text: #
  # Licensed to the Apache Software Foundation (ASF) under one or more
  # contributor license agreements.  See the NOTICE file distributed with
  # this work for additional information regarding copyright ownership.
  # The ASF licenses this file to You under the Apache License, Version 2.0
  # (the "License"); you may not use this file except in compliance with
  # the License.  You may obtain a copy of the License at
  #
  #    http://www.apache.org/licenses/LICENSE-2.0
  #
  # Unless required by applicable law or agreed to in writing, software
  # distributed under the License is distributed on an "AS IS" BASIS,
  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  # See the License for the specific language governing permissions and
  # limitations under the License.
  #
  
  """
  MLlib utilities for linear algebra. For dense vectors, MLlib
  uses the NumPy `array` type, so you can simply pass NumPy arrays
  around. For sparse vectors, users can construct a :class:`SparseVector`
  object from MLlib or pass SciPy `scipy.sparse` column vectors if
  SciPy is available in their environment.
  """
  
  import sys
  import array
  import struct
  
  import numpy as np
  
  from pyspark import since
  from pyspark.ml import linalg as newlinalg
  from pyspark.sql.types import (
      UserDefinedType,
      StructField,
      StructType,
      ArrayType,
      DoubleType,
      IntegerType,
      ByteType,
      BooleanType,
  )
  
  
  __all__ = [
      "Vector",
      "DenseVector",
      "SparseVector",
      "Vectors",
      "Matrix",
      "DenseMatrix",
      "SparseMatrix",
      "Matrices",
      "QRDecomposition",
  ]
  
  
  # Check whether we have SciPy. MLlib works without it too, but if we have it, some methods,
  # such as _dot and _serialize_double_vector, start to support scipy.sparse matrices.
  
  try:
      import scipy.sparse
  
      _have_scipy = True
  except:
      # No SciPy in environment, but that's okay
      _have_scipy = False
  
  
  def _convert_to_vector(l):
      if isinstance(l, Vector):
          return l
      elif type(l) in (array.array, np.array, np.ndarray, list, tuple, range):
          return DenseVector(l)
      elif _have_scipy and scipy.sparse.issparse(l):
          assert l.shape[1] == 1, "Expected column vector"
          # Make sure the converted csc_matrix has sorted indices.
          csc = l.tocsc()
          if not csc.has_sorted_indices:
              csc.sort_indices()
          return SparseVector(l.shape[0], csc.indices, csc.data)
      else:
          raise TypeError("Cannot convert type %s into Vector" % type(l))
  
  
  def _vector_size(v):
      """
      Returns the size of the vector.
  
      Examples
      --------
      >>> _vector_size([1., 2., 3.])
      3
      >>> _vector_size((1., 2., 3.))
      3
      >>> _vector_size(array.array('d', [1., 2., 3.]))
      3
      >>> _vector_size(np.zeros(3))
      3
      >>> _vector_size(np.zeros((3, 1)))
      3
      >>> _vector_size(np.zeros((1, 3)))
      Traceback (most recent call last):
          ...
      ValueError: Cannot treat an ndarray of shape (1, 3) as a vector
      """
      if isinstance(v, Vector):
          return len(v)
      elif type(v) in (array.array, list, tuple, range):
          return len(v)
      elif type(v) == np.ndarray:
          if v.ndim == 1 or (v.ndim == 2 and v.shape[1] == 1):
              return len(v)
          else:
              raise ValueError("Cannot treat an ndarray of shape %s as a vector" % str(v.shape))
      elif _have_scipy and scipy.sparse.issparse(v):
          assert v.shape[1] == 1, "Expected column vector"
          return v.shape[0]
      else:
          raise TypeError("Cannot treat type %s as a vector" % type(v))
  
  
  def _format_float(f, digits=4):
      s = str(round(f, digits))
      if "." in s:
          s = s[: s.index(".") + 1 + digits]
      return s
  
  
  def _format_float_list(l):
      return [_format_float(x) for x in l]
  
  
  def _double_to_long_bits(value):
      if np.isnan(value):
          value = float("nan")
      # pack double into 64 bits, then unpack as long int
      return struct.unpack("Q", struct.pack("d", value))[0]
  
  
  class VectorUDT(UserDefinedType):
      """
      SQL user-defined type (UDT) for Vector.
      """
  
      @classmethod
      def sqlType(cls):
          return StructType(
              [
                  StructField("type", ByteType(), False),
                  StructField("size", IntegerType(), True),
                  StructField("indices", ArrayType(IntegerType(), False), True),
                  StructField("values", ArrayType(DoubleType(), False), True),
              ]
          )
  
      @classmethod
      def module(cls):
          return "pyspark.mllib.linalg"
  
      @classmethod
      def scalaUDT(cls):
          return "org.apache.spark.mllib.linalg.VectorUDT"
  
      def serialize(self, obj):
          if isinstance(obj, SparseVector):
              indices = [int(i) for i in obj.indices]
              values = [float(v) for v in obj.values]
              return (0, obj.size, indices, values)
          elif isinstance(obj, DenseVector):
              values = [float(v) for v in obj]
              return (1, None, None, values)
          else:
              raise TypeError("cannot serialize %r of type %r" % (obj, type(obj)))
  
      def deserialize(self, datum):
          assert (
              len(datum) == 4
          ), "VectorUDT.deserialize given row with length %d but requires 4" % len(datum)
          tpe = datum[0]
          if tpe == 0:
              return SparseVector(datum[1], datum[2], datum[3])
          elif tpe == 1:
              return DenseVector(datum[3])
          else:
              raise ValueError("do not recognize type %r" % tpe)
  
      def simpleString(self):
          return "vector"
  
  
  class MatrixUDT(UserDefinedType):
      """
      SQL user-defined type (UDT) for Matrix.
      """
  
