def body(k, state):
pivot, perm, a = state
m_idx = jnp.arange(m)
n_idx = jnp.arange(n)
if jnp.issubdtype(a.dtype, jnp.complexfloating):
t = a[:, k]
magnitude = jnp.abs(jnp.real(t)) + jnp.abs(jnp.imag(t))
else:
magnitude = jnp.abs(a[:, k])
i = jnp.argmax(jnp.where(m_idx >= k, magnitude, -jnp.inf))
pivot = ops.index_update(pivot, ops.index[k], i)
a = ops.index_update(a, ops.index[[k, i], ], a[[i, k], ])
perm = ops.index_update(perm, ops.index[[i, k], ], perm[[k, i], ])
# a[k+1:, k] /= a[k, k], adapted for loop-invariant shapes
x = a[k, k]
a = ops.index_update(a, ops.index[:, k],
jnp.where(m_idx > k, a[:, k] / x, a[:, k]))
# a[k+1:, k+1:] -= jnp.outer(a[k+1:, k], a[k, k+1:])
a = a - jnp.where(
(m_idx[:, None] > k) & (n_idx > k), jnp.outer(a[:, k], a[k, :]),
jnp.array(0, dtype=a.dtype))
return pivot, perm, a
def _rsf2scf_iter(i, TZ):
m = N - i
T, Z = TZ
T, Z = lax.cond(
jnp.abs(T[m, m - 1]) > eps *
(jnp.abs(T[m - 1, m - 1]) + jnp.abs(T[m, m])), _update_T_Z,
lambda m, T, Z: (T, Z), m, T, Z)
T = T.at[m, m - 1].set(0.0)
return T, Z
def _algorithm_11_1_1(F, T):
# Algorithm 11.1.1 from Golub and Van Loan "Matrix Computations"
N = T.shape[0]
minden = jnp.abs(T[0, 0])
def _outer_loop(p, F_minden):
_, F, minden = lax.fori_loop(1, N-p+1, _inner_loop, (p, *F_minden))
return F, minden
def _inner_loop(i, p_F_minden):
p, F, minden = p_F_minden
j = i+p
s = T[i-1, j-1] * (F[j-1, j-1] - F[i-1, i-1])
T_row, T_col = T[i-1], T[:, j-1]
F_row, F_col = F[i-1], F[:, j-1]
ind = (jnp.arange(N) >= i) & (jnp.arange(N) < j-1)
val = (jnp.where(ind, T_row, 0) @ jnp.where(ind, F_col, 0) -
jnp.where(ind, F_row, 0) @ jnp.where(ind, T_col, 0))
s = s + val
den = T[j-1, j-1] - T[i-1, i-1]
s = jnp.where(den != 0, s / den, s)
F = F.at[i-1, j-1].set(s)
minden = jnp.minimum(minden, jnp.abs(den))
return p, F, minden
return lax.fori_loop(1, N, _outer_loop, (F, minden))
def _slogdet_lu(a):
dtype = lax.dtype(a)
lu, pivot, _ = lax_linalg.lu(a)
diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
is_zero = jnp.any(diag == jnp.array(0, dtype=dtype), axis=-1)
iota = lax.expand_dims(jnp.arange(a.shape[-1]), range(pivot.ndim - 1))
parity = jnp.count_nonzero(pivot != iota, axis=-1)
if jnp.iscomplexobj(a):
sign = jnp.prod(diag / jnp.abs(diag), axis=-1)
else:
sign = jnp.array(1, dtype=dtype)
parity = parity + jnp.count_nonzero(diag < 0, axis=-1)
sign = jnp.where(is_zero, jnp.array(0, dtype=dtype),
sign * jnp.array(-2 * (parity % 2) + 1, dtype=dtype))
logdet = jnp.where(is_zero, jnp.array(-jnp.inf, dtype=dtype),
jnp.sum(jnp.log(jnp.abs(diag)), axis=-1))
return sign, jnp.real(logdet)
def slogdet(a):
a = _promote_arg_dtypes(jnp.asarray(a))
dtype = lax.dtype(a)
a_shape = jnp.shape(a)
if len(a_shape) < 2 or a_shape[-1] != a_shape[-2]:
msg = "Argument to slogdet() must have shape [..., n, n], got {}"
raise ValueError(msg.format(a_shape))
lu, pivot, _ = lax_linalg.lu(a)
diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
is_zero = jnp.any(diag == jnp.array(0, dtype=dtype), axis=-1)
