _findLapackRoutine('gesv', t)
      if isComplexType(t):
          lapack_routine = lapack_lite.zgesv
      else:
          lapack_routine = lapack_lite.dgesv
      a, b = _fastCopyAndTranspose(t, a, b)
      a, b = _to_native_byte_order(a, b)
      pivots = zeros(n_eq, fortran_int)
      results = lapack_routine(n_eq, n_rhs, a, n_eq, pivots, b, n_eq, 0)
      if results['info'] > 0:
          raise LinAlgError, 'Singular matrix'
      if one_eq:
          return wrap(b.ravel().astype(result_t))
      else:
          return wrap(b.transpose().astype(result_t))
  
  
  def tensorinv(a, ind=2):
      """
      Compute the 'inverse' of an N-dimensional array.
  
      The result is an inverse for `a` relative to the tensordot operation
      ``tensordot(a, b, ind)``, i. e., up to floating-point accuracy,
      ``tensordot(tensorinv(a), a, ind)`` is the "identity" tensor for the
      tensordot operation.
  
      Parameters
      ----------
      a : array_like
          Tensor to 'invert'. Its shape must be 'square', i. e.,
          ``prod(a.shape[:ind]) == prod(a.shape[ind:])``.
      ind : int, optional
          Number of first indices that are involved in the inverse sum.
          Must be a positive integer, default is 2.
  
      Returns
      -------
      b : ndarray
          `a`'s tensordot inverse, shape ``a.shape[:ind] + a.shape[ind:]``.
  
      Raises
      ------
      LinAlgError
          If `a` is singular or not 'square' (in the above sense).
  
      See Also
      --------
      tensordot, tensorsolve
  
      Examples
      --------
      >>> a = np.eye(4*6)
      >>> a.shape = (4, 6, 8, 3)
      >>> ainv = np.linalg.tensorinv(a, ind=2)
      >>> ainv.shape
      (8, 3, 4, 6)
      >>> b = np.random.randn(4, 6)
      >>> np.allclose(np.tensordot(ainv, b), np.linalg.tensorsolve(a, b))
      True
  
      >>> a = np.eye(4*6)
      >>> a.shape = (24, 8, 3)
      >>> ainv = np.linalg.tensorinv(a, ind=1)
      >>> ainv.shape
      (8, 3, 24)
      >>> b = np.random.randn(24)
      >>> np.allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b))
      True
  
      """
      a = asarray(a)
      oldshape = a.shape
      prod = 1
      if ind > 0:
          invshape = oldshape[ind:] + oldshape[:ind]
          for k in oldshape[ind:]:
              prod *= k
      else:
          raise ValueError, "Invalid ind argument."
      a = a.reshape(prod, -1)
      ia = inv(a)
      return ia.reshape(*invshape)
  
  
  # Matrix inversion
  
  def inv(a):
      """
      Compute the (multiplicative) inverse of a matrix.
  
      Given a square matrix `a`, return the matrix `ainv` satisfying
      ``dot(a, ainv) = dot(ainv, a) = eye(a.shape[0])``.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          Matrix to be inverted.
  
      Returns
      -------
      ainv : ndarray or matrix, shape (M, M)
          (Multiplicative) inverse of the matrix `a`.
  
      Raises
      ------
      LinAlgError
          If `a` is singular or not square.
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> a = np.array([[1., 2.], [3., 4.]])
      >>> ainv = LA.inv(a)
      >>> np.allclose(np.dot(a, ainv), np.eye(2))
      True
      >>> np.allclose(np.dot(ainv, a), np.eye(2))
      True
  
      If a is a matrix object, then the return value is a matrix as well:
  
      >>> ainv = LA.inv(np.matrix(a))
      >>> ainv
      matrix([[-2. ,  1. ],
              [ 1.5, -0.5]])
  
      """
      a, wrap = _makearray(a)
      return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))
  
  
  # Cholesky decomposition
  
  def cholesky(a):
      """
      Cholesky decomposition.
  
      Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`,
      where `L` is lower-triangular and .H is the conjugate transpose operator
      (which is the ordinary transpose if `a` is real-valued).  `a` must be
      Hermitian (symmetric if real-valued) and positive-definite.  Only `L` is
      actually returned.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          Hermitian (symmetric if all elements are real), positive-definite
          input matrix.
  
      Returns
      -------
      L : ndarray, or matrix object if `a` is, shape (M, M)
          Lower-triangular Cholesky factor of a.
  
      Raises
      ------
      LinAlgError
         If the decomposition fails, for example, if `a` is not
         positive-definite.
  
      Notes
      -----
      The Cholesky decomposition is often used as a fast way of solving
  
      .. math:: A \\mathbf{x} = \\mathbf{b}
  
      (when `A` is both Hermitian/symmetric and positive-definite).
  
      First, we solve for :math:`\\mathbf{y}` in
  
      .. math:: L \\mathbf{y} = \\mathbf{b},
  
      and then for :math:`\\mathbf{x}` in
  
      .. math:: L.H \\mathbf{x} = \\mathbf{y}.
  
      Examples
      --------
      >>> A = np.array([[1,-2j],[2j,5]])
      >>> A
      array([[ 1.+0.j,  0.-2.j],
             [ 0.+2.j,  5.+0.j]])
      >>> L = np.linalg.cholesky(A)
      >>> L
      array([[ 1.+0.j,  0.+0.j],
             [ 0.+2.j,  1.+0.j]])
      >>> np.dot(L, L.T.conj()) # verify that L * L.H = A
      array([[ 1.+0.j,  0.-2.j],
             [ 0.+2.j,  5.+0.j]])
      >>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
      >>> np.linalg.cholesky(A) # an ndarray object is returned
      array([[ 1.+0.j,  0.+0.j],
             [ 0.+2.j,  1.+0.j]])
      >>> # But a matrix object is returned if A is a matrix object
      >>> LA.cholesky(np.matrix(A))
      matrix([[ 1.+0.j,  0.+0.j],
              [ 0.+2.j,  1.+0.j]])
  
      """
      a, wrap = _makearray(a)
      _assertRank2(a)
      _assertSquareness(a)
      t, result_t = _commonType(a)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      m = a.shape[0]
      n = a.shape[1]
      if isComplexType(t):
          lapack_routine = lapack_lite.zpotrf
      else:
          lapack_routine = lapack_lite.dpotrf
      results = lapack_routine(_L, n, a, m, 0)
      if results['info'] > 0:
          raise LinAlgError, 'Matrix is not positive definite - \
          Cholesky decomposition cannot be computed'
      s = triu(a, k=0).transpose()
      if (s.dtype != result_t):
          s = s.astype(result_t)
      return wrap(s)
  
  # QR decompostion
  
  def qr(a, mode='full'):
      """
      Compute the qr factorization of a matrix.
  