      @classmethod
      def sqlType(cls):
          return StructType(
              [
                  StructField("type", ByteType(), False),
                  StructField("numRows", IntegerType(), False),
                  StructField("numCols", IntegerType(), False),
                  StructField("colPtrs", ArrayType(IntegerType(), False), True),
                  StructField("rowIndices", ArrayType(IntegerType(), False), True),
                  StructField("values", ArrayType(DoubleType(), False), True),
                  StructField("isTransposed", BooleanType(), False),
              ]
          )
  
      @classmethod
      def module(cls):
          return "pyspark.mllib.linalg"
  
      @classmethod
      def scalaUDT(cls):
          return "org.apache.spark.mllib.linalg.MatrixUDT"
  
      def serialize(self, obj):
          if isinstance(obj, SparseMatrix):
              colPtrs = [int(i) for i in obj.colPtrs]
              rowIndices = [int(i) for i in obj.rowIndices]
              values = [float(v) for v in obj.values]
              return (
                  0,
                  obj.numRows,
                  obj.numCols,
                  colPtrs,
                  rowIndices,
                  values,
                  bool(obj.isTransposed),
              )
          elif isinstance(obj, DenseMatrix):
              values = [float(v) for v in obj.values]
              return (1, obj.numRows, obj.numCols, None, None, values, bool(obj.isTransposed))
          else:
              raise TypeError("cannot serialize type %r" % (type(obj)))
  
      def deserialize(self, datum):
          assert (
              len(datum) == 7
          ), "MatrixUDT.deserialize given row with length %d but requires 7" % len(datum)
          tpe = datum[0]
          if tpe == 0:
              return SparseMatrix(*datum[1:])
          elif tpe == 1:
              return DenseMatrix(datum[1], datum[2], datum[5], datum[6])
          else:
              raise ValueError("do not recognize type %r" % tpe)
  
      def simpleString(self):
          return "matrix"
  
  
  class Vector(object):
  
      __UDT__ = VectorUDT()
  
      """
      Abstract class for DenseVector and SparseVector
      """
  
      def toArray(self):
          """
          Convert the vector into an numpy.ndarray
  
          Returns
          -------
          :py:class:`numpy.ndarray`
          """
          raise NotImplementedError
  
      def asML(self):
          """
          Convert this vector to the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          Returns
          -------
          :py:class:`pyspark.ml.linalg.Vector`
          """
          raise NotImplementedError
  
  
  class DenseVector(Vector):
      """
      A dense vector represented by a value array. We use numpy array for
      storage and arithmetics will be delegated to the underlying numpy
      array.
  
      Examples
      --------
      >>> v = Vectors.dense([1.0, 2.0])
      >>> u = Vectors.dense([3.0, 4.0])
      >>> v + u
      DenseVector([4.0, 6.0])
      >>> 2 - v
      DenseVector([1.0, 0.0])
      >>> v / 2
      DenseVector([0.5, 1.0])
      >>> v * u
      DenseVector([3.0, 8.0])
      >>> u / v
      DenseVector([3.0, 2.0])
      >>> u % 2
      DenseVector([1.0, 0.0])
      >>> -v
      DenseVector([-1.0, -2.0])
      """
  
      def __init__(self, ar):
          if isinstance(ar, bytes):
              ar = np.frombuffer(ar, dtype=np.float64)
          elif not isinstance(ar, np.ndarray):
              ar = np.array(ar, dtype=np.float64)
          if ar.dtype != np.float64:
              ar = ar.astype(np.float64)
          self.array = ar
  
      @staticmethod
      def parse(s):
          """
          Parse string representation back into the DenseVector.
  
          Examples
          --------
          >>> DenseVector.parse(' [ 0.0,1.0,2.0,  3.0]')
          DenseVector([0.0, 1.0, 2.0, 3.0])
          """
          start = s.find("[")
          if start == -1:
              raise ValueError("Array should start with '['.")
          end = s.find("]")
          if end == -1:
              raise ValueError("Array should end with ']'.")
          s = s[start + 1 : end]
  
          try:
              values = [float(val) for val in s.split(",") if val]
          except ValueError:
              raise ValueError("Unable to parse values from %s" % s)
          return DenseVector(values)
  
      def __reduce__(self):
          return DenseVector, (self.array.tostring(),)
  
      def numNonzeros(self):
          """
          Number of nonzero elements. This scans all active values and count non zeros
          """
          return np.count_nonzero(self.array)
  
      def norm(self, p):
          """
          Calculates the norm of a DenseVector.
  
          Examples
          --------
          >>> a = DenseVector([0, -1, 2, -3])
          >>> a.norm(2)
          3.7...
          >>> a.norm(1)
          6.0
          """
          return np.linalg.norm(self.array, p)
  
      def dot(self, other):
          """
          Compute the dot product of two Vectors. We support
          (Numpy array, list, SparseVector, or SciPy sparse)
          and a target NumPy array that is either 1- or 2-dimensional.
          Equivalent to calling numpy.dot of the two vectors.
  