parity = jnp.count_nonzero(pivot != jnp.arange(a_shape[-1]), axis=-1)
if jnp.iscomplexobj(a):
sign = jnp.prod(diag / jnp.abs(diag), axis=-1)
else:
sign = jnp.array(1, dtype=dtype)
parity = parity + jnp.count_nonzero(diag < 0, axis=-1)
sign = jnp.where(is_zero, jnp.array(0, dtype=dtype),
sign * jnp.array(-2 * (parity % 2) + 1, dtype=dtype))
logdet = jnp.where(is_zero, jnp.array(-jnp.inf, dtype=dtype),
jnp.sum(jnp.log(jnp.abs(diag)), axis=-1))
return sign, jnp.real(logdet)
def _inner_loop(i, p_F_minden):
p, F, minden = p_F_minden
j = i+p
s = T[i-1, j-1] * (F[j-1, j-1] - F[i-1, i-1])
T_row, T_col = T[i-1], T[:, j-1]
F_row, F_col = F[i-1], F[:, j-1]
ind = (jnp.arange(N) >= i) & (jnp.arange(N) < j-1)
val = (jnp.where(ind, T_row, 0) @ jnp.where(ind, F_col, 0) -
jnp.where(ind, F_row, 0) @ jnp.where(ind, T_col, 0))
s = s + val
den = T[j-1, j-1] - T[i-1, i-1]
s = jnp.where(den != 0, s / den, s)
F = F.at[i-1, j-1].set(s)
minden = jnp.minimum(minden, jnp.abs(den))
return p, F, minden
def _slogdet_qr(a):
# Implementation of slogdet using QR decomposition. One reason we might prefer
# QR decomposition is that it is more amenable to a fast batched
# implementation on TPU because of the lack of row pivoting.
if jnp.issubdtype(lax.dtype(a), jnp.complexfloating):
raise NotImplementedError("slogdet method='qr' not implemented for complex "
"inputs")
n = a.shape[-1]
a, taus = lax_linalg.geqrf(a)
# The determinant of a triangular matrix is the product of its diagonal
# elements. We are working in log space, so we compute the magnitude as the
# the trace of the log-absolute values, and we compute the sign separately.
log_abs_det = jnp.trace(jnp.log(jnp.abs(a)), axis1=-2, axis2=-1)
sign_diag = jnp.prod(jnp.sign(jnp.diagonal(a, axis1=-2, axis2=-1)), axis=-1)
# The determinant of a Householder reflector is -1. So whenever we actually
# made a reflection (tau != 0), multiply the result by -1.
sign_taus = jnp.prod(jnp.where(taus[..., :(n-1)] != 0, -1, 1), axis=-1).astype(sign_diag.dtype)
return sign_diag * sign_taus, log_abs_det
def norm(x,
ord=None,
axis: Union[None, Tuple[int, ...], int] = None,
keepdims=False):
x = _promote_arg_dtypes(jnp.asarray(x))
x_shape = jnp.shape(x)
ndim = len(x_shape)
if axis is None:
# NumPy has an undocumented behavior that admits arbitrary rank inputs if
# `ord` is None: https://github.com/numpy/numpy/issues/14215
if ord is None:
return jnp.sqrt(
jnp.sum(jnp.real(x * jnp.conj(x)), keepdims=keepdims))
axis = tuple(range(ndim))
elif isinstance(axis, tuple):
axis = tuple(canonicalize_axis(x, ndim) for x in axis)
else:
axis = (canonicalize_axis(axis, ndim), )
num_axes = len(axis)
if num_axes == 1:
if ord is None or ord == 2:
return jnp.sqrt(
jnp.sum(jnp.real(x * jnp.conj(x)),
axis=axis,
keepdims=keepdims))
elif ord == jnp.inf:
return jnp.amax(jnp.abs(x), axis=axis, keepdims=keepdims)
elif ord == -jnp.inf:
return jnp.amin(jnp.abs(x), axis=axis, keepdims=keepdims)
elif ord == 0:
return jnp.sum(x != 0,
dtype=jnp.finfo(lax.dtype(x)).dtype,
axis=axis,
keepdims=keepdims)
elif ord == 1:
