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import itertools
import functools
import operator
import numpy as np
from math import prod
from . import _bspl # type: ignore[attr-defined]
import scipy.sparse.linalg as ssl
from scipy.sparse import csr_array
from ._bsplines import _not_a_knot
__all__ = ["NdBSpline"]
def _get_dtype(dtype):
"""Return np.complex128 for complex dtypes, np.float64 otherwise."""
if np.issubdtype(dtype, np.complexfloating):
return np.complex128
else:
return np.float64
class NdBSpline:
"""Tensor product spline object.
The value at point ``xp = (x1, x2, ..., xN)`` is evaluated as a linear
combination of products of one-dimensional b-splines in each of the ``N``
dimensions::
c[i1, i2, ..., iN] * B(x1; i1, t1) * B(x2; i2, t2) * ... * B(xN; iN, tN)
Here ``B(x; i, t)`` is the ``i``-th b-spline defined by the knot vector
``t`` evaluated at ``x``.
Parameters
----------
t : tuple of 1D ndarrays
knot vectors in directions 1, 2, ... N,
``len(t[i]) == n[i] + k + 1``
c : ndarray, shape (n1, n2, ..., nN, ...)
b-spline coefficients
k : int or length-d tuple of integers
spline degrees.
A single integer is interpreted as having this degree for
all dimensions.
extrapolate : bool, optional
Whether to extrapolate out-of-bounds inputs, or return `nan`.
Default is to extrapolate.
Attributes
----------
t : tuple of ndarrays
Knots vectors.
c : ndarray
Coefficients of the tensor-product spline.
k : tuple of integers
Degrees for each dimension.
extrapolate : bool, optional
Whether to extrapolate or return nans for out-of-bounds inputs.
Defaults to true.
Methods
-------
__call__
design_matrix
See Also
--------
BSpline : a one-dimensional B-spline object
NdPPoly : an N-dimensional piecewise tensor product polynomial
"""
def __init__(self, t, c, k, *, extrapolate=None):
self._k, self._indices_k1d, (self._t, self._len_t) = _preprocess_inputs(k, t)
if extrapolate is None:
extrapolate = True
self.extrapolate = bool(extrapolate)
self.c = np.asarray(c)
ndim = self._t.shape[0] # == len(self.t)
if self.c.ndim < ndim:
raise ValueError(f"Coefficients must be at least {ndim}-dimensional.")
for d in range(ndim):
td = self.t[d]
kd = self.k[d]
n = td.shape[0] - kd - 1
if self.c.shape[d] != n:
raise ValueError(f"Knots, coefficients and degree in dimension"
f" {d} are inconsistent:"
f" got {self.c.shape[d]} coefficients for"
f" {len(td)} knots, need at least {n} for"
f" k={k}.")
dt = _get_dtype(self.c.dtype)
self.c = np.ascontiguousarray(self.c, dtype=dt)
@property
def k(self):
return tuple(self._k)
@property
def t(self):
# repack the knots into a tuple
return tuple(self._t[d, :self._len_t[d]] for d in range(self._t.shape[0]))
def __call__(self, xi, *, nu=None, extrapolate=None):
"""Evaluate the tensor product b-spline at ``xi``.
Parameters
----------
xi : array_like, shape(..., ndim)
The coordinates to evaluate the interpolator at.
This can be a list or tuple of ndim-dimensional points
or an array with the shape (num_points, ndim).
nu : array_like, optional, shape (ndim,)
Orders of derivatives to evaluate. Each must be non-negative.
Defaults to the zeroth derivivative.
extrapolate : bool, optional
Whether to exrapolate based on first and last intervals in each
dimension, or return `nan`. Default is to ``self.extrapolate``.
Returns
-------
values : ndarray, shape ``xi.shape[:-1] + self.c.shape[ndim:]``
Interpolated values at ``xi``
"""
ndim = self._t.shape[0] # == len(self.t)
if extrapolate is None:
extrapolate = self.extrapolate
extrapolate = bool(extrapolate)
if nu is None:
nu = np.zeros((ndim,), dtype=np.intc)
else:
nu = np.asarray(nu, dtype=np.intc)
if nu.ndim != 1 or nu.shape[0] != ndim:
raise ValueError(
f"invalid number of derivative orders {nu = } for "
f"ndim = {len(self.t)}.")