          Examples
          --------
          >>> dense = DenseVector(array.array('d', [1., 2.]))
          >>> dense.dot(dense)
          5.0
          >>> dense.dot(SparseVector(2, [0, 1], [2., 1.]))
          4.0
          >>> dense.dot(range(1, 3))
          5.0
          >>> dense.dot(np.array(range(1, 3)))
          5.0
          >>> dense.dot([1.,])
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          >>> dense.dot(np.reshape([1., 2., 3., 4.], (2, 2), order='F'))
          array([  5.,  11.])
          >>> dense.dot(np.reshape([1., 2., 3.], (3, 1), order='F'))
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          """
          if type(other) == np.ndarray:
              if other.ndim > 1:
                  assert len(self) == other.shape[0], "dimension mismatch"
              return np.dot(self.array, other)
          elif _have_scipy and scipy.sparse.issparse(other):
              assert len(self) == other.shape[0], "dimension mismatch"
              return other.transpose().dot(self.toArray())
          else:
              assert len(self) == _vector_size(other), "dimension mismatch"
              if isinstance(other, SparseVector):
                  return other.dot(self)
              elif isinstance(other, Vector):
                  return np.dot(self.toArray(), other.toArray())
              else:
                  return np.dot(self.toArray(), other)
  
      def squared_distance(self, other):
          """
          Squared distance of two Vectors.
  
          Examples
          --------
          >>> dense1 = DenseVector(array.array('d', [1., 2.]))
          >>> dense1.squared_distance(dense1)
          0.0
          >>> dense2 = np.array([2., 1.])
          >>> dense1.squared_distance(dense2)
          2.0
          >>> dense3 = [2., 1.]
          >>> dense1.squared_distance(dense3)
          2.0
          >>> sparse1 = SparseVector(2, [0, 1], [2., 1.])
          >>> dense1.squared_distance(sparse1)
          2.0
          >>> dense1.squared_distance([1.,])
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          >>> dense1.squared_distance(SparseVector(1, [0,], [1.,]))
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          """
          assert len(self) == _vector_size(other), "dimension mismatch"
          if isinstance(other, SparseVector):
              return other.squared_distance(self)
          elif _have_scipy and scipy.sparse.issparse(other):
              return _convert_to_vector(other).squared_distance(self)
  
          if isinstance(other, Vector):
              other = other.toArray()
          elif not isinstance(other, np.ndarray):
              other = np.array(other)
          diff = self.toArray() - other
          return np.dot(diff, diff)
  
      def toArray(self):
          """
          Returns an numpy.ndarray
          """
          return self.array
  
      def asML(self):
          """
          Convert this vector to the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          .. versionadded:: 2.0.0
  
          Returns
          -------
          :py:class:`pyspark.ml.linalg.DenseVector`
          """
          return newlinalg.DenseVector(self.array)
  
      @property
      def values(self):
          """
          Returns a list of values
          """
          return self.array
  
      def __getitem__(self, item):
          return self.array[item]
  
      def __len__(self):
          return len(self.array)
  
      def __str__(self):
          return "[" + ",".join([str(v) for v in self.array]) + "]"
  
      def __repr__(self):
          return "DenseVector([%s])" % (", ".join(_format_float(i) for i in self.array))
  
      def __eq__(self, other):
          if isinstance(other, DenseVector):
              return np.array_equal(self.array, other.array)
          elif isinstance(other, SparseVector):
              if len(self) != other.size:
                  return False
              return Vectors._equals(list(range(len(self))), self.array, other.indices, other.values)
          return False
  
      def __ne__(self, other):
          return not self == other
  
      def __hash__(self):
          size = len(self)
          result = 31 + size
          nnz = 0
          i = 0
          while i < size and nnz < 128:
              if self.array[i] != 0:
                  result = 31 * result + i
                  bits = _double_to_long_bits(self.array[i])
                  result = 31 * result + (bits ^ (bits >> 32))
                  nnz += 1
              i += 1
          return result
  
      def __getattr__(self, item):
          return getattr(self.array, item)
  
      def __neg__(self):
          return DenseVector(-self.array)
  
      def _delegate(op):
          def func(self, other):
              if isinstance(other, DenseVector):
                  other = other.array
              return DenseVector(getattr(self.array, op)(other))
  
          return func
  
      __add__ = _delegate("__add__")
      __sub__ = _delegate("__sub__")
      __mul__ = _delegate("__mul__")
      __div__ = _delegate("__div__")
      __truediv__ = _delegate("__truediv__")
      __mod__ = _delegate("__mod__")
      __radd__ = _delegate("__radd__")
      __rsub__ = _delegate("__rsub__")
      __rmul__ = _delegate("__rmul__")
      __rdiv__ = _delegate("__rdiv__")
      __rtruediv__ = _delegate("__rtruediv__")
      __rmod__ = _delegate("__rmod__")
  
  
  class SparseVector(Vector):
      """
      A simple sparse vector class for passing data to MLlib. Users may
      alternatively pass SciPy's {scipy.sparse} data types.
      """
  
      def __init__(self, size, *args):
          """
          Create a sparse vector, using either a dictionary, a list of
          (index, value) pairs, or two separate arrays of indices and
          values (sorted by index).
  
          Parameters
          ----------
          size : int
              Size of the vector.
          args
              Active entries, as a dictionary {index: value, ...},
              a list of tuples [(index, value), ...], or a list of strictly
              increasing indices and a list of corresponding values [index, ...],
              [value, ...]. Inactive entries are treated as zeros.
  