      Factor the matrix `a` as *qr*, where `q` is orthonormal and `r` is
      upper-triangular.
  
      Parameters
      ----------
      a : array_like
          Matrix to be factored, of shape (M, N).
      mode : {'full', 'r', 'economic'}, optional
          Specifies the values to be returned. 'full' is the default.
          Economic mode is slightly faster then 'r' mode if only `r` is needed.
  
      Returns
      -------
      q : ndarray of float or complex, optional
          The orthonormal matrix, of shape (M, K). Only returned if
          ``mode='full'``.
      r : ndarray of float or complex, optional
          The upper-triangular matrix, of shape (K, N) with K = min(M, N).
          Only returned when ``mode='full'`` or ``mode='r'``.
      a2 : ndarray of float or complex, optional
          Array of shape (M, N), only returned when ``mode='economic``'.
          The  diagonal and the upper triangle of `a2` contains `r`, while
          the rest of the matrix is undefined.
  
      Raises
      ------
      LinAlgError
          If factoring fails.
  
      Notes
      -----
      This is an interface to the LAPACK routines dgeqrf, zgeqrf,
      dorgqr, and zungqr.
  
      For more information on the qr factorization, see for example:
      http://en.wikipedia.org/wiki/QR_factorization
  
      Subclasses of `ndarray` are preserved, so if `a` is of type `matrix`,
      all the return values will be matrices too.
  
      Examples
      --------
      >>> a = np.random.randn(9, 6)
      >>> q, r = np.linalg.qr(a)
      >>> np.allclose(a, np.dot(q, r))  # a does equal qr
      True
      >>> r2 = np.linalg.qr(a, mode='r')
      >>> r3 = np.linalg.qr(a, mode='economic')
      >>> np.allclose(r, r2)  # mode='r' returns the same r as mode='full'
      True
      >>> # But only triu parts are guaranteed equal when mode='economic'
      >>> np.allclose(r, np.triu(r3[:6,:6], k=0))
      True
  
      Example illustrating a common use of `qr`: solving of least squares
      problems
  
      What are the least-squares-best `m` and `y0` in ``y = y0 + mx`` for
      the following data: {(0,1), (1,0), (1,2), (2,1)}. (Graph the points
      and you'll see that it should be y0 = 0, m = 1.)  The answer is provided
      by solving the over-determined matrix equation ``Ax = b``, where::
  
        A = array([[0, 1], [1, 1], [1, 1], [2, 1]])
        x = array([[y0], [m]])
        b = array([[1], [0], [2], [1]])
  
      If A = qr such that q is orthonormal (which is always possible via
      Gram-Schmidt), then ``x = inv(r) * (q.T) * b``.  (In numpy practice,
      however, we simply use `lstsq`.)
  
      >>> A = np.array([[0, 1], [1, 1], [1, 1], [2, 1]])
      >>> A
      array([[0, 1],
             [1, 1],
             [1, 1],
             [2, 1]])
      >>> b = np.array([1, 0, 2, 1])
      >>> q, r = LA.qr(A)
      >>> p = np.dot(q.T, b)
      >>> np.dot(LA.inv(r), p)
      array([  1.1e-16,   1.0e+00])
  
      """
      a, wrap = _makearray(a)
      _assertRank2(a)
      m, n = a.shape
      t, result_t = _commonType(a)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      mn = min(m, n)
      tau = zeros((mn,), t)
      if isComplexType(t):
          lapack_routine = lapack_lite.zgeqrf
          routine_name = 'zgeqrf'
      else:
          lapack_routine = lapack_lite.dgeqrf
          routine_name = 'dgeqrf'
  
      # calculate optimal size of work data 'work'
      lwork = 1
      work = zeros((lwork,), t)
      results = lapack_routine(m, n, a, m, tau, work, -1, 0)
      if results['info'] != 0:
          raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
  
      # do qr decomposition
      lwork = int(abs(work[0]))
      work = zeros((lwork,), t)
      results = lapack_routine(m, n, a, m, tau, work, lwork, 0)
  
      if results['info'] != 0:
          raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
  
      #  economic mode. Isn't actually economic.
      if mode[0] == 'e':
          if t != result_t :
              a = a.astype(result_t)
          return a.T
  
      #  generate r
      r = _fastCopyAndTranspose(result_t, a[:,:mn])
      for i in range(mn):
          r[i,:i].fill(0.0)
  
      #  'r'-mode, that is, calculate only r
      if mode[0] == 'r':
          return r
  
      #  from here on: build orthonormal matrix q from a
  
      if isComplexType(t):
          lapack_routine = lapack_lite.zungqr
          routine_name = 'zungqr'
      else:
          lapack_routine = lapack_lite.dorgqr
          routine_name = 'dorgqr'
  
      # determine optimal lwork
      lwork = 1
      work = zeros((lwork,), t)
      results = lapack_routine(m, mn, mn, a, m, tau, work, -1, 0)
      if results['info'] != 0:
          raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
  
      # compute q
      lwork = int(abs(work[0]))
      work = zeros((lwork,), t)
      results = lapack_routine(m, mn, mn, a, m, tau, work, lwork, 0)
      if results['info'] != 0:
          raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
  
      q = _fastCopyAndTranspose(result_t, a[:mn,:])
  
      return wrap(q), wrap(r)
  
  
  # Eigenvalues
  
  
  def eigvals(a):
      """
      Compute the eigenvalues of a general matrix.
  
      Main difference between `eigvals` and `eig`: the eigenvectors aren't
      returned.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          A complex- or real-valued matrix whose eigenvalues will be computed.
  
      Returns
      -------
      w : ndarray, shape (M,)
          The eigenvalues, each repeated according to its multiplicity.
          They are not necessarily ordered, nor are they necessarily
          real for real matrices.
  
      Raises
      ------
      LinAlgError
          If the eigenvalue computation does not converge.
  
      See Also
      --------
      eig : eigenvalues and right eigenvectors of general arrays
      eigvalsh : eigenvalues of symmetric or Hermitian arrays.
      eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays.
  
      Notes
      -----
      This is a simple interface to the LAPACK routines dgeev and zgeev
      that sets those routines' flags to return only the eigenvalues of
      general real and complex arrays, respectively.
  
