def _lu_jvp_rule(primals, tangents):
a, = primals
a_dot, = tangents
lu, pivots, permutation = lu_p.bind(a)
a_shape = jnp.shape(a)
m, n = a_shape[-2:]
dtype = lax.dtype(a)
k = min(m, n)
batch_dims = a_shape[:-2]
iotas = jnp.ix_(*(lax.iota(jnp.int32, b) for b in batch_dims + (1, )))
x = a_dot[iotas[:-1] + (permutation, slice(None))]
# Differentiation of Matrix Functionals Using Triangular Factorization
# F. R. De Hoog, R. S. Anderssen, and M. A. Lukas
#
# LU = A
# ==> L'U + LU' = A'
# ==> inv(L) . L' + U' . inv(U) = inv(L) A' inv(U)
# ==> L' = L . tril(inv(L) . A' . inv(U), -1)
# U' = triu(inv(L) . A' . inv(U)) . U
ndims = len(a_shape)
l_padding = [(0, 0, 0)] * ndims
l_padding[-1] = (0, m - k, 0)
zero = jnp._constant_like(lu, 0)
l = lax.pad(jnp.tril(lu[..., :, :k], -1), zero, l_padding)
l = l + jnp.eye(m, m, dtype=dtype)
u_eye = lax.pad(jnp.eye(n - k, n - k, dtype=dtype), zero,
((k, 0, 0), (k, 0, 0)))
u_padding = [(0, 0, 0)] * ndims
u_padding[-2] = (0, n - k, 0)
u = lax.pad(jnp.triu(lu[..., :k, :]), zero, u_padding) + u_eye
la = triangular_solve(l,
x,
left_side=True,
transpose_a=False,
lower=True,
unit_diagonal=True)
lau = triangular_solve(u,
la,
left_side=False,
transpose_a=False,
lower=False)
l_dot = jnp.matmul(l, jnp.tril(lau, -1))
u_dot = jnp.matmul(jnp.triu(lau), u)
lu_dot = l_dot + u_dot
return (lu, pivots, permutation), (lu_dot, ad_util.Zero.from_value(pivots),
ad_util.Zero.from_value(permutation))
def triangular_solve_jvp_rule_a(g_a, ans, a, b, left_side, lower, transpose_a,
conjugate_a, unit_diagonal):
m, n = b.shape[-2:]
k = 1 if unit_diagonal else 0
g_a = jnp.tril(g_a, k=-k) if lower else jnp.triu(g_a, k=k)
g_a = lax.neg(g_a)
g_a = jnp.swapaxes(g_a, -1, -2) if transpose_a else g_a
g_a = jnp.conj(g_a) if conjugate_a else g_a
dot = partial(lax.dot if g_a.ndim == 2 else lax.batch_matmul,
precision=lax.Precision.HIGHEST)
def a_inverse(rhs):
return triangular_solve(a, rhs, left_side, lower, transpose_a,
conjugate_a, unit_diagonal)
# triangular_solve is about the same cost as matrix multplication (~n^2 FLOPs
# for matrix/vector inputs). Order these operations in whichever order is
# cheaper.
if left_side:
assert g_a.shape[-2:] == a.shape[-2:] == (m, m) and ans.shape[-2:] == (
m, n)
if m > n:
return a_inverse(dot(g_a, ans)) # A^{-1} (∂A X)
else:
return dot(a_inverse(g_a), ans) # (A^{-1} ∂A) X
else:
assert g_a.shape[-2:] == a.shape[-2:] == (n, n) and ans.shape[-2:] == (
m, n)
if m < n:
return a_inverse(dot(ans, g_a)) # (X ∂A) A^{-1}
else:
return dot(ans, a_inverse(g_a)) # X (∂A A^{-1})
def funm(A, func, disp=True):
A = jnp.asarray(A)
if A.ndim != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected square array_like input')
T, Z = schur(A)
T, Z = rsf2csf(T, Z)
F = jnp.diag(func(jnp.diag(T)))
F = F.astype(T.dtype.char)
F, minden = _algorithm_11_1_1(F, T)
F = Z @ F @ Z.conj().T
if disp:
return F
if F.dtype.char.lower() == 'e':
tol = jnp.finfo(jnp.float16).eps
if F.dtype.char.lower() == 'f':
tol = jnp.finfo(jnp.float32).eps
else:
tol = jnp.finfo(jnp.float64).eps
minden = jnp.where(minden == 0.0, tol, minden)
err = jnp.where(jnp.any(jnp.isinf(F)), jnp.inf, jnp.minimum(1, jnp.maximum(
tol, (tol / minden) * norm(jnp.triu(T, 1), 1))))
return F, err
def _lu(a, permute_l):
a = np_linalg._promote_arg_dtypes(jnp.asarray(a))
lu, pivots, permutation = lax_linalg.lu(a)
dtype = lax.dtype(a)
m, n = jnp.shape(a)
p = jnp.real(jnp.array(permutation == jnp.arange(m)[:, None], dtype=dtype))
k = min(m, n)
l = jnp.tril(lu, -1)[:, :k] + jnp.eye(m, k, dtype=dtype)
u = jnp.triu(lu)[:k, :]
if permute_l:
return jnp.matmul(p, l), u
else:
return p, l, u
def _lu(a, permute_l):
a, = _promote_dtypes_inexact(jnp.asarray(a))
lu, _, permutation = lax_linalg.lu(a)
dtype = lax.dtype(a)
m, n = jnp.shape(a)
p = jnp.real(
jnp.array(permutation[None, :] == jnp.arange(
m, dtype=permutation.dtype)[:, None],
dtype=dtype))
k = min(m, n)
l = jnp.tril(lu, -1)[:, :k] + jnp.eye(m, k, dtype=dtype)
u = jnp.triu(lu)[:k, :]
if permute_l:
return jnp.matmul(p, l), u
else:
return p, l, u
def triu(m, k=0):
return jnp.triu(m, k)