          Examples
          --------
          >>> SparseVector(4, {1: 1.0, 3: 5.5})
          SparseVector(4, {1: 1.0, 3: 5.5})
          >>> SparseVector(4, [(1, 1.0), (3, 5.5)])
          SparseVector(4, {1: 1.0, 3: 5.5})
          >>> SparseVector(4, [1, 3], [1.0, 5.5])
          SparseVector(4, {1: 1.0, 3: 5.5})
          """
          self.size = int(size)
          """ Size of the vector. """
          assert 1 <= len(args) <= 2, "must pass either 2 or 3 arguments"
          if len(args) == 1:
              pairs = args[0]
              if type(pairs) == dict:
                  pairs = pairs.items()
              pairs = sorted(pairs)
              self.indices = np.array([p[0] for p in pairs], dtype=np.int32)
              """ A list of indices corresponding to active entries. """
              self.values = np.array([p[1] for p in pairs], dtype=np.float64)
              """ A list of values corresponding to active entries. """
          else:
              if isinstance(args[0], bytes):
                  assert isinstance(args[1], bytes), "values should be string too"
                  if args[0]:
                      self.indices = np.frombuffer(args[0], np.int32)
                      self.values = np.frombuffer(args[1], np.float64)
                  else:
                      # np.frombuffer() doesn't work well with empty string in older version
                      self.indices = np.array([], dtype=np.int32)
                      self.values = np.array([], dtype=np.float64)
              else:
                  self.indices = np.array(args[0], dtype=np.int32)
                  self.values = np.array(args[1], dtype=np.float64)
              assert len(self.indices) == len(self.values), "index and value arrays not same length"
              for i in range(len(self.indices) - 1):
                  if self.indices[i] >= self.indices[i + 1]:
                      raise TypeError(
                          "Indices %s and %s are not strictly increasing"
                          % (self.indices[i], self.indices[i + 1])
                      )
  
      def numNonzeros(self):
          """
          Number of nonzero elements. This scans all active values and count non zeros.
          """
          return np.count_nonzero(self.values)
  
      def norm(self, p):
          """
          Calculates the norm of a SparseVector.
  
          Examples
          --------
          >>> a = SparseVector(4, [0, 1], [3., -4.])
          >>> a.norm(1)
          7.0
          >>> a.norm(2)
          5.0
          """
          return np.linalg.norm(self.values, p)
  
      def __reduce__(self):
          return (SparseVector, (self.size, self.indices.tostring(), self.values.tostring()))
  
      @staticmethod
      def parse(s):
          """
          Parse string representation back into the SparseVector.
  
          Examples
          --------
          >>> SparseVector.parse(' (4, [0,1 ],[ 4.0,5.0] )')
          SparseVector(4, {0: 4.0, 1: 5.0})
          """
          start = s.find("(")
          if start == -1:
              raise ValueError("Tuple should start with '('")
          end = s.find(")")
          if end == -1:
              raise ValueError("Tuple should end with ')'")
          s = s[start + 1 : end].strip()
  
          size = s[: s.find(",")]
          try:
              size = int(size)
          except ValueError:
              raise ValueError("Cannot parse size %s." % size)
  
          ind_start = s.find("[")
          if ind_start == -1:
              raise ValueError("Indices array should start with '['.")
          ind_end = s.find("]")
          if ind_end == -1:
              raise ValueError("Indices array should end with ']'")
          new_s = s[ind_start + 1 : ind_end]
          ind_list = new_s.split(",")
          try:
              indices = [int(ind) for ind in ind_list if ind]
          except ValueError:
              raise ValueError("Unable to parse indices from %s." % new_s)
          s = s[ind_end + 1 :].strip()
  
          val_start = s.find("[")
          if val_start == -1:
              raise ValueError("Values array should start with '['.")
          val_end = s.find("]")
          if val_end == -1:
              raise ValueError("Values array should end with ']'.")
          val_list = s[val_start + 1 : val_end].split(",")
          try:
              values = [float(val) for val in val_list if val]
          except ValueError:
              raise ValueError("Unable to parse values from %s." % s)
          return SparseVector(size, indices, values)
  
      def dot(self, other):
          """
          Dot product with a SparseVector or 1- or 2-dimensional Numpy array.
  
          Examples
          --------
          >>> a = SparseVector(4, [1, 3], [3.0, 4.0])
          >>> a.dot(a)
          25.0
          >>> a.dot(array.array('d', [1., 2., 3., 4.]))
          22.0
          >>> b = SparseVector(4, [2], [1.0])
          >>> a.dot(b)
          0.0
          >>> a.dot(np.array([[1, 1], [2, 2], [3, 3], [4, 4]]))
          array([ 22.,  22.])
          >>> a.dot([1., 2., 3.])
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          >>> a.dot(np.array([1., 2.]))
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          >>> a.dot(DenseVector([1., 2.]))
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          >>> a.dot(np.zeros((3, 2)))
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          """
  
          if isinstance(other, np.ndarray):
              if other.ndim not in [2, 1]:
                  raise ValueError("Cannot call dot with %d-dimensional array" % other.ndim)
              assert len(self) == other.shape[0], "dimension mismatch"
              return np.dot(self.values, other[self.indices])
  
          assert len(self) == _vector_size(other), "dimension mismatch"
  
          if isinstance(other, DenseVector):
              return np.dot(other.array[self.indices], self.values)
  
          elif isinstance(other, SparseVector):
              # Find out common indices.
              self_cmind = np.in1d(self.indices, other.indices, assume_unique=True)
              self_values = self.values[self_cmind]
              if self_values.size == 0:
                  return 0.0
              else:
                  other_cmind = np.in1d(other.indices, self.indices, assume_unique=True)
                  return np.dot(self_values, other.values[other_cmind])
  
          else:
              return self.dot(_convert_to_vector(other))
  
      def squared_distance(self, other):
          """
          Squared distance from a SparseVector or 1-dimensional NumPy array.
  