      Examples
      --------
      Illustration, using the fact that the eigenvalues of a diagonal matrix
      are its diagonal elements, that multiplying a matrix on the left
      by an orthogonal matrix, `Q`, and on the right by `Q.T` (the transpose
      of `Q`), preserves the eigenvalues of the "middle" matrix.  In other words,
      if `Q` is orthogonal, then ``Q * A * Q.T`` has the same eigenvalues as
      ``A``:
  
      >>> from numpy import linalg as LA
      >>> x = np.random.random()
      >>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]])
      >>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :])
      (1.0, 1.0, 0.0)
  
      Now multiply a diagonal matrix by Q on one side and by Q.T on the other:
  
      >>> D = np.diag((-1,1))
      >>> LA.eigvals(D)
      array([-1.,  1.])
      >>> A = np.dot(Q, D)
      >>> A = np.dot(A, Q.T)
      >>> LA.eigvals(A)
      array([ 1., -1.])
  
      """
      a, wrap = _makearray(a)
      _assertRank2(a)
      _assertSquareness(a)
      _assertFinite(a)
      t, result_t = _commonType(a)
      real_t = _linalgRealType(t)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      n = a.shape[0]
      dummy = zeros((1,), t)
      if isComplexType(t):
          lapack_routine = lapack_lite.zgeev
          w = zeros((n,), t)
          rwork = zeros((n,), real_t)
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _N, n, a, n, w,
                                   dummy, 1, dummy, 1, work, -1, rwork, 0)
          lwork = int(abs(work[0]))
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _N, n, a, n, w,
                                   dummy, 1, dummy, 1, work, lwork, rwork, 0)
      else:
          lapack_routine = lapack_lite.dgeev
          wr = zeros((n,), t)
          wi = zeros((n,), t)
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _N, n, a, n, wr, wi,
                                   dummy, 1, dummy, 1, work, -1, 0)
          lwork = int(work[0])
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _N, n, a, n, wr, wi,
                                   dummy, 1, dummy, 1, work, lwork, 0)
          if all(wi == 0.):
              w = wr
              result_t = _realType(result_t)
          else:
              w = wr+1j*wi
              result_t = _complexType(result_t)
      if results['info'] > 0:
          raise LinAlgError, 'Eigenvalues did not converge'
      return w.astype(result_t)
  
  
  def eigvalsh(a, UPLO='L'):
      """
      Compute the eigenvalues of a Hermitian or real symmetric matrix.
  
      Main difference from eigh: the eigenvectors are not computed.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          A complex- or real-valued matrix whose eigenvalues are to be
          computed.
      UPLO : {'L', 'U'}, optional
          Specifies whether the calculation is done with the lower triangular
          part of `a` ('L', default) or the upper triangular part ('U').
  
      Returns
      -------
      w : ndarray, shape (M,)
          The eigenvalues, not necessarily ordered, each repeated according to
          its multiplicity.
  
      Raises
      ------
      LinAlgError
          If the eigenvalue computation does not converge.
  
      See Also
      --------
      eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays.
      eigvals : eigenvalues of general real or complex arrays.
      eig : eigenvalues and right eigenvectors of general real or complex
            arrays.
  
      Notes
      -----
      This is a simple interface to the LAPACK routines dsyevd and zheevd
      that sets those routines' flags to return only the eigenvalues of
      real symmetric and complex Hermitian arrays, respectively.
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> a = np.array([[1, -2j], [2j, 5]])
      >>> LA.eigvalsh(a)
      array([ 0.17157288+0.j,  5.82842712+0.j])
  
      """
      UPLO = asbytes(UPLO)
      a, wrap = _makearray(a)
      _assertRank2(a)
      _assertSquareness(a)
      t, result_t = _commonType(a)
      real_t = _linalgRealType(t)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      n = a.shape[0]
      liwork = 5*n+3
      iwork = zeros((liwork,), fortran_int)
      if isComplexType(t):
          lapack_routine = lapack_lite.zheevd
          w = zeros((n,), real_t)
          lwork = 1
          work = zeros((lwork,), t)
          lrwork = 1
          rwork = zeros((lrwork,), real_t)
          results = lapack_routine(_N, UPLO, n, a, n, w, work, -1,
                                   rwork, -1, iwork, liwork,  0)
          lwork = int(abs(work[0]))
          work = zeros((lwork,), t)
          lrwork = int(rwork[0])
          rwork = zeros((lrwork,), real_t)
          results = lapack_routine(_N, UPLO, n, a, n, w, work, lwork,
                                  rwork, lrwork, iwork, liwork,  0)
      else:
          lapack_routine = lapack_lite.dsyevd
          w = zeros((n,), t)
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(_N, UPLO, n, a, n, w, work, -1,
                                   iwork, liwork, 0)
          lwork = int(work[0])
          work = zeros((lwork,), t)
          results = lapack_routine(_N, UPLO, n, a, n, w, work, lwork,
                                   iwork, liwork, 0)
      if results['info'] > 0:
          raise LinAlgError, 'Eigenvalues did not converge'
      return w.astype(result_t)
  
  def _convertarray(a):
      t, result_t = _commonType(a)
      a = _fastCT(a.astype(t))
      return a, t, result_t
  
  
  # Eigenvectors
  
  
  def eig(a):
      """
      Compute the eigenvalues and right eigenvectors of a square array.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          A square array of real or complex elements.
  
      Returns
      -------
      w : ndarray, shape (M,)
          The eigenvalues, each repeated according to its multiplicity.
          The eigenvalues are not necessarily ordered, nor are they
          necessarily real for real arrays (though for real arrays
          complex-valued eigenvalues should occur in conjugate pairs).
  
      v : ndarray, shape (M, M)
          The normalized (unit "length") eigenvectors, such that the
          column ``v[:,i]`` is the eigenvector corresponding to the
          eigenvalue ``w[i]``.
  
      Raises
      ------
      LinAlgError
          If the eigenvalue computation does not converge.
  
      See Also
      --------
      eigvalsh : eigenvalues of a symmetric or Hermitian (conjugate symmetric)
         array.
  
      eigvals : eigenvalues of a non-symmetric array.
  
      Notes
      -----
      This is a simple interface to the LAPACK routines dgeev and zgeev
      which compute the eigenvalues and eigenvectors of, respectively,
      general real- and complex-valued square arrays.
  
      The number `w` is an eigenvalue of `a` if there exists a vector
      `v` such that ``dot(a,v) = w * v``. Thus, the arrays `a`, `w`, and
      `v` satisfy the equations ``dot(a[i,:], v[i]) = w[i] * v[:,i]``
      for :math:`i \\in \\{0,...,M-1\\}`.
  
      The array `v` of eigenvectors may not be of maximum rank, that is, some
      of the columns may be linearly dependent, although round-off error may
      obscure that fact. If the eigenvalues are all different, then theoretically
      the eigenvectors are linearly independent. Likewise, the (complex-valued)
      matrix of eigenvectors `v` is unitary if the matrix `a` is normal, i.e.,
      if ``dot(a, a.H) = dot(a.H, a)``, where `a.H` denotes the conjugate
      transpose of `a`.
  