# Numpy has a special case for ord == 1 as an optimization. We don't
# really need the optimization (XLA could do it for us), but the Numpy
# code has slightly different type promotion semantics, so we need a
# special case too.
return jnp.sum(jnp.abs(x), axis=axis, keepdims=keepdims)
else:
abs_x = jnp.abs(x)
ord = lax._const(abs_x, ord)
out = jnp.sum(abs_x**ord, axis=axis, keepdims=keepdims)
return jnp.power(out, 1. / ord)
elif num_axes == 2:
row_axis, col_axis = cast(Tuple[int, ...], axis)
if ord is None or ord in ('f', 'fro'):
return jnp.sqrt(
jnp.sum(jnp.real(x * jnp.conj(x)),
axis=axis,
keepdims=keepdims))
elif ord == 1:
if not keepdims and col_axis > row_axis:
col_axis -= 1
return jnp.amax(jnp.sum(jnp.abs(x),
axis=row_axis,
keepdims=keepdims),
axis=col_axis,
keepdims=keepdims)
elif ord == -1:
if not keepdims and col_axis > row_axis:
col_axis -= 1
return jnp.amin(jnp.sum(jnp.abs(x),
axis=row_axis,
keepdims=keepdims),
axis=col_axis,
keepdims=keepdims)
elif ord == jnp.inf:
if not keepdims and row_axis > col_axis:
row_axis -= 1
return jnp.amax(jnp.sum(jnp.abs(x),
axis=col_axis,
keepdims=keepdims),
axis=row_axis,
keepdims=keepdims)
elif ord == -jnp.inf:
if not keepdims and row_axis > col_axis:
row_axis -= 1
return jnp.amin(jnp.sum(jnp.abs(x),
axis=col_axis,
keepdims=keepdims),
axis=row_axis,
keepdims=keepdims)
elif ord in ('nuc', 2, -2):
x = jnp.moveaxis(x, axis, (-2, -1))
if ord == 2:
reducer = jnp.amax
elif ord == -2:
reducer = jnp.amin
else:
reducer = jnp.sum
y = reducer(svd(x, compute_uv=False), axis=-1)
if keepdims:
result_shape = list(x_shape)
result_shape[axis[0]] = 1
result_shape[axis[1]] = 1
y = jnp.reshape(y, result_shape)
return y
else:
raise ValueError("Invalid order '{}' for matrix norm.".format(ord))
else:
raise ValueError(
"Invalid axis values ({}) for jnp.linalg.norm.".format(axis))
def eigh_tridiagonal(d,
e,
*,
eigvals_only=False,
select='a',
select_range=None,
tol=None):
if not eigvals_only:
raise NotImplementedError(
"Calculation of eigenvectors is not implemented")
def _sturm(alpha, beta_sq, pivmin, alpha0_perturbation, x):
"""Implements the Sturm sequence recurrence."""
n = alpha.shape[0]
zeros = jnp.zeros(x.shape, dtype=jnp.int32)
ones = jnp.ones(x.shape, dtype=jnp.int32)
# The first step in the Sturm sequence recurrence
# requires special care if x is equal to alpha[0].
def sturm_step0():
q = alpha[0] - x
count = jnp.where(q < 0, ones, zeros)
q = jnp.where(alpha[0] == x, alpha0_perturbation, q)
return q, count
# Subsequent steps all take this form:
def sturm_step(i, q, count):
q = alpha[i] - beta_sq[i - 1] / q - x
count = jnp.where(q <= pivmin, count + 1, count)
q = jnp.where(q <= pivmin, jnp.minimum(q, -pivmin), q)
return q, count
# The first step initializes q and count.
q, count = sturm_step0()
# Peel off ((n-1) % blocksize) steps from the main loop, so we can run
# the bulk of the iterations unrolled by a factor of blocksize.
blocksize = 16
i = 1
peel = (n - 1) % blocksize
unroll_cnt = peel
def unrolled_steps(args):
start, q, count = args
for j in range(unroll_cnt):
q, count = sturm_step(start + j, q, count)
return start + unroll_cnt, q, count
i, q, count = unrolled_steps((i, q, count))