          Examples
          --------
          >>> a = SparseVector(4, [1, 3], [3.0, 4.0])
          >>> a.squared_distance(a)
          0.0
          >>> a.squared_distance(array.array('d', [1., 2., 3., 4.]))
          11.0
          >>> a.squared_distance(np.array([1., 2., 3., 4.]))
          11.0
          >>> b = SparseVector(4, [2], [1.0])
          >>> a.squared_distance(b)
          26.0
          >>> b.squared_distance(a)
          26.0
          >>> b.squared_distance([1., 2.])
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          >>> b.squared_distance(SparseVector(3, [1,], [1.0,]))
          Traceback (most recent call last):
              ...
          AssertionError: dimension mismatch
          """
          assert len(self) == _vector_size(other), "dimension mismatch"
  
          if isinstance(other, np.ndarray) or isinstance(other, DenseVector):
              if isinstance(other, np.ndarray) and other.ndim != 1:
                  raise ValueError(
                      "Cannot call squared_distance with %d-dimensional array" % other.ndim
                  )
              if isinstance(other, DenseVector):
                  other = other.array
              sparse_ind = np.zeros(other.size, dtype=bool)
              sparse_ind[self.indices] = True
              dist = other[sparse_ind] - self.values
              result = np.dot(dist, dist)
  
              other_ind = other[~sparse_ind]
              result += np.dot(other_ind, other_ind)
              return result
  
          elif isinstance(other, SparseVector):
              result = 0.0
              i, j = 0, 0
              while i < len(self.indices) and j < len(other.indices):
                  if self.indices[i] == other.indices[j]:
                      diff = self.values[i] - other.values[j]
                      result += diff * diff
                      i += 1
                      j += 1
                  elif self.indices[i] < other.indices[j]:
                      result += self.values[i] * self.values[i]
                      i += 1
                  else:
                      result += other.values[j] * other.values[j]
                      j += 1
              while i < len(self.indices):
                  result += self.values[i] * self.values[i]
                  i += 1
              while j < len(other.indices):
                  result += other.values[j] * other.values[j]
                  j += 1
              return result
          else:
              return self.squared_distance(_convert_to_vector(other))
  
      def toArray(self):
          """
          Returns a copy of this SparseVector as a 1-dimensional NumPy array.
          """
          arr = np.zeros((self.size,), dtype=np.float64)
          arr[self.indices] = self.values
          return arr
  
      def asML(self):
          """
          Convert this vector to the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          .. versionadded:: 2.0.0
  
          Returns
          -------
          :py:class:`pyspark.ml.linalg.SparseVector`
          """
          return newlinalg.SparseVector(self.size, self.indices, self.values)
  
      def __len__(self):
          return self.size
  
      def __str__(self):
          inds = "[" + ",".join([str(i) for i in self.indices]) + "]"
          vals = "[" + ",".join([str(v) for v in self.values]) + "]"
          return "(" + ",".join((str(self.size), inds, vals)) + ")"
  
      def __repr__(self):
          inds = self.indices
          vals = self.values
          entries = ", ".join(
              ["{0}: {1}".format(inds[i], _format_float(vals[i])) for i in range(len(inds))]
          )
          return "SparseVector({0}, {{{1}}})".format(self.size, entries)
  
      def __eq__(self, other):
          if isinstance(other, SparseVector):
              return (
                  other.size == self.size
                  and np.array_equal(other.indices, self.indices)
                  and np.array_equal(other.values, self.values)
              )
          elif isinstance(other, DenseVector):
              if self.size != len(other):
                  return False
              return Vectors._equals(self.indices, self.values, list(range(len(other))), other.array)
          return False
  
      def __getitem__(self, index):
          inds = self.indices
          vals = self.values
          if not isinstance(index, int):
              raise TypeError("Indices must be of type integer, got type %s" % type(index))
  
          if index >= self.size or index < -self.size:
              raise IndexError("Index %d out of bounds." % index)
          if index < 0:
              index += self.size
  
          if (inds.size == 0) or (index > inds.item(-1)):
              return 0.0
  
          insert_index = np.searchsorted(inds, index)
          row_ind = inds[insert_index]
          if row_ind == index:
              return vals[insert_index]
          return 0.0
  
      def __ne__(self, other):
          return not self.__eq__(other)
  
      def __hash__(self):
          result = 31 + self.size
          nnz = 0
          i = 0
          while i < len(self.values) and nnz < 128:
              if self.values[i] != 0:
                  result = 31 * result + int(self.indices[i])
                  bits = _double_to_long_bits(self.values[i])
                  result = 31 * result + (bits ^ (bits >> 32))
                  nnz += 1
              i += 1
          return result
  
  
  class Vectors(object):
  
      """
      Factory methods for working with vectors.
  
      Notes
      -----
      Dense vectors are simply represented as NumPy array objects,
      so there is no need to covert them for use in MLlib. For sparse vectors,
      the factory methods in this class create an MLlib-compatible type, or users
      can pass in SciPy's `scipy.sparse` column vectors.
      """
  
      @staticmethod
      def sparse(size, *args):
          """
          Create a sparse vector, using either a dictionary, a list of
          (index, value) pairs, or two separate arrays of indices and
          values (sorted by index).
  
          Parameters
          ----------
          size : int
              Size of the vector.
          args
              Non-zero entries, as a dictionary, list of tuples,
              or two sorted lists containing indices and values.
  
          Examples
          --------
          >>> Vectors.sparse(4, {1: 1.0, 3: 5.5})
          SparseVector(4, {1: 1.0, 3: 5.5})
          >>> Vectors.sparse(4, [(1, 1.0), (3, 5.5)])
          SparseVector(4, {1: 1.0, 3: 5.5})
          >>> Vectors.sparse(4, [1, 3], [1.0, 5.5])
          SparseVector(4, {1: 1.0, 3: 5.5})
          """
          return SparseVector(size, *args)
  
      @staticmethod
      def dense(*elements):
          """
          Create a dense vector of 64-bit floats from a Python list or numbers.
  