      Finally, it is emphasized that `v` consists of the *right* (as in
      right-hand side) eigenvectors of `a`.  A vector `y` satisfying
      ``dot(y.T, a) = z * y.T`` for some number `z` is called a *left*
      eigenvector of `a`, and, in general, the left and right eigenvectors
      of a matrix are not necessarily the (perhaps conjugate) transposes
      of each other.
  
      References
      ----------
      G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL,
      Academic Press, Inc., 1980, Various pp.
  
      Examples
      --------
      >>> from numpy import linalg as LA
  
      (Almost) trivial example with real e-values and e-vectors.
  
      >>> w, v = LA.eig(np.diag((1, 2, 3)))
      >>> w; v
      array([ 1.,  2.,  3.])
      array([[ 1.,  0.,  0.],
             [ 0.,  1.,  0.],
             [ 0.,  0.,  1.]])
  
      Real matrix possessing complex e-values and e-vectors; note that the
      e-values are complex conjugates of each other.
  
      >>> w, v = LA.eig(np.array([[1, -1], [1, 1]]))
      >>> w; v
      array([ 1. + 1.j,  1. - 1.j])
      array([[ 0.70710678+0.j        ,  0.70710678+0.j        ],
             [ 0.00000000-0.70710678j,  0.00000000+0.70710678j]])
  
      Complex-valued matrix with real e-values (but complex-valued e-vectors);
      note that a.conj().T = a, i.e., a is Hermitian.
  
      >>> a = np.array([[1, 1j], [-1j, 1]])
      >>> w, v = LA.eig(a)
      >>> w; v
      array([  2.00000000e+00+0.j,   5.98651912e-36+0.j]) # i.e., {2, 0}
      array([[ 0.00000000+0.70710678j,  0.70710678+0.j        ],
             [ 0.70710678+0.j        ,  0.00000000+0.70710678j]])
  
      Be careful about round-off error!
  
      >>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
      >>> # Theor. e-values are 1 +/- 1e-9
      >>> w, v = LA.eig(a)
      >>> w; v
      array([ 1.,  1.])
      array([[ 1.,  0.],
             [ 0.,  1.]])
  
      """
      a, wrap = _makearray(a)
      _assertRank2(a)
      _assertSquareness(a)
      _assertFinite(a)
      a, t, result_t = _convertarray(a) # convert to double or cdouble type
      a = _to_native_byte_order(a)
      real_t = _linalgRealType(t)
      n = a.shape[0]
      dummy = zeros((1,), t)
      if isComplexType(t):
          # Complex routines take different arguments
          lapack_routine = lapack_lite.zgeev
          w = zeros((n,), t)
          v = zeros((n, n), t)
          lwork = 1
          work = zeros((lwork,), t)
          rwork = zeros((2*n,), real_t)
          results = lapack_routine(_N, _V, n, a, n, w,
                                   dummy, 1, v, n, work, -1, rwork, 0)
          lwork = int(abs(work[0]))
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _V, n, a, n, w,
                                   dummy, 1, v, n, work, lwork, rwork, 0)
      else:
          lapack_routine = lapack_lite.dgeev
          wr = zeros((n,), t)
          wi = zeros((n,), t)
          vr = zeros((n, n), t)
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _V, n, a, n, wr, wi,
                                    dummy, 1, vr, n, work, -1, 0)
          lwork = int(work[0])
          work = zeros((lwork,), t)
          results = lapack_routine(_N, _V, n, a, n, wr, wi,
                                    dummy, 1, vr, n, work, lwork, 0)
          if all(wi == 0.0):
              w = wr
              v = vr
              result_t = _realType(result_t)
          else:
              w = wr+1j*wi
              v = array(vr, w.dtype)
              ind = flatnonzero(wi != 0.0)      # indices of complex e-vals
              for i in range(len(ind)//2):
                  v[ind[2*i]] = vr[ind[2*i]] + 1j*vr[ind[2*i+1]]
                  v[ind[2*i+1]] = vr[ind[2*i]] - 1j*vr[ind[2*i+1]]
              result_t = _complexType(result_t)
  
      if results['info'] > 0:
          raise LinAlgError, 'Eigenvalues did not converge'
      vt = v.transpose().astype(result_t)
      return w.astype(result_t), wrap(vt)
  
  
  def eigh(a, UPLO='L'):
      """
      Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.
  
      Returns two objects, a 1-D array containing the eigenvalues of `a`, and
      a 2-D square array or matrix (depending on the input type) of the
      corresponding eigenvectors (in columns).
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          A complex Hermitian or real symmetric matrix.
      UPLO : {'L', 'U'}, optional
          Specifies whether the calculation is done with the lower triangular
          part of `a` ('L', default) or the upper triangular part ('U').
  
      Returns
      -------
      w : ndarray, shape (M,)
          The eigenvalues, not necessarily ordered.
      v : ndarray, or matrix object if `a` is, shape (M, M)
          The column ``v[:, i]`` is the normalized eigenvector corresponding
          to the eigenvalue ``w[i]``.
  
      Raises
      ------
      LinAlgError
          If the eigenvalue computation does not converge.
  
      See Also
      --------
      eigvalsh : eigenvalues of symmetric or Hermitian arrays.
      eig : eigenvalues and right eigenvectors for non-symmetric arrays.
      eigvals : eigenvalues of non-symmetric arrays.
  
      Notes
      -----
      This is a simple interface to the LAPACK routines dsyevd and zheevd,
      which compute the eigenvalues and eigenvectors of real symmetric and
      complex Hermitian arrays, respectively.
  
      The eigenvalues of real symmetric or complex Hermitian matrices are
      always real. [1]_ The array `v` of (column) eigenvectors is unitary
      and `a`, `w`, and `v` satisfy the equations
      ``dot(a, v[:, i]) = w[i] * v[:, i]``.
  
