def _scatter_impl(x, y, scatter_op, treedef, static_idx, dynamic_idx,
indices_are_sorted, unique_indices, normalize_indices):
dtype = lax.dtype(x)
x, y = jnp._promote_dtypes(x, y)
idx = jnp._merge_static_and_dynamic_indices(treedef, static_idx,
dynamic_idx)
indexer = jnp._index_to_gather(jnp.shape(x),
idx,
normalize_indices=normalize_indices)
# Broadcast `y` to the slice output shape.
y = jnp.broadcast_to(y, tuple(indexer.slice_shape))
# Collapse any `None`/`jnp.newaxis` dimensions.
y = jnp.squeeze(y, axis=indexer.newaxis_dims)
if indexer.reversed_y_dims:
y = lax.rev(y, indexer.reversed_y_dims)
# Transpose the gather dimensions into scatter dimensions (cf.
# lax._gather_transpose_rule)
dnums = lax.ScatterDimensionNumbers(
update_window_dims=indexer.dnums.offset_dims,
inserted_window_dims=indexer.dnums.collapsed_slice_dims,
scatter_dims_to_operand_dims=indexer.dnums.start_index_map)
out = scatter_op(x,
indexer.gather_indices,
y,
dnums,
indices_are_sorted=indices_are_sorted,
unique_indices=unique_indices)
return lax.convert_element_type(out, dtype)
def matrix_power(a, n):
a = _promote_arg_dtypes(jnp.asarray(a))
if a.ndim < 2:
raise TypeError("{}-dimensional array given. Array must be at least "
"two-dimensional".format(a.ndim))
if a.shape[-2] != a.shape[-1]:
raise TypeError("Last 2 dimensions of the array must be square")
try:
n = operator.index(n)
except TypeError as err:
raise TypeError(
"exponent must be an integer, got {}".format(n)) from err
if n == 0:
return jnp.broadcast_to(jnp.eye(a.shape[-2], dtype=a.dtype), a.shape)
elif n < 0:
a = inv(a)
n = np.abs(n)
if n == 1:
return a
elif n == 2:
return a @ a
elif n == 3:
return (a @ a) @ a
z = result = None
while n > 0:
z = a if z is None else (z @ z)
n, bit = divmod(n, 2)
if bit:
result = z if result is None else (result @ z)
return result
def wrapped(*args):
error_context = ("on vectorized function with excluded={!r} and "
"signature={!r}".format(excluded, signature))
excluded_func, args = _apply_excluded(pyfunc, excluded, args)
args = tuple(map(jnp.asarray, args))
if signature is not None:
input_core_dims, output_core_dims = _parse_gufunc_signature(signature)
else:
input_core_dims = [()] * len(args)
output_core_dims = None
broadcast_shape, dim_sizes = _parse_input_dimensions(
args, input_core_dims, error_context)
checked_func = _check_output_dims(
excluded_func, dim_sizes, output_core_dims, error_context)
# Rather than broadcasting all arguments to full broadcast shapes, prefer
# expanding dimensions using vmap when possible. By pushing broadcasting
# into vmap, we can make use of more efficient batching rules for
# primitives where only some arguments are batched (e.g., for
# lax_linalg.triangular_solve).
vec_args = []
vmap_counts = []
for arg, core_dims in zip(args, input_core_dims):