          Examples
          --------
          >>> Vectors.dense([1, 2, 3])
          DenseVector([1.0, 2.0, 3.0])
          >>> Vectors.dense(1.0, 2.0)
          DenseVector([1.0, 2.0])
          """
          if len(elements) == 1 and not isinstance(elements[0], (float, int)):
              # it's list, numpy.array or other iterable object.
              elements = elements[0]
          return DenseVector(elements)
  
      @staticmethod
      def fromML(vec):
          """
          Convert a vector from the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          .. versionadded:: 2.0.0
  
          Parameters
          ----------
          vec : :py:class:`pyspark.ml.linalg.Vector`
  
          Returns
          -------
          :py:class:`pyspark.mllib.linalg.Vector`
          """
          if isinstance(vec, newlinalg.DenseVector):
              return DenseVector(vec.array)
          elif isinstance(vec, newlinalg.SparseVector):
              return SparseVector(vec.size, vec.indices, vec.values)
          else:
              raise TypeError("Unsupported vector type %s" % type(vec))
  
      @staticmethod
      def stringify(vector):
          """
          Converts a vector into a string, which can be recognized by
          Vectors.parse().
  
          Examples
          --------
          >>> Vectors.stringify(Vectors.sparse(2, [1], [1.0]))
          '(2,[1],[1.0])'
          >>> Vectors.stringify(Vectors.dense([0.0, 1.0]))
          '[0.0,1.0]'
          """
          return str(vector)
  
      @staticmethod
      def squared_distance(v1, v2):
          """
          Squared distance between two vectors.
          a and b can be of type SparseVector, DenseVector, np.ndarray
          or array.array.
  
          Examples
          --------
          >>> a = Vectors.sparse(4, [(0, 1), (3, 4)])
          >>> b = Vectors.dense([2, 5, 4, 1])
          >>> a.squared_distance(b)
          51.0
          """
          v1, v2 = _convert_to_vector(v1), _convert_to_vector(v2)
          return v1.squared_distance(v2)
  
      @staticmethod
      def norm(vector, p):
          """
          Find norm of the given vector.
          """
          return _convert_to_vector(vector).norm(p)
  
      @staticmethod
      def parse(s):
          """Parse a string representation back into the Vector.
  
          Examples
          --------
          >>> Vectors.parse('[2,1,2 ]')
          DenseVector([2.0, 1.0, 2.0])
          >>> Vectors.parse(' ( 100,  [0],  [2])')
          SparseVector(100, {0: 2.0})
          """
          if s.find("(") == -1 and s.find("[") != -1:
              return DenseVector.parse(s)
          elif s.find("(") != -1:
              return SparseVector.parse(s)
          else:
              raise ValueError("Cannot find tokens '[' or '(' from the input string.")
  
      @staticmethod
      def zeros(size):
          return DenseVector(np.zeros(size))
  
      @staticmethod
      def _equals(v1_indices, v1_values, v2_indices, v2_values):
          """
          Check equality between sparse/dense vectors,
          v1_indices and v2_indices assume to be strictly increasing.
          """
          v1_size = len(v1_values)
          v2_size = len(v2_values)
          k1 = 0
          k2 = 0
          all_equal = True
          while all_equal:
              while k1 < v1_size and v1_values[k1] == 0:
                  k1 += 1
              while k2 < v2_size and v2_values[k2] == 0:
                  k2 += 1
  
              if k1 >= v1_size or k2 >= v2_size:
                  return k1 >= v1_size and k2 >= v2_size
  
              all_equal = v1_indices[k1] == v2_indices[k2] and v1_values[k1] == v2_values[k2]
              k1 += 1
              k2 += 1
          return all_equal
  
  
  class Matrix(object):
  
      __UDT__ = MatrixUDT()
  
      """
      Represents a local matrix.
      """
  
      def __init__(self, numRows, numCols, isTransposed=False):
          self.numRows = numRows
          self.numCols = numCols
          self.isTransposed = isTransposed
  
      def toArray(self):
          """
          Returns its elements in a NumPy ndarray.
          """
          raise NotImplementedError
  
      def asML(self):
          """
          Convert this matrix to the new mllib-local representation.
          This does NOT copy the data; it copies references.
          """
          raise NotImplementedError
  
      @staticmethod
      def _convert_to_array(array_like, dtype):
          """
          Convert Matrix attributes which are array-like or buffer to array.
          """
          if isinstance(array_like, bytes):
              return np.frombuffer(array_like, dtype=dtype)
          return np.asarray(array_like, dtype=dtype)
  
  
  class DenseMatrix(Matrix):
      """
      Column-major dense matrix.
      """
  
      def __init__(self, numRows, numCols, values, isTransposed=False):
          Matrix.__init__(self, numRows, numCols, isTransposed)
          values = self._convert_to_array(values, np.float64)
          assert len(values) == numRows * numCols
          self.values = values
  
      def __reduce__(self):
          return DenseMatrix, (
              self.numRows,
              self.numCols,
              self.values.tostring(),
              int(self.isTransposed),
          )
  
      def __str__(self):
          """
          Pretty printing of a DenseMatrix
  
          Examples
          --------
          >>> dm = DenseMatrix(2, 2, range(4))
          >>> print(dm)
          DenseMatrix([[ 0.,  2.],
                       [ 1.,  3.]])
          >>> dm = DenseMatrix(2, 2, range(4), isTransposed=True)
          >>> print(dm)
          DenseMatrix([[ 0.,  1.],
                       [ 2.,  3.]])
          """
          # Inspired by __repr__ in scipy matrices.
          array_lines = repr(self.toArray()).splitlines()
  