      References
      ----------
      .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
             FL, Academic Press, Inc., 1980, pg. 222.
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> a = np.array([[1, -2j], [2j, 5]])
      >>> a
      array([[ 1.+0.j,  0.-2.j],
             [ 0.+2.j,  5.+0.j]])
      >>> w, v = LA.eigh(a)
      >>> w; v
      array([ 0.17157288,  5.82842712])
      array([[-0.92387953+0.j        , -0.38268343+0.j        ],
             [ 0.00000000+0.38268343j,  0.00000000-0.92387953j]])
  
      >>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
      array([2.77555756e-17 + 0.j, 0. + 1.38777878e-16j])
      >>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair
      array([ 0.+0.j,  0.+0.j])
  
      >>> A = np.matrix(a) # what happens if input is a matrix object
      >>> A
      matrix([[ 1.+0.j,  0.-2.j],
              [ 0.+2.j,  5.+0.j]])
      >>> w, v = LA.eigh(A)
      >>> w; v
      array([ 0.17157288,  5.82842712])
      matrix([[-0.92387953+0.j        , -0.38268343+0.j        ],
              [ 0.00000000+0.38268343j,  0.00000000-0.92387953j]])
  
      """
      UPLO = asbytes(UPLO)
      a, wrap = _makearray(a)
      _assertRank2(a)
      _assertSquareness(a)
      t, result_t = _commonType(a)
      real_t = _linalgRealType(t)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      n = a.shape[0]
      liwork = 5*n+3
      iwork = zeros((liwork,), fortran_int)
      if isComplexType(t):
          lapack_routine = lapack_lite.zheevd
          w = zeros((n,), real_t)
          lwork = 1
          work = zeros((lwork,), t)
          lrwork = 1
          rwork = zeros((lrwork,), real_t)
          results = lapack_routine(_V, UPLO, n, a, n, w, work, -1,
                                   rwork, -1, iwork, liwork,  0)
          lwork = int(abs(work[0]))
          work = zeros((lwork,), t)
          lrwork = int(rwork[0])
          rwork = zeros((lrwork,), real_t)
          results = lapack_routine(_V, UPLO, n, a, n, w, work, lwork,
                                   rwork, lrwork, iwork, liwork,  0)
      else:
          lapack_routine = lapack_lite.dsyevd
          w = zeros((n,), t)
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(_V, UPLO, n, a, n, w, work, -1,
                  iwork, liwork, 0)
          lwork = int(work[0])
          work = zeros((lwork,), t)
          results = lapack_routine(_V, UPLO, n, a, n, w, work, lwork,
                  iwork, liwork, 0)
      if results['info'] > 0:
          raise LinAlgError, 'Eigenvalues did not converge'
      at = a.transpose().astype(result_t)
      return w.astype(_realType(result_t)), wrap(at)
  
  
  # Singular value decomposition
  
  def svd(a, full_matrices=1, compute_uv=1):
      """
      Singular Value Decomposition.
  
      Factors the matrix `a` as ``u * np.diag(s) * v``, where `u` and `v`
      are unitary and `s` is a 1-d array of `a`'s singular values.
  
      Parameters
      ----------
      a : array_like
          A real or complex matrix of shape (`M`, `N`) .
      full_matrices : bool, optional
          If True (default), `u` and `v` have the shapes (`M`, `M`) and
          (`N`, `N`), respectively.  Otherwise, the shapes are (`M`, `K`)
          and (`K`, `N`), respectively, where `K` = min(`M`, `N`).
      compute_uv : bool, optional
          Whether or not to compute `u` and `v` in addition to `s`.  True
          by default.
  
      Returns
      -------
      u : ndarray
          Unitary matrix.  The shape of `u` is (`M`, `M`) or (`M`, `K`)
          depending on value of ``full_matrices``.
      s : ndarray
          The singular values, sorted so that ``s[i] >= s[i+1]``.  `s` is
          a 1-d array of length min(`M`, `N`).
      v : ndarray
          Unitary matrix of shape (`N`, `N`) or (`K`, `N`), depending on
          ``full_matrices``.
  
      Raises
      ------
      LinAlgError
          If SVD computation does not converge.
  
      Notes
      -----
      The SVD is commonly written as ``a = U S V.H``.  The `v` returned
      by this function is ``V.H`` and ``u = U``.
  
      If ``U`` is a unitary matrix, it means that it
      satisfies ``U.H = inv(U)``.
  
      The rows of `v` are the eigenvectors of ``a.H a``. The columns
      of `u` are the eigenvectors of ``a a.H``.  For row ``i`` in
      `v` and column ``i`` in `u`, the corresponding eigenvalue is
      ``s[i]**2``.
  
      If `a` is a `matrix` object (as opposed to an `ndarray`), then so
      are all the return values.
  
      Examples
      --------
      >>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)
  
      Reconstruction based on full SVD:
  
      >>> U, s, V = np.linalg.svd(a, full_matrices=True)
      >>> U.shape, V.shape, s.shape
      ((9, 6), (6, 6), (6,))
      >>> S = np.zeros((9, 6), dtype=complex)
      >>> S[:6, :6] = np.diag(s)
      >>> np.allclose(a, np.dot(U, np.dot(S, V)))
      True
  
      Reconstruction based on reduced SVD:
  
      >>> U, s, V = np.linalg.svd(a, full_matrices=False)
      >>> U.shape, V.shape, s.shape
      ((9, 6), (6, 6), (6,))
      >>> S = np.diag(s)
      >>> np.allclose(a, np.dot(U, np.dot(S, V)))
      True
  
      """
      a, wrap = _makearray(a)
      _assertRank2(a)
      _assertNonEmpty(a)
      m, n = a.shape
      t, result_t = _commonType(a)
      real_t = _linalgRealType(t)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      s = zeros((min(n, m),), real_t)
      if compute_uv:
          if full_matrices:
              nu = m
              nvt = n
              option = _A
          else:
              nu = min(n, m)
              nvt = min(n, m)
              option = _S
          u = zeros((nu, m), t)
          vt = zeros((n, nvt), t)
      else:
          option = _N
          nu = 1
          nvt = 1
          u = empty((1, 1), t)
          vt = empty((1, 1), t)
  
      iwork = zeros((8*min(m, n),), fortran_int)
      if isComplexType(t):
          lapack_routine = lapack_lite.zgesdd
          rwork = zeros((5*min(m, n)*min(m, n) + 5*min(m, n),), real_t)
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                   work, -1, rwork, iwork, 0)
          lwork = int(abs(work[0]))
          work = zeros((lwork,), t)
          results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                   work, lwork, rwork, iwork, 0)
      else:
          lapack_routine = lapack_lite.dgesdd
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                   work, -1, iwork, 0)
          lwork = int(work[0])
          work = zeros((lwork,), t)
          results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                   work, lwork, iwork, 0)
      if results['info'] > 0:
          raise LinAlgError, 'SVD did not converge'
      s = s.astype(_realType(result_t))
      if compute_uv:
          u = u.transpose().astype(result_t)
          vt = vt.transpose().astype(result_t)
          return wrap(u), s, wrap(vt)
      else:
          return s
  
  def cond(x, p=None):
      """
      Compute the condition number of a matrix.
  
      This function is capable of returning the condition number using
      one of seven different norms, depending on the value of `p` (see
      Parameters below).
  