# Explicitly broadcast the dimensions already found on each argument,
# because these dimensiosns might be of size 1, which vmap doesn't
# handle.
# TODO(shoyer): Consider squeezing out size 1 dimensions instead, and
# doing all vectorization with vmap? This *might* be a little more
# efficient but would require more careful book-keeping.
core_shape = tuple(dim_sizes[dim] for dim in core_dims)
full_shape = broadcast_shape + core_shape
vec_shape = full_shape[-arg.ndim:] if arg.ndim else ()
vec_arg = jnp.broadcast_to(arg, vec_shape)
vec_args.append(vec_arg)
vmap_count = len(vec_shape) - len(core_shape)
vmap_counts.append(vmap_count)
vectorized_func = checked_func
while any(vmap_counts):
in_axes = tuple(0 if c > 0 else None for c in vmap_counts)
vmap_counts = [max(c - 1, 0) for c in vmap_counts]
vectorized_func = api.vmap(vectorized_func, in_axes)
return vectorized_func(*vec_args)
def _empty_svd(a, *, full_matrices, compute_uv):
batch_shape = a.shape[:-2]
m, n = a.shape[-2:]
s = jnp.empty(batch_shape + (0,), dtype=lax_internal._complex_basetype(a.dtype))
if not compute_uv:
return (s,)
if full_matrices:
size = max(m, n)
u = jnp.broadcast_to(jnp.eye(size, dtype=a.dtype), batch_shape + (size, size))
else:
u = jnp.empty(batch_shape + (m, n), dtype=a.dtype)
v = jnp.empty(batch_shape + (0, 0), dtype=a.dtype)
if m < n:
u, v = v, u
return s, u, v
def _scatter_impl(x, y, scatter_op, treedef, static_idx, dynamic_idx,
indices_are_sorted, unique_indices, mode, normalize_indices):
dtype = lax.dtype(x)
weak_type = dtypes.is_weakly_typed(x)
if dtype != dtypes.result_type(x, y):
# TODO(jakevdp): change this to an error after the deprecation period.
warnings.warn(
"scatter inputs have incompatible types: cannot safely cast "
f"value from dtype={lax.dtype(y)} to dtype={lax.dtype(x)}. "
"In future JAX releases this will result in an error.",
FutureWarning)
idx = jnp._merge_static_and_dynamic_indices(treedef, static_idx,
dynamic_idx)
indexer = jnp._index_to_gather(jnp.shape(x),
idx,
normalize_indices=normalize_indices)
# Avoid calling scatter if the slice shape is empty, both as a fast path and
# to handle cases like zeros(0)[array([], int32)].
if core.is_empty_shape(indexer.slice_shape):
return x
x, y = jnp._promote_dtypes(x, y)
# Broadcast `y` to the slice output shape.
y = jnp.broadcast_to(y, tuple(indexer.slice_shape))
# Collapse any `None`/`jnp.newaxis` dimensions.
y = jnp.squeeze(y, axis=indexer.newaxis_dims)
if indexer.reversed_y_dims:
y = lax.rev(y, indexer.reversed_y_dims)
# Transpose the gather dimensions into scatter dimensions (cf.
# lax._gather_transpose_rule)
dnums = lax.ScatterDimensionNumbers(
update_window_dims=indexer.dnums.offset_dims,
inserted_window_dims=indexer.dnums.collapsed_slice_dims,
scatter_dims_to_operand_dims=indexer.dnums.start_index_map)
out = scatter_op(x,
indexer.gather_indices,
y,
dnums,
indices_are_sorted=indexer.indices_are_sorted
or indices_are_sorted,
unique_indices=indexer.unique_indices or unique_indices,
mode=mode)
return lax_internal._convert_element_type(out, dtype, weak_type)
def _scatter_impl(x, y, scatter_op, treedef, static_idx, dynamic_idx,
indices_are_sorted, unique_indices, mode, normalize_indices):
dtype = lax.dtype(x)
weak_type = dtypes.is_weakly_typed(x)
idx = jnp._merge_static_and_dynamic_indices(treedef, static_idx,
dynamic_idx)
indexer = jnp._index_to_gather(jnp.shape(x),
idx,
normalize_indices=normalize_indices)
# Avoid calling scatter if the slice shape is empty, both as a fast path and
# to handle cases like zeros(0)[array([], int32)].
if core.is_empty_shape(indexer.slice_shape):
return x
x, y = jnp._promote_dtypes(x, y)
# Broadcast `y` to the slice output shape.
y = jnp.broadcast_to(y, tuple(indexer.slice_shape))
# Collapse any `None`/`jnp.newaxis` dimensions.
y = jnp.squeeze(y, axis=indexer.newaxis_dims)
if indexer.reversed_y_dims:
y = lax.rev(y, indexer.reversed_y_dims)
