flopscope.numpy.linalg.multi_dot

multi_dot chains flops.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(flops.dot, arrays)

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.

output:ndarray

Returns the dot product of the supplied arrays.

• we.flops.dot dot multiplication with two arguments.

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

multi_dot allows you to write:

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

instead of:

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