      Parameters
      ----------
      x : array_like, shape (M, N)
          The matrix whose condition number is sought.
      p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional
          Order of the norm:
  
          =====  ============================
          p      norm for matrices
          =====  ============================
          None   2-norm, computed directly using the ``SVD``
          'fro'  Frobenius norm
          inf    max(sum(abs(x), axis=1))
          -inf   min(sum(abs(x), axis=1))
          1      max(sum(abs(x), axis=0))
          -1     min(sum(abs(x), axis=0))
          2      2-norm (largest sing. value)
          -2     smallest singular value
          =====  ============================
  
          inf means the numpy.inf object, and the Frobenius norm is
          the root-of-sum-of-squares norm.
  
      Returns
      -------
      c : {float, inf}
          The condition number of the matrix. May be infinite.
  
      See Also
      --------
      numpy.linalg.linalg.norm
  
      Notes
      -----
      The condition number of `x` is defined as the norm of `x` times the
      norm of the inverse of `x` [1]_; the norm can be the usual L2-norm
      (root-of-sum-of-squares) or one of a number of other matrix norms.
  
      References
      ----------
      .. [1] G. Strang, *Linear Algebra and Its Applications*, Orlando, FL,
             Academic Press, Inc., 1980, pg. 285.
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])
      >>> a
      array([[ 1,  0, -1],
             [ 0,  1,  0],
             [ 1,  0,  1]])
      >>> LA.cond(a)
      1.4142135623730951
      >>> LA.cond(a, 'fro')
      3.1622776601683795
      >>> LA.cond(a, np.inf)
      2.0
      >>> LA.cond(a, -np.inf)
      1.0
      >>> LA.cond(a, 1)
      2.0
      >>> LA.cond(a, -1)
      1.0
      >>> LA.cond(a, 2)
      1.4142135623730951
      >>> LA.cond(a, -2)
      0.70710678118654746
      >>> min(LA.svd(a, compute_uv=0))*min(LA.svd(LA.inv(a), compute_uv=0))
      0.70710678118654746
  
      """
      x = asarray(x) # in case we have a matrix
      if p is None:
          s = svd(x,compute_uv=False)
          return s[0]/s[-1]
      else:
          return norm(x,p)*norm(inv(x),p)
  
  
  def matrix_rank(M, tol=None):
      """
      Return matrix rank of array using SVD method
  
      Rank of the array is the number of SVD singular values of the
      array that are greater than `tol`.
  
      Parameters
      ----------
      M : array_like
          array of <=2 dimensions
      tol : {None, float}
         threshold below which SVD values are considered zero. If `tol` is
         None, and ``S`` is an array with singular values for `M`, and
         ``eps`` is the epsilon value for datatype of ``S``, then `tol` is
         set to ``S.max() * eps``.
  
      Notes
      -----
      Golub and van Loan [1]_ define "numerical rank deficiency" as using
      tol=eps*S[0] (where S[0] is the maximum singular value and thus the
      2-norm of the matrix). This is one definition of rank deficiency,
      and the one we use here.  When floating point roundoff is the main
      concern, then "numerical rank deficiency" is a reasonable choice. In
      some cases you may prefer other definitions. The most useful measure
      of the tolerance depends on the operations you intend to use on your
      matrix. For example, if your data come from uncertain measurements
      with uncertainties greater than floating point epsilon, choosing a
      tolerance near that uncertainty may be preferable.  The tolerance
      may be absolute if the uncertainties are absolute rather than
      relative.
  
      References
      ----------
      .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*.
         Baltimore: Johns Hopkins University Press, 1996.
  
      Examples
      --------
      >>> matrix_rank(np.eye(4)) # Full rank matrix
      4
      >>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix
      >>> matrix_rank(I)
      3
      >>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
      1
      >>> matrix_rank(np.zeros((4,)))
      0
  
      """
      M = asarray(M)
      if M.ndim > 2:
          raise TypeError('array should have 2 or fewer dimensions')
      if M.ndim < 2:
          return int(not all(M==0))
      S = svd(M, compute_uv=False)
      if tol is None:
          tol = S.max() * finfo(S.dtype).eps
      return sum(S > tol)
  
  
  # Generalized inverse
  
  def pinv(a, rcond=1e-15 ):
      """
      Compute the (Moore-Penrose) pseudo-inverse of a matrix.
  
      Calculate the generalized inverse of a matrix using its
      singular-value decomposition (SVD) and including all
      *large* singular values.
  
      Parameters
      ----------
      a : array_like, shape (M, N)
        Matrix to be pseudo-inverted.
      rcond : float
        Cutoff for small singular values.
        Singular values smaller (in modulus) than
        `rcond` * largest_singular_value (again, in modulus)
        are set to zero.
  
      Returns
      -------
      B : ndarray, shape (N, M)
        The pseudo-inverse of `a`. If `a` is a `matrix` instance, then so
        is `B`.
  
      Raises
      ------
      LinAlgError
        If the SVD computation does not converge.
  
      Notes
      -----
      The pseudo-inverse of a matrix A, denoted :math:`A^+`, is
      defined as: "the matrix that 'solves' [the least-squares problem]
      :math:`Ax = b`," i.e., if :math:`\\bar{x}` is said solution, then
      :math:`A^+` is that matrix such that :math:`\\bar{x} = A^+b`.
  
      It can be shown that if :math:`Q_1 \\Sigma Q_2^T = A` is the singular
      value decomposition of A, then
      :math:`A^+ = Q_2 \\Sigma^+ Q_1^T`, where :math:`Q_{1,2}` are
      orthogonal matrices, :math:`\\Sigma` is a diagonal matrix consisting
      of A's so-called singular values, (followed, typically, by
      zeros), and then :math:`\\Sigma^+` is simply the diagonal matrix
      consisting of the reciprocals of A's singular values
      (again, followed by zeros). [1]_
  
      References
      ----------
      .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
             FL, Academic Press, Inc., 1980, pp. 139-142.
  
      Examples
      --------
      The following example checks that ``a * a+ * a == a`` and
      ``a+ * a * a+ == a+``:
  
      >>> a = np.random.randn(9, 6)
      >>> B = np.linalg.pinv(a)
      >>> np.allclose(a, np.dot(a, np.dot(B, a)))
      True
      >>> np.allclose(B, np.dot(B, np.dot(a, B)))
      True
  
      """
      a, wrap = _makearray(a)
      _assertNonEmpty(a)
      a = a.conjugate()
      u, s, vt = svd(a, 0)
      m = u.shape[0]
      n = vt.shape[1]
      cutoff = rcond*maximum.reduce(s)
      for i in range(min(n, m)):
          if s[i] > cutoff:
              s[i] = 1./s[i]
          else:
              s[i] = 0.;
      res = dot(transpose(vt), multiply(s[:, newaxis],transpose(u)))
      return wrap(res)
  
  # Determinant
  
  def slogdet(a):
      """
      Compute the sign and (natural) logarithm of the determinant of an array.
  