          # We need to adjust six spaces which is the difference in number
          # of letters between "DenseMatrix" and "array"
          x = "\n".join([(" " * 6 + line) for line in array_lines[1:]])
          return array_lines[0].replace("array", "DenseMatrix") + "\n" + x
  
      def __repr__(self):
          """
          Representation of a DenseMatrix
  
          Examples
          --------
          >>> dm = DenseMatrix(2, 2, range(4))
          >>> dm
          DenseMatrix(2, 2, [0.0, 1.0, 2.0, 3.0], False)
          """
          # If the number of values are less than seventeen then return as it is.
          # Else return first eight values and last eight values.
          if len(self.values) < 17:
              entries = _format_float_list(self.values)
          else:
              entries = (
                  _format_float_list(self.values[:8]) + ["..."] + _format_float_list(self.values[-8:])
              )
  
          entries = ", ".join(entries)
          return "DenseMatrix({0}, {1}, [{2}], {3})".format(
              self.numRows, self.numCols, entries, self.isTransposed
          )
  
      def toArray(self):
          """
          Return an numpy.ndarray
  
          Examples
          --------
          >>> m = DenseMatrix(2, 2, range(4))
          >>> m.toArray()
          array([[ 0.,  2.],
                 [ 1.,  3.]])
          """
          if self.isTransposed:
              return np.asfortranarray(self.values.reshape((self.numRows, self.numCols)))
          else:
              return self.values.reshape((self.numRows, self.numCols), order="F")
  
      def toSparse(self):
          """Convert to SparseMatrix"""
          if self.isTransposed:
              values = np.ravel(self.toArray(), order="F")
          else:
              values = self.values
          indices = np.nonzero(values)[0]
          colCounts = np.bincount(indices // self.numRows)
          colPtrs = np.cumsum(np.hstack((0, colCounts, np.zeros(self.numCols - colCounts.size))))
          values = values[indices]
          rowIndices = indices % self.numRows
  
          return SparseMatrix(self.numRows, self.numCols, colPtrs, rowIndices, values)
  
      def asML(self):
          """
          Convert this matrix to the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          .. versionadded:: 2.0.0
  
          Returns
          -------
          :py:class:`pyspark.ml.linalg.DenseMatrix`
          """
          return newlinalg.DenseMatrix(self.numRows, self.numCols, self.values, self.isTransposed)
  
      def __getitem__(self, indices):
          i, j = indices
          if i < 0 or i >= self.numRows:
              raise IndexError("Row index %d is out of range [0, %d)" % (i, self.numRows))
          if j >= self.numCols or j < 0:
              raise IndexError("Column index %d is out of range [0, %d)" % (j, self.numCols))
  
          if self.isTransposed:
              return self.values[i * self.numCols + j]
          else:
              return self.values[i + j * self.numRows]
  
      def __eq__(self, other):
          if self.numRows != other.numRows or self.numCols != other.numCols:
              return False
          if isinstance(other, SparseMatrix):
              return np.all(self.toArray() == other.toArray())
  
          self_values = np.ravel(self.toArray(), order="F")
          other_values = np.ravel(other.toArray(), order="F")
          return np.all(self_values == other_values)
  
  
  class SparseMatrix(Matrix):
      """Sparse Matrix stored in CSC format."""
  
      def __init__(self, numRows, numCols, colPtrs, rowIndices, values, isTransposed=False):
          Matrix.__init__(self, numRows, numCols, isTransposed)
          self.colPtrs = self._convert_to_array(colPtrs, np.int32)
          self.rowIndices = self._convert_to_array(rowIndices, np.int32)
          self.values = self._convert_to_array(values, np.float64)
  
          if self.isTransposed:
              if self.colPtrs.size != numRows + 1:
                  raise ValueError(
                      "Expected colPtrs of size %d, got %d." % (numRows + 1, self.colPtrs.size)
                  )
          else:
              if self.colPtrs.size != numCols + 1:
                  raise ValueError(
                      "Expected colPtrs of size %d, got %d." % (numCols + 1, self.colPtrs.size)
                  )
          if self.rowIndices.size != self.values.size:
              raise ValueError(
                  "Expected rowIndices of length %d, got %d."
                  % (self.rowIndices.size, self.values.size)
              )
  
      def __str__(self):
          """
          Pretty printing of a SparseMatrix
  
          Examples
          --------
          >>> sm1 = SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2, 3, 4])
          >>> print(sm1)
          2 X 2 CSCMatrix
          (0,0) 2.0
          (1,0) 3.0
          (1,1) 4.0
          >>> sm1 = SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2, 3, 4], True)
          >>> print(sm1)
          2 X 2 CSRMatrix
          (0,0) 2.0
          (0,1) 3.0
          (1,1) 4.0
          """
          spstr = "{0} X {1} ".format(self.numRows, self.numCols)
          if self.isTransposed:
              spstr += "CSRMatrix\n"
          else:
              spstr += "CSCMatrix\n"
  
          cur_col = 0
          smlist = []
  