# Transpose the gather dimensions into scatter dimensions (cf.
# lax._gather_transpose_rule)
dnums = lax.ScatterDimensionNumbers(
update_window_dims=indexer.dnums.offset_dims,
inserted_window_dims=indexer.dnums.collapsed_slice_dims,
scatter_dims_to_operand_dims=indexer.dnums.start_index_map)
out = scatter_op(x,
indexer.gather_indices,
y,
dnums,
indices_are_sorted=indexer.indices_are_sorted
or indices_are_sorted,
unique_indices=indexer.unique_indices or unique_indices,
mode=mode)
return lax._convert_element_type(out, dtype, weak_type)
def _solve(a, b):
_check_solve_shapes(a, b)
# Broadcast leading dimensions of b to the shape of a, as is required by
# custom_linear_solve.
out_shape = tuple(d_a if d_b == 1 else d_b
for d_a, d_b in zip(a.shape[:-1] + (1,), b.shape))
b = jnp.broadcast_to(b, out_shape)
# With custom_linear_solve, we can reuse the same factorization when
# computing sensitivities. This is considerably faster.
lu_, _, permutation = lu(lax.stop_gradient(a))
custom_solve = partial(
lax.custom_linear_solve,
lambda x: _matvec_multiply(a, x),
solve=lambda _, x: lu_solve(lu_, permutation, x, trans=0),
transpose_solve=lambda _, x: lu_solve(lu_, permutation, x, trans=1))
if a.ndim == b.ndim + 1:
# b.shape == [..., m]
return custom_solve(b)
else:
# b.shape == [..., m, k]
return api.vmap(custom_solve, b.ndim - 1, max(a.ndim, b.ndim) - 1)(b)
def _cofactor_solve(a, b):
"""Equivalent to det(a)*solve(a, b) for nonsingular mat.
Intermediate function used for jvp and vjp of det.
This function borrows heavily from jax.numpy.linalg.solve and
jax.numpy.linalg.slogdet to compute the gradient of the determinant
in a way that is well defined even for low rank matrices.
This function handles two different cases:
* rank(a) == n or n-1
* rank(a) < n-1
For rank n-1 matrices, the gradient of the determinant is a rank 1 matrix.
Rather than computing det(a)*solve(a, b), which would return NaN, we work
directly with the LU decomposition. If a = p @ l @ u, then
det(a)*solve(a, b) =
prod(diag(u)) * u^-1 @ l^-1 @ p^-1 b =
prod(diag(u)) * triangular_solve(u, solve(p @ l, b))
If a is rank n-1, then the lower right corner of u will be zero and the
triangular_solve will fail.
Let x = solve(p @ l, b) and y = det(a)*solve(a, b).
Then y_{n}
x_{n} / u_{nn} * prod_{i=1...n}(u_{ii}) =
x_{n} * prod_{i=1...n-1}(u_{ii})
So by replacing the lower-right corner of u with prod_{i=1...n-1}(u_{ii})^-1
we can avoid the triangular_solve failing.
To correctly compute the rest of y_{i} for i != n, we simply multiply
x_{i} by det(a) for all i != n, which will be zero if rank(a) = n-1.
For the second case, a check is done on the matrix to see if `solve`
returns NaN or Inf, and gives a matrix of zeros as a result, as the
gradient of the determinant of a matrix with rank less than n-1 is 0.
This will still return the correct value for rank n-1 matrices, as the check
is applied *after* the lower right corner of u has been updated.
Args:
a: A square matrix or batch of matrices, possibly singular.
b: A matrix, or batch of matrices of the same dimension as a.