      If an array has a very small or very large determinant, than a call to
      `det` may overflow or underflow. This routine is more robust against such
      issues, because it computes the logarithm of the determinant rather than
      the determinant itself.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          Input array.
  
      Returns
      -------
      sign : float or complex
          A number representing the sign of the determinant. For a real matrix,
          this is 1, 0, or -1. For a complex matrix, this is a complex number
          with absolute value 1 (i.e., it is on the unit circle), or else 0.
      logdet : float
          The natural log of the absolute value of the determinant.
  
      If the determinant is zero, then `sign` will be 0 and `logdet` will be
      -Inf. In all cases, the determinant is equal to `sign * np.exp(logdet)`.
  
      Notes
      -----
      The determinant is computed via LU factorization using the LAPACK
      routine z/dgetrf.
  
      .. versionadded:: 2.0.0.
  
      Examples
      --------
      The determinant of a 2-D array [[a, b], [c, d]] is ad - bc:
  
      >>> a = np.array([[1, 2], [3, 4]])
      >>> (sign, logdet) = np.linalg.slogdet(a)
      >>> (sign, logdet)
      (-1, 0.69314718055994529)
      >>> sign * np.exp(logdet)
      -2.0
  
      This routine succeeds where ordinary `det` does not:
  
      >>> np.linalg.det(np.eye(500) * 0.1)
      0.0
      >>> np.linalg.slogdet(np.eye(500) * 0.1)
      (1, -1151.2925464970228)
  
      See Also
      --------
      det
  
      """
      a = asarray(a)
      _assertRank2(a)
      _assertSquareness(a)
      t, result_t = _commonType(a)
      a = _fastCopyAndTranspose(t, a)
      a = _to_native_byte_order(a)
      n = a.shape[0]
      if isComplexType(t):
          lapack_routine = lapack_lite.zgetrf
      else:
          lapack_routine = lapack_lite.dgetrf
      pivots = zeros((n,), fortran_int)
      results = lapack_routine(n, n, a, n, pivots, 0)
      info = results['info']
      if (info < 0):
          raise TypeError, "Illegal input to Fortran routine"
      elif (info > 0):
          return (t(0.0), _realType(t)(-Inf))
      sign = 1. - 2. * (add.reduce(pivots != arange(1, n + 1)) % 2)
      d = diagonal(a)
      absd = absolute(d)
      sign *= multiply.reduce(d / absd)
      log(absd, absd)
      logdet = add.reduce(absd, axis=-1)
      return sign, logdet
  
  def det(a):
      """
      Compute the determinant of an array.
  
      Parameters
      ----------
      a : array_like, shape (M, M)
          Input array.
  
      Returns
      -------
      det : ndarray
          Determinant of `a`.
  
      Notes
      -----
      The determinant is computed via LU factorization using the LAPACK
      routine z/dgetrf.
  
      Examples
      --------
      The determinant of a 2-D array [[a, b], [c, d]] is ad - bc:
  
      >>> a = np.array([[1, 2], [3, 4]])
      >>> np.linalg.det(a)
      -2.0
  
      See Also
      --------
      slogdet : Another way to representing the determinant, more suitable
        for large matrices where underflow/overflow may occur.
  
      """
      sign, logdet = slogdet(a)
      return sign * exp(logdet)
  
  # Linear Least Squares
  
  def lstsq(a, b, rcond=-1):
      """
      Return the least-squares solution to a linear matrix equation.
  
      Solves the equation `a x = b` by computing a vector `x` that
      minimizes the Euclidean 2-norm `|| b - a x ||^2`.  The equation may
      be under-, well-, or over- determined (i.e., the number of
      linearly independent rows of `a` can be less than, equal to, or
      greater than its number of linearly independent columns).  If `a`
      is square and of full rank, then `x` (but for round-off error) is
      the "exact" solution of the equation.
  
      Parameters
      ----------
      a : array_like, shape (M, N)
          "Coefficient" matrix.
      b : array_like, shape (M,) or (M, K)
          Ordinate or "dependent variable" values. If `b` is two-dimensional,
          the least-squares solution is calculated for each of the `K` columns
          of `b`.
      rcond : float, optional
          Cut-off ratio for small singular values of `a`.
          Singular values are set to zero if they are smaller than `rcond`
          times the largest singular value of `a`.
  
      Returns
      -------
      x : ndarray, shape (N,) or (N, K)
          Least-squares solution.  The shape of `x` depends on the shape of
          `b`.
      residues : ndarray, shape (), (1,), or (K,)
          Sums of residues; squared Euclidean 2-norm for each column in
          ``b - a*x``.
          If the rank of `a` is < N or > M, this is an empty array.
          If `b` is 1-dimensional, this is a (1,) shape array.
          Otherwise the shape is (K,).
      rank : int
          Rank of matrix `a`.
      s : ndarray, shape (min(M,N),)
          Singular values of `a`.
  
      Raises
      ------
      LinAlgError
          If computation does not converge.
  
      Notes
      -----
      If `b` is a matrix, then all array results are returned as matrices.
  
      Examples
      --------
      Fit a line, ``y = mx + c``, through some noisy data-points:
  
      >>> x = np.array([0, 1, 2, 3])
      >>> y = np.array([-1, 0.2, 0.9, 2.1])
  
      By examining the coefficients, we see that the line should have a
      gradient of roughly 1 and cut the y-axis at, more or less, -1.
  
      We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]``
      and ``p = [[m], [c]]``.  Now use `lstsq` to solve for `p`:
  
      >>> A = np.vstack([x, np.ones(len(x))]).T
      >>> A
      array([[ 0.,  1.],
             [ 1.,  1.],
             [ 2.,  1.],
             [ 3.,  1.]])
  
      >>> m, c = np.linalg.lstsq(A, y)[0]
      >>> print m, c
      1.0 -0.95
  
      Plot the data along with the fitted line:
  
      >>> import matplotlib.pyplot as plt
      >>> plt.plot(x, y, 'o', label='Original data', markersize=10)
      >>> plt.plot(x, m*x + c, 'r', label='Fitted line')
      >>> plt.legend()
      >>> plt.show()
  
