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# 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.

  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 \(A_{10 imes 100}, B_{100 imes 5}, C_{5 imes 50}\).

  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:

      >>> import numpy as np
      >>> 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)