# numpy.linalg.multi_dot

linalg.multi_dot(arrays, *, out=None)

  Compute the dot product of two or more arrays in a single function call, while automatically selecting the fastest evaluation order.

  `multi_dot` chains `numpy.dot` and uses optimal parenthesization of the matrices [1] [2]. Depending on the shapes of the matrices, this can speed up the multiplication a lot.

  If the first argument is 1-D it is treated as a row vector. If the last argument is 1-D it is treated as a column vector. The other arguments must be 2-D.

  Think of `multi_dot` as:

      def multi_dot(arrays): return functools.reduce(np.dot, arrays)

  Parameters
    arrays sequence of array_like

      If the first argument is 1-D it is treated as row vector. If the last argument is 1-D it is treated as column vector. The other arguments must be 2-D.

    out ndarray, optional

      Output argument. This must have the exact kind that would be returned if it was not used. In particular, it must have the right type, must be C-contiguous, and its dtype must be the dtype that would be returned for dot(a, b). This is a performance feature. Therefore, if these conditions are not met, an exception is raised, instead of attempting to be flexible.

      New in version 1.19.0.

  Returns
    output ndarray

      Returns the dot product of the supplied arrays.

  See also

  `numpy.dot`

    dot multiplication with two arguments.

  Notes

  The cost for a matrix multiplication can be calculated with the following function:

      def cost(A, B):
          return A.shape[0] * A.shape[1] * B.shape[1]

  Assume we have three matrices \(A_{10x100}, B_{100x5}, C_{5x50}\).

  The costs for the two different parenthesizations are as follows:

      cost((AB)C) = 10*100*5 + 10*5*50   = 5000 + 2500   = 7500
      cost(A(BC)) = 10*100*50 + 100*5*50 = 50000 + 25000 = 75000

  References

  1

    Cormen, “Introduction to Algorithms”, Chapter 15.2, p. 370-378

  2

    https://en.wikipedia.org/wiki/Matrix_chain_multiplication

  Examples

  `multi_dot` allows you to write:

      >>> from numpy.linalg import multi_dot
      >>> # Prepare some data
      >>> A = np.random.random((10000, 100))
      >>> B = np.random.random((100, 1000))
      >>> C = np.random.random((1000, 5))
      >>> D = np.random.random((5, 333))
      >>> # the actual dot multiplication
      >>> _ = multi_dot([A, B, C, D])

  instead of:

      >>> _ = np.dot(np.dot(np.dot(A, B), C), D)
      >>> # or
      >>> _ = A.dot(B).dot(C).dot(D)

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