# Run the remaining steps of the Sturm sequence using a partially
# unrolled while loop.
unroll_cnt = blocksize
def cond(iqc):
i, q, count = iqc
return jnp.less(i, n)
_, _, count = lax.while_loop(cond, unrolled_steps, (i, q, count))
return count
alpha = jnp.asarray(d)
beta = jnp.asarray(e)
supported_dtypes = (jnp.float32, jnp.float64, jnp.complex64,
jnp.complex128)
if alpha.dtype != beta.dtype:
raise TypeError(
"diagonal and off-diagonal values must have same dtype, "
f"got {alpha.dtype} and {beta.dtype}")
if alpha.dtype not in supported_dtypes or beta.dtype not in supported_dtypes:
raise TypeError(
"Only float32 and float64 inputs are supported as inputs "
"to jax.scipy.linalg.eigh_tridiagonal, got "
f"{alpha.dtype} and {beta.dtype}")
n = alpha.shape[0]
if n <= 1:
return jnp.real(alpha)
if jnp.issubdtype(alpha.dtype, jnp.complexfloating):
alpha = jnp.real(alpha)
beta_sq = jnp.real(beta * jnp.conj(beta))
beta_abs = jnp.sqrt(beta_sq)
else:
beta_abs = jnp.abs(beta)
beta_sq = jnp.square(beta)
# Estimate the largest and smallest eigenvalues of T using the Gershgorin
# circle theorem.
off_diag_abs_row_sum = jnp.concatenate(
[beta_abs[:1], beta_abs[:-1] + beta_abs[1:], beta_abs[-1:]], axis=0)
lambda_est_max = jnp.amax(alpha + off_diag_abs_row_sum)
lambda_est_min = jnp.amin(alpha - off_diag_abs_row_sum)
# Upper bound on 2-norm of T.
t_norm = jnp.maximum(jnp.abs(lambda_est_min), jnp.abs(lambda_est_max))
# Compute the smallest allowed pivot in the Sturm sequence to avoid
# overflow.
finfo = np.finfo(alpha.dtype)
one = np.ones([], dtype=alpha.dtype)
safemin = np.maximum(one / finfo.max, (one + finfo.eps) * finfo.tiny)
pivmin = safemin * jnp.maximum(1, jnp.amax(beta_sq))
alpha0_perturbation = jnp.square(finfo.eps * beta_abs[0])
abs_tol = finfo.eps * t_norm
if tol is not None:
abs_tol = jnp.maximum(tol, abs_tol)
# In the worst case, when the absolute tolerance is eps*lambda_est_max and
# lambda_est_max = -lambda_est_min, we have to take as many bisection steps
# as there are bits in the mantissa plus 1.
# The proof is left as an exercise to the reader.
max_it = finfo.nmant + 1
# Determine the indices of the desired eigenvalues, based on select and
# select_range.
if select == 'a':
target_counts = jnp.arange(n, dtype=jnp.int32)
elif select == 'i':
if select_range[0] > select_range[1]:
raise ValueError('Got empty index range in select_range.')
target_counts = jnp.arange(select_range[0],
select_range[1] + 1,
dtype=jnp.int32)
elif select == 'v':
# TODO(phawkins): requires dynamic shape support.
raise NotImplementedError("eigh_tridiagonal(..., select='v') is not "
"implemented")
else:
raise ValueError("'select must have a value in {'a', 'i', 'v'}.")
# Run binary search for all desired eigenvalues in parallel, starting from
# the interval lightly wider than the estimated
# [lambda_est_min, lambda_est_max].
fudge = 2.1 # We widen starting interval the Gershgorin interval a bit.
norm_slack = jnp.array(n, alpha.dtype) * fudge * finfo.eps * t_norm
lower = lambda_est_min - norm_slack - 2 * fudge * pivmin
upper = lambda_est_max + norm_slack + fudge * pivmin
# Pre-broadcast the scalars used in the Sturm sequence for improved
# performance.
target_shape = jnp.shape(target_counts)
lower = jnp.broadcast_to(lower, shape=target_shape)
upper = jnp.broadcast_to(upper, shape=target_shape)
mid = 0.5 * (upper + lower)
pivmin = jnp.broadcast_to(pivmin, target_shape)
alpha0_perturbation = jnp.broadcast_to(alpha0_perturbation, target_shape)
# Start parallel binary searches.
def cond(args):
i, lower, _, upper = args
return jnp.logical_and(jnp.less(i, max_it),
jnp.less(abs_tol, jnp.amax(upper - lower)))
def body(args):
i, lower, mid, upper = args
counts = _sturm(alpha, beta_sq, pivmin, alpha0_perturbation, mid)
lower = jnp.where(counts <= target_counts, mid, lower)
upper = jnp.where(counts > target_counts, mid, upper)
mid = 0.5 * (lower + upper)
return i + 1, lower, mid, upper
_, _, mid, _ = lax.while_loop(cond, body, (0, lower, mid, upper))
return mid
def _mirror_index_fixer(index, size):
s = size - 1 # Half-wavelength of triangular wave
# Scaled, integer-valued version of the triangular wave |x - round(x)|
return jnp.abs((index + s) % (2 * s) - s)