Returns:
det(a) and cofactor(a)^T*b, aka adjugate(a)*b
"""
a = _promote_arg_dtypes(jnp.asarray(a))
b = _promote_arg_dtypes(jnp.asarray(b))
a_shape = jnp.shape(a)
b_shape = jnp.shape(b)
a_ndims = len(a_shape)
if not (a_ndims >= 2 and a_shape[-1] == a_shape[-2]
and b_shape[-2:] == a_shape[-2:]):
msg = ("The arguments to _cofactor_solve must have shapes "
"a=[..., m, m] and b=[..., m, m]; got a={} and b={}")
raise ValueError(msg.format(a_shape, b_shape))
if a_shape[-1] == 1:
return a[..., 0, 0], b
# lu contains u in the upper triangular matrix and l in the strict lower
# triangular matrix.
# The diagonal of l is set to ones without loss of generality.
lu, pivots, permutation = lax_linalg.lu(a)
dtype = lax.dtype(a)
batch_dims = lax.broadcast_shapes(lu.shape[:-2], b.shape[:-2])
x = jnp.broadcast_to(b, batch_dims + b.shape[-2:])
lu = jnp.broadcast_to(lu, batch_dims + lu.shape[-2:])
# Compute (partial) determinant, ignoring last diagonal of LU
diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
parity = jnp.count_nonzero(pivots != jnp.arange(a_shape[-1]), axis=-1)
sign = jnp.asarray(-2 * (parity % 2) + 1, dtype=dtype)
# partial_det[:, -1] contains the full determinant and
# partial_det[:, -2] contains det(u) / u_{nn}.
partial_det = jnp.cumprod(diag, axis=-1) * sign[..., None]
lu = lu.at[..., -1, -1].set(1.0 / partial_det[..., -2])
permutation = jnp.broadcast_to(permutation, batch_dims + (a_shape[-1], ))
iotas = jnp.ix_(*(lax.iota(jnp.int32, b) for b in batch_dims + (1, )))
# filter out any matrices that are not full rank
d = jnp.ones(x.shape[:-1], x.dtype)
d = lax_linalg.triangular_solve(lu, d, left_side=True, lower=False)
d = jnp.any(jnp.logical_or(jnp.isnan(d), jnp.isinf(d)), axis=-1)
d = jnp.tile(d[..., None, None], d.ndim * (1, ) + x.shape[-2:])
x = jnp.where(d, jnp.zeros_like(x), x) # first filter
x = x[iotas[:-1] + (permutation, slice(None))]
x = lax_linalg.triangular_solve(lu,
x,
left_side=True,
lower=True,
unit_diagonal=True)
x = jnp.concatenate(
(x[..., :-1, :] * partial_det[..., -1, None, None], x[..., -1:, :]),
axis=-2)
x = lax_linalg.triangular_solve(lu, x, left_side=True, lower=False)
x = jnp.where(d, jnp.zeros_like(x), x) # second filter
return partial_det[..., -1], x
def polygamma(n, x):
assert jnp.issubdtype(lax.dtype(n), jnp.integer)
n, x = _promote_args_inexact("polygamma", n, x)
shape = lax.broadcast_shapes(n.shape, x.shape)
return _polygamma(jnp.broadcast_to(n, shape), jnp.broadcast_to(x, shape))
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 _gen_associated_legendre(l_max: int, x: jnp.ndarray,
is_normalized: bool) -> jnp.ndarray:
r"""Computes associated Legendre functions (ALFs) of the first kind.
The ALFs of the first kind are used in spherical harmonics. The spherical
harmonic of degree `l` and order `m` can be written as
`Y_l^m(θ, φ) = N_l^m * P_l^m(cos(θ)) * exp(i m φ)`, where `N_l^m` is the
normalization factor and θ and φ are the colatitude and longitude,
repectively. `N_l^m` is chosen in the way that the spherical harmonics form
a set of orthonormal basis function of L^2(S^2). For the computational
efficiency of spherical harmonics transform, the normalization factor is
used in the computation of the ALFs. In addition, normalizing `P_l^m`
avoids overflow/underflow and achieves better numerical stability. Three
recurrence relations are used in the computation.
