Created
December 5, 2016 06:39
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@ray.remote | |
def lu_decomp_invert(lu_decomp): | |
"""Takes the inverse of each of the components in the P, L, U | |
decomposition. Needed as a helper function for the block-level LU decomp. | |
""" | |
return tuple(np.linalg.inv(x) for x in lu_decomp) | |
@ray.remote(num_return_vals=3) | |
def block_lu(a, block_size=100): | |
"""Returns the LU decomposition of a square matrix. | |
Parameters | |
---------- | |
a : array_like | |
Returns | |
------- | |
p : array_like | |
l : array_like | |
u : array_like | |
""" | |
if a.shape[0] <= block_size or a.shape[1] <= block_size: | |
return ray.get(ra.linalg.lu.remote(a)) | |
p, l, u = np.zeros(a.shape), np.zeros(a.shape), np.zeros(a.shape) | |
num_blocks = int(np.ceil(float(a.shape[0]) / block_size)) | |
# Compute all single-node LU decompositions in parallel | |
block_decomps_remote = [] | |
a_modified = a | |
for idx in range(num_blocks): | |
a11 = a_modified[:block_size, :block_size] | |
block_decomps_remote.append(ra.linalg.lu.remote(a11)) | |
a12 = a_modified[:block_size, block_size:] | |
a21 = a_modified[block_size:, :block_size] | |
# Compute the Schur complements | |
a22 = a_modified[block_size:, block_size:] | |
a_modified = a22 - np.dot(a21, np.dot(np.linalg.inv(a11), a12)) | |
block_decomps = ray.get(block_decomps_remote) | |
# Compute the inverses for each of the LU components in parallel | |
block_decomp_inverses = ray.get([lu_decomp_invert.remote(decomp) | |
for decomp in block_decomps_remote]) | |
for idx, (p11, _, _) in enumerate(block_decomps): | |
block_low = block_size * idx | |
block_high = block_low + block_size | |
p[block_low:block_high, block_low:block_high] = p11 | |
# Perform coalescing | |
for idx, (plu, plu_inverse) in enumerate(zip(block_decomps, | |
block_decomp_inverses)): | |
p11, l11, u11 = plu | |
p11_inverse, l11_inverse, u11_inverse = plu_inverse | |
block_low = block_size * idx | |
block_high = block_low + block_size | |
# p[block_low:block_high, block_low:block_high] = p11 | |
if idx < num_blocks - 1: | |
a12 = a[:block_size, block_size:] | |
a21 = a[block_size:, :block_size] | |
a22 = a[block_size:, block_size:] | |
# Inverse of the permutation matrix is just its transpose | |
p22_inverse = p[block_high:, block_high:].T | |
# p22 = p[block_high:, block_high:] | |
# TODO(kvchen): Change this to use distributed block | |
# multiplication. Not sure if it'll be any more efficient. | |
# u12 = np.dot(l11_inverse, np.dot(p11_inverse, a12)) | |
# l21 = np.dot(p22_inverse, np.dot(a21, u11_inverse)) | |
u12, l21 = ray.get([ | |
ra.dot.remote(l11_inverse, ra.dot.remote(p11_inverse, a12)), | |
ra.dot.remote(p22_inverse, ra.dot.remote(a21, u11_inverse)), | |
]) | |
l[block_high:, block_low:block_high] = l21 | |
u[block_low:block_high, block_high:] = u12 | |
# Take the Schur complement and recurse | |
a = a22 - np.dot(l21, u12) | |
l[block_low:block_high, block_low:block_high] = l11 | |
u[block_low:block_high, block_low:block_high] = u11 | |
return p, l, u |
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