jax.numpy.linalg.multi_dot(arrays, *, precision=None)[source]#

Efficiently compute matrix products between a sequence of arrays.

JAX implementation of numpy.linalg.multi_dot().

JAX internally uses the opt_einsum library to compute the most efficient operation order.

  • arrays (Sequence[jax.typing.ArrayLike]) – sequence of arrays. All must be two-dimensional, except the first and last which may be one-dimensional.

  • precision (str | Precision | tuple[str, str] | tuple[Precision, Precision] | None) – either None (default), which means the default precision for the backend, a Precision enum value (Precision.DEFAULT, Precision.HIGH or Precision.HIGHEST).


an array representing the equivalent of reduce(jnp.matmul, arrays), but evaluated in the optimal order.

Return type:


This function exists because the cost of computing sequences of matmul operations can differ vastly depending on the order in which the operations are evaluated. For a single matmul, the number of floating point operations (flops) required to compute a matrix product can be approximated this way:

>>> def approx_flops(x, y):
...   # for 2D x and y, with x.shape[1] == y.shape[0]
...   return 2 * x.shape[0] * x.shape[1] * y.shape[1]

Suppose we have three matrices that we’d like to multiply in sequence:

>>> key1, key2, key3 = jax.random.split(jax.random.key(0), 3)
>>> x = jax.random.normal(key1, shape=(200, 5))
>>> y = jax.random.normal(key2, shape=(5, 100))
>>> z = jax.random.normal(key3, shape=(100, 10))

Because of associativity of matrix products, there are two orders in which we might evaluate the product x @ y @ z, and both produce equivalent outputs up to floating point precision:

>>> result1 = (x @ y) @ z
>>> result2 = x @ (y @ z)
>>> jnp.allclose(result1, result2, atol=1E-4)
Array(True, dtype=bool)

But the computational cost of these differ greatly:

>>> print("(x @ y) @ z flops:", approx_flops(x, y) + approx_flops(x @ y, z))
(x @ y) @ z flops: 600000
>>> print("x @ (y @ z) flops:", approx_flops(y, z) + approx_flops(x, y @ z))
x @ (y @ z) flops: 30000

The second approach is about 20x more efficient in terms of estimated flops!

multi_dot is a function that will automatically choose the fastest computational path for such problems:

>>> result3 = jnp.linalg.multi_dot([x, y, z])
>>> jnp.allclose(result1, result3, atol=1E-4)
Array(True, dtype=bool)

We can use JAX’s Ahead-of-time lowering and compilation tools to estimate the total flops of each approach, and confirm that multi_dot is choosing the more efficient option:

>>> jax.jit(lambda x, y, z: (x @ y) @ z).lower(x, y, z).cost_analysis()['flops']
>>> jax.jit(lambda x, y, z: x @ (y @ z)).lower(x, y, z).cost_analysis()['flops']
>>> jax.jit(jnp.linalg.multi_dot).lower([x, y, z]).cost_analysis()['flops']