Args:
l_max: The maximum degree of the associated Legendre function. Both the
degrees and orders are `[0, 1, 2, ..., l_max]`.
x: A vector of type `float32`, `float64` containing the sampled points in
spherical coordinates, at which the ALFs are computed; `x` is essentially
`cos(θ)`. For the numerical integration used by the spherical harmonics
transforms, `x` contains the quadrature points in the interval of
`[-1, 1]`. There are several approaches to provide the quadrature points:
Gauss-Legendre method (`scipy.special.roots_legendre`), Gauss-Chebyshev
method (`scipy.special.roots_chebyu`), and Driscoll & Healy
method (Driscoll, James R., and Dennis M. Healy. "Computing Fourier
transforms and convolutions on the 2-sphere." Advances in applied
mathematics 15, no. 2 (1994): 202-250.). The Gauss-Legendre quadrature
points are nearly equal-spaced along θ and provide exact discrete
orthogonality, (P^m)^T W P_m = I, where `T` represents the transpose
operation, `W` is a diagonal matrix containing the quadrature weights,
and `I` is the identity matrix. The Gauss-Chebyshev points are equally
spaced, which only provide approximate discrete orthogonality. The
Driscoll & Healy qudarture points are equally spaced and provide the
exact discrete orthogonality. The number of sampling points is required to
be twice as the number of frequency points (modes) in the Driscoll & Healy
approach, which enables FFT and achieves a fast spherical harmonics
transform.
is_normalized: True if the associated Legendre functions are normalized.
With normalization, `N_l^m` is applied such that the spherical harmonics
form a set of orthonormal basis functions of L^2(S^2).
Returns:
The 3D array of shape `(l_max + 1, l_max + 1, len(x))` containing the values
of the ALFs at `x`; the dimensions in the sequence of order, degree, and
evalution points.
"""
p = jnp.zeros((l_max + 1, l_max + 1, x.shape[0]))
a_idx = jnp.arange(1, l_max + 1)
b_idx = jnp.arange(l_max)
if is_normalized:
initial_value = 0.5 / jnp.sqrt(jnp.pi) # The initial value p(0,0).
f_a = jnp.cumprod(-1 * jnp.sqrt(1.0 + 0.5 / a_idx))
f_b = jnp.sqrt(2.0 * b_idx + 3.0)
else:
initial_value = 1.0 # The initial value p(0,0).
f_a = jnp.cumprod(1.0 - 2.0 * a_idx)
f_b = 2.0 * b_idx + 1.0
p = p.at[(0, 0)].set(initial_value)
# Compute the diagonal entries p(l,l) with recurrence.
y = jnp.cumprod(jnp.broadcast_to(jnp.sqrt(1.0 - x * x),
(l_max, x.shape[0])),
axis=0)
p_diag = initial_value * jnp.einsum('i,ij->ij', f_a, y)
diag_indices = jnp.diag_indices(l_max + 1)
p = p.at[(diag_indices[0][1:], diag_indices[1][1:])].set(p_diag)
# Compute the off-diagonal entries with recurrence.
p_offdiag = jnp.einsum('ij,ij->ij', jnp.einsum('i,j->ij', f_b, x),
p[jnp.diag_indices(l_max)])
offdiag_indices = (diag_indices[0][:l_max], diag_indices[1][:l_max] + 1)
p = p.at[offdiag_indices].set(p_offdiag)
# Compute the remaining entries with recurrence.
d0_mask_3d, d1_mask_3d = _gen_recurrence_mask(l_max,
is_normalized=is_normalized)
def body_fun(i, p_val):
coeff_0 = d0_mask_3d[i]
coeff_1 = d1_mask_3d[i]
h = (jnp.einsum(
'ij,ijk->ijk', coeff_0,
jnp.einsum('ijk,k->ijk', jnp.roll(p_val, shift=1, axis=1), x)) -
jnp.einsum('ij,ijk->ijk', coeff_1, jnp.roll(
p_val, shift=2, axis=1)))
p_val = p_val + h
return p_val
if l_max > 1:
p = lax.fori_loop(lower=2,
upper=l_max + 1,
body_fun=body_fun,
init_val=p)
return p