Possible speedup for linalg.solve

Looking at the code

Line 20 in 7f0ae77

@export

, linalg.solve is implemented by explicitly calculating the inverse times a vector. Is there a reason not to use something similar to scipy.linalg.solve to achieve this?

import cola
import numpy as np
import scipy as sp
import timeit

A = np.random.randn(2,2)
A = A @ A.T
v = np.random.randn(2)

cola_A = cola.ops.Dense(A)

print(timeit.timeit(lambda: cola.linalg.solve(cola_A, v), number=10000))
print(timeit.timeit(lambda: sp.linalg.solve(A, v), number=10000))

the outputs are

14.14389366703108
0.0880865000654012

I could also try to submit a pull request later. Thanks

Hi @ksnxr ,

Thanks for opening the pull request.

solve(A) (with the default Auto algorithm) uses one of Cholesky, LU, CG, GMRES to solve the linear system.
See inverse/inv.py#L71-L89).
None of them form the inverse matrix explicitly, but for the 2x2 example cola will call the LU solve.

The matrix listed is very small, but the runtime differences are a bit surprising so we should take a look into it.

I played around a bit. There are expressions for "solve" for different backends e.g.

cola/cola/backends/jax_fns.py

Line 56 in 7f0ae77

solve = jnp.linalg.solve

. Directly replacing the solve function with the below seems to make the running times comparable to scipy version.

@export
def solve(A, b, alg=Auto()):
    """ Computes Linear solve of a linear operator. Equivalent to cola.inv

    Args:
        A (LinearOperator): The linear operator to compute the inverse of.
        b (Array): The right hand side of the linear system of shape (d, ) or (d, k)
        alg (Algorithm, optional): The algorithm to use for the solves.

    Returns:
        Array: The solution of the linear system of shape (d, ) or (d, k)

    Example:
        >>> A = MyLinearOperator()
        >>> x = cola.solve(A, b, alg=Auto(max_iters=10, pbar=True))
    """
    xnp = A.xnp
    return xnp.solve(A.to_dense(), b)

Looking at the code for inv, it seems complicated to properly treat the individual cases. I might leave it for now