          # Display first 16 values.
          if len(self.values) <= 16:
              zipindval = zip(self.rowIndices, self.values)
          else:
              zipindval = zip(self.rowIndices[:16], self.values[:16])
          for i, (rowInd, value) in enumerate(zipindval):
              if self.colPtrs[cur_col + 1] <= i:
                  cur_col += 1
              if self.isTransposed:
                  smlist.append("({0},{1}) {2}".format(cur_col, rowInd, _format_float(value)))
              else:
                  smlist.append("({0},{1}) {2}".format(rowInd, cur_col, _format_float(value)))
          spstr += "\n".join(smlist)
  
          if len(self.values) > 16:
              spstr += "\n.." * 2
          return spstr
  
      def __repr__(self):
          """
          Representation of a SparseMatrix
  
          Examples
          --------
          >>> sm1 = SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2, 3, 4])
          >>> sm1
          SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2.0, 3.0, 4.0], False)
          """
          rowIndices = list(self.rowIndices)
          colPtrs = list(self.colPtrs)
  
          if len(self.values) <= 16:
              values = _format_float_list(self.values)
  
          else:
              values = (
                  _format_float_list(self.values[:8]) + ["..."] + _format_float_list(self.values[-8:])
              )
              rowIndices = rowIndices[:8] + ["..."] + rowIndices[-8:]
  
          if len(self.colPtrs) > 16:
              colPtrs = colPtrs[:8] + ["..."] + colPtrs[-8:]
  
          values = ", ".join(values)
          rowIndices = ", ".join([str(ind) for ind in rowIndices])
          colPtrs = ", ".join([str(ptr) for ptr in colPtrs])
          return "SparseMatrix({0}, {1}, [{2}], [{3}], [{4}], {5})".format(
              self.numRows, self.numCols, colPtrs, rowIndices, values, self.isTransposed
          )
  
      def __reduce__(self):
          return SparseMatrix, (
              self.numRows,
              self.numCols,
              self.colPtrs.tostring(),
              self.rowIndices.tostring(),
              self.values.tostring(),
              int(self.isTransposed),
          )
  
      def __getitem__(self, indices):
          i, j = indices
          if i < 0 or i >= self.numRows:
              raise IndexError("Row index %d is out of range [0, %d)" % (i, self.numRows))
          if j < 0 or j >= self.numCols:
              raise IndexError("Column index %d is out of range [0, %d)" % (j, self.numCols))
  
          # If a CSR matrix is given, then the row index should be searched
          # for in ColPtrs, and the column index should be searched for in the
          # corresponding slice obtained from rowIndices.
          if self.isTransposed:
              j, i = i, j
  
          colStart = self.colPtrs[j]
          colEnd = self.colPtrs[j + 1]
          nz = self.rowIndices[colStart:colEnd]
          ind = np.searchsorted(nz, i) + colStart
          if ind < colEnd and self.rowIndices[ind] == i:
              return self.values[ind]
          else:
              return 0.0
  
      def toArray(self):
          """
          Return an numpy.ndarray
          """
          A = np.zeros((self.numRows, self.numCols), dtype=np.float64, order="F")
          for k in range(self.colPtrs.size - 1):
              startptr = self.colPtrs[k]
              endptr = self.colPtrs[k + 1]
              if self.isTransposed:
                  A[k, self.rowIndices[startptr:endptr]] = self.values[startptr:endptr]
              else:
                  A[self.rowIndices[startptr:endptr], k] = self.values[startptr:endptr]
          return A
  
      def toDense(self):
          densevals = np.ravel(self.toArray(), order="F")
          return DenseMatrix(self.numRows, self.numCols, densevals)
  
      def asML(self):
          """
          Convert this matrix to the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          .. versionadded:: 2.0.0
  
          Returns
          -------
          :py:class:`pyspark.ml.linalg.SparseMatrix`
          """
          return newlinalg.SparseMatrix(
              self.numRows,
              self.numCols,
              self.colPtrs,
              self.rowIndices,
              self.values,
              self.isTransposed,
          )
  
      # TODO: More efficient implementation:
      def __eq__(self, other):
          return np.all(self.toArray() == other.toArray())
  
  
  class Matrices(object):
      @staticmethod
      def dense(numRows, numCols, values):
          """
          Create a DenseMatrix
          """
          return DenseMatrix(numRows, numCols, values)
  
      @staticmethod
      def sparse(numRows, numCols, colPtrs, rowIndices, values):
          """
          Create a SparseMatrix
          """
          return SparseMatrix(numRows, numCols, colPtrs, rowIndices, values)
  
      @staticmethod
      def fromML(mat):
          """
          Convert a matrix from the new mllib-local representation.
          This does NOT copy the data; it copies references.
  
          .. versionadded:: 2.0.0
  
          Parameters
          ----------
          mat : :py:class:`pyspark.ml.linalg.Matrix`
  
          Returns
          -------
          :py:class:`pyspark.mllib.linalg.Matrix`
          """
          if isinstance(mat, newlinalg.DenseMatrix):
              return DenseMatrix(mat.numRows, mat.numCols, mat.values, mat.isTransposed)
          elif isinstance(mat, newlinalg.SparseMatrix):
              return SparseMatrix(
                  mat.numRows, mat.numCols, mat.colPtrs, mat.rowIndices, mat.values, mat.isTransposed
              )
          else:
              raise TypeError("Unsupported matrix type %s" % type(mat))
  
  
  class QRDecomposition(object):
      """
      Represents QR factors.
      """
  
      def __init__(self, Q, R):
          self._Q = Q
          self._R = R
  
      @property
      @since("2.0.0")
      def Q(self):
          """
          An orthogonal matrix Q in a QR decomposition.
          May be null if not computed.
          """
          return self._Q
  
      @property
      @since("2.0.0")
      def R(self):
          """
          An upper triangular matrix R in a QR decomposition.
          """
          return self._R
  
  
  def _test():
      import doctest
      import numpy
  
      try:
          # Numpy 1.14+ changed it's string format.
          numpy.set_printoptions(legacy="1.13")
      except TypeError:
          pass
      (failure_count, test_count) = doctest.testmod(optionflags=doctest.ELLIPSIS)
      if failure_count:
          sys.exit(-1)
  
  
  if __name__ == "__main__":
      _test()
  