      """
      import math
      a, _ = _makearray(a)
      b, wrap = _makearray(b)
      is_1d = len(b.shape) == 1
      if is_1d:
          b = b[:, newaxis]
      _assertRank2(a, b)
      m  = a.shape[0]
      n  = a.shape[1]
      n_rhs = b.shape[1]
      ldb = max(n, m)
      if m != b.shape[0]:
          raise LinAlgError, 'Incompatible dimensions'
      t, result_t = _commonType(a, b)
      result_real_t = _realType(result_t)
      real_t = _linalgRealType(t)
      bstar = zeros((ldb, n_rhs), t)
      bstar[:b.shape[0],:n_rhs] = b.copy()
      a, bstar = _fastCopyAndTranspose(t, a, bstar)
      a, bstar = _to_native_byte_order(a, bstar)
      s = zeros((min(m, n),), real_t)
      nlvl = max( 0, int( math.log( float(min(m, n))/2. ) ) + 1 )
      iwork = zeros((3*min(m, n)*nlvl+11*min(m, n),), fortran_int)
      if isComplexType(t):
          lapack_routine = lapack_lite.zgelsd
          lwork = 1
          rwork = zeros((lwork,), real_t)
          work = zeros((lwork,), t)
          results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                   0, work, -1, rwork, iwork, 0)
          lwork = int(abs(work[0]))
          rwork = zeros((lwork,), real_t)
          a_real = zeros((m, n), real_t)
          bstar_real = zeros((ldb, n_rhs,), real_t)
          results = lapack_lite.dgelsd(m, n, n_rhs, a_real, m,
                                       bstar_real, ldb, s, rcond,
                                       0, rwork, -1, iwork, 0)
          lrwork = int(rwork[0])
          work = zeros((lwork,), t)
          rwork = zeros((lrwork,), real_t)
          results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                   0, work, lwork, rwork, iwork, 0)
      else:
          lapack_routine = lapack_lite.dgelsd
          lwork = 1
          work = zeros((lwork,), t)
          results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                   0, work, -1, iwork, 0)
          lwork = int(work[0])
          work = zeros((lwork,), t)
          results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                   0, work, lwork, iwork, 0)
      if results['info'] > 0:
          raise LinAlgError, 'SVD did not converge in Linear Least Squares'
      resids = array([], result_real_t)
      if is_1d:
          x = array(ravel(bstar)[:n], dtype=result_t, copy=True)
          if results['rank'] == n and m > n:
              if isComplexType(t):
                  resids = array([sum(abs(ravel(bstar)[n:])**2)],
                                 dtype=result_real_t)
              else:
                  resids = array([sum((ravel(bstar)[n:])**2)],
                                 dtype=result_real_t)
      else:
          x = array(transpose(bstar)[:n,:], dtype=result_t, copy=True)
          if results['rank'] == n and m > n:
              if isComplexType(t):
                  resids = sum(abs(transpose(bstar)[n:,:])**2, axis=0).astype(
                      result_real_t)
              else:
                  resids = sum((transpose(bstar)[n:,:])**2, axis=0).astype(
                      result_real_t)
  
      st = s[:min(n, m)].copy().astype(result_real_t)
      return wrap(x), wrap(resids), results['rank'], st
  
  def norm(x, ord=None):
      """
      Matrix or vector norm.
  
      This function is able to return one of seven different matrix norms,
      or one of an infinite number of vector norms (described below), depending
      on the value of the ``ord`` parameter.
  
      Parameters
      ----------
      x : array_like, shape (M,) or (M, N)
          Input array.
      ord : {non-zero int, inf, -inf, 'fro'}, optional
          Order of the norm (see table under ``Notes``). inf means numpy's
          `inf` object.
  
      Returns
      -------
      n : float
          Norm of the matrix or vector.
  
      Notes
      -----
      For values of ``ord <= 0``, the result is, strictly speaking, not a
      mathematical 'norm', but it may still be useful for various numerical
      purposes.
  
      The following norms can be calculated:
  
      =====  ============================  ==========================
      ord    norm for matrices             norm for vectors
      =====  ============================  ==========================
      None   Frobenius norm                2-norm
      'fro'  Frobenius norm                --
      inf    max(sum(abs(x), axis=1))      max(abs(x))
      -inf   min(sum(abs(x), axis=1))      min(abs(x))
      0      --                            sum(x != 0)
      1      max(sum(abs(x), axis=0))      as below
      -1     min(sum(abs(x), axis=0))      as below
      2      2-norm (largest sing. value)  as below
      -2     smallest singular value       as below
      other  --                            sum(abs(x)**ord)**(1./ord)
      =====  ============================  ==========================
  
      The Frobenius norm is given by [1]_:
  
          :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
  
      References
      ----------
      .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
             Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> a = np.arange(9) - 4
      >>> a
      array([-4, -3, -2, -1,  0,  1,  2,  3,  4])
      >>> b = a.reshape((3, 3))
      >>> b
      array([[-4, -3, -2],
             [-1,  0,  1],
             [ 2,  3,  4]])
  
      >>> LA.norm(a)
      7.745966692414834
      >>> LA.norm(b)
      7.745966692414834
      >>> LA.norm(b, 'fro')
      7.745966692414834
      >>> LA.norm(a, np.inf)
      4
      >>> LA.norm(b, np.inf)
      9
      >>> LA.norm(a, -np.inf)
      0
      >>> LA.norm(b, -np.inf)
      2
  
      >>> LA.norm(a, 1)
      20
      >>> LA.norm(b, 1)
      7
      >>> LA.norm(a, -1)
      -4.6566128774142013e-010
      >>> LA.norm(b, -1)
      6
      >>> LA.norm(a, 2)
      7.745966692414834
      >>> LA.norm(b, 2)
      7.3484692283495345
  
      >>> LA.norm(a, -2)
      nan
      >>> LA.norm(b, -2)
      1.8570331885190563e-016
      >>> LA.norm(a, 3)
      5.8480354764257312
      >>> LA.norm(a, -3)
      nan
  
      """
      x = asarray(x)
      if ord is None: # check the default case first and handle it immediately
          return sqrt(add.reduce((x.conj() * x).ravel().real))
  
      nd = x.ndim
      if nd == 1:
          if ord == Inf:
              return abs(x).max()
          elif ord == -Inf:
              return abs(x).min()
          elif ord == 0:
              return (x != 0).sum() # Zero norm
          elif ord == 1:
              return abs(x).sum() # special case for speedup
          elif ord == 2:
              return sqrt(((x.conj()*x).real).sum()) # special case for speedup
          else:
              try:
                  ord + 1
              except TypeError:
                  raise ValueError, "Invalid norm order for vectors."
              return ((abs(x)**ord).sum())**(1.0/ord)
      elif nd == 2:
          if ord == 2:
              return svd(x, compute_uv=0).max()
          elif ord == -2:
              return svd(x, compute_uv=0).min()
          elif ord == 1:
              return abs(x).sum(axis=0).max()
          elif ord == Inf:
              return abs(x).sum(axis=1).max()
          elif ord == -1:
              return abs(x).sum(axis=0).min()
          elif ord == -Inf:
              return abs(x).sum(axis=1).min()
          elif ord in ['fro','f']:
              return sqrt(add.reduce((x.conj() * x).real.ravel()))
          else:
              raise ValueError, "Invalid norm order for matrices."
      else:
          raise ValueError, "Improper number of dimensions to norm."
  