if any(nu < 0):
raise ValueError(f"derivatives must be positive, got {nu = }")
# prepare xi : shape (..., m1, ..., md) -> (1, m1, ..., md)
xi = np.asarray(xi, dtype=float)
xi_shape = xi.shape
xi = xi.reshape(-1, xi_shape[-1])
xi = np.ascontiguousarray(xi)
if xi_shape[-1] != ndim:
raise ValueError(f"Shapes: xi.shape={xi_shape} and ndim={ndim}")
# complex -> double
was_complex = self.c.dtype.kind == 'c'
cc = self.c
if was_complex and self.c.ndim == ndim:
# make sure that core dimensions are intact, and complex->float
# size doubling only adds a trailing dimension
cc = self.c[..., None]
cc = cc.view(float)
# prepare the coefficients: flatten the trailing dimensions
c1 = cc.reshape(cc.shape[:ndim] + (-1,))
c1r = c1.ravel()
# replacement for np.ravel_multi_index for indexing of `c1`:
_strides_c1 = np.asarray([s // c1.dtype.itemsize
for s in c1.strides], dtype=np.intp)
num_c_tr = c1.shape[-1] # # of trailing coefficients
out = np.empty(xi.shape[:-1] + (num_c_tr,), dtype=c1.dtype)
_bspl.evaluate_ndbspline(xi,
self._t,
self._len_t,
self._k,
nu,
extrapolate,
c1r,
num_c_tr,
_strides_c1,
self._indices_k1d,
out,)
out = out.view(self.c.dtype)
return out.reshape(xi_shape[:-1] + self.c.shape[ndim:])
@classmethod
def design_matrix(cls, xvals, t, k, extrapolate=True):
"""Construct the design matrix as a CSR format sparse array.
Parameters
----------
xvals : ndarray, shape(npts, ndim)
Data points. ``xvals[j, :]`` gives the ``j``-th data point as an
``ndim``-dimensional array.
t : tuple of 1D ndarrays, length-ndim
Knot vectors in directions 1, 2, ... ndim,
k : int
B-spline degree.
extrapolate : bool, optional
Whether to extrapolate out-of-bounds values of raise a `ValueError`
Returns
-------
design_matrix : a CSR array
Each row of the design matrix corresponds to a value in `xvals` and
contains values of b-spline basis elements which are non-zero
at this value.
"""
xvals = np.asarray(xvals, dtype=float)
ndim = xvals.shape[-1]
if len(t) != ndim:
raise ValueError(
f"Data and knots are inconsistent: len(t) = {len(t)} for "
f" {ndim = }."
)
# tabulate the flat indices for iterating over the (k+1)**ndim subarray
k, _indices_k1d, (_t, len_t) = _preprocess_inputs(k, t)
# Precompute the shape and strides of the 'coefficients array'.
# This would have been the NdBSpline coefficients; in the present context
# this is a helper to compute the indices into the colocation matrix.
c_shape = tuple(len_t[d] - k[d] - 1 for d in range(ndim))
# The strides of the coeffs array: the computation is equivalent to
# >>> cstrides = [s // 8 for s in np.empty(c_shape).strides]
cs = c_shape[1:] + (1,)
cstrides = np.cumprod(cs[::-1], dtype=np.intp)[::-1].copy()
# heavy lifting happens here
data, indices, indptr = _bspl._colloc_nd(xvals,
_t,
len_t,
k,
_indices_k1d,
cstrides)
return csr_array((data, indices, indptr))
def _preprocess_inputs(k, t_tpl):
"""Helpers: validate and preprocess NdBSpline inputs.
Parameters
----------
k : int or tuple
Spline orders
t_tpl : tuple or array-likes
Knots.
"""
# 1. Make sure t_tpl is a tuple
if not isinstance(t_tpl, tuple):
raise ValueError(f"Expect `t` to be a tuple of array-likes. "
f"Got {t_tpl} instead."
)
# 2. Make ``k`` a tuple of integers
ndim = len(t_tpl)
try:
len(k)
except TypeError:
# make k a tuple
k = (k,)*ndim
k = np.asarray([operator.index(ki) for ki in k], dtype=np.int32)
if len(k) != ndim:
raise ValueError(f"len(t) = {len(t_tpl)} != {len(k) = }.")
# 3. Validate inputs
ndim = len(t_tpl)
for d in range(ndim):
td = np.asarray(t_tpl[d])
kd = k[d]
n = td.shape[0] - kd - 1
if kd < 0:
raise ValueError(f"Spline degree in dimension {d} cannot be"
f" negative.")
if td.ndim != 1:
raise ValueError(f"Knot vector in dimension {d} must be"
f" one-dimensional.")
if n < kd + 1:
raise ValueError(f"Need at least {2*kd + 2} knots for degree"
f" {kd} in dimension {d}.")
if (np.diff(td) < 0).any():
raise ValueError(f"Knots in dimension {d} must be in a"
f" non-decreasing order.")
if len(np.unique(td[kd:n + 1])) < 2:
raise ValueError(f"Need at least two internal knots in"
f" dimension {d}.")
if not np.isfinite(td).all():
raise ValueError(f"Knots in dimension {d} should not have"
f" nans or infs.")
# 4. tabulate the flat indices for iterating over the (k+1)**ndim subarray
# non-zero b-spline elements
shape = tuple(kd + 1 for kd in k)
indices = np.unravel_index(np.arange(prod(shape)), shape)
_indices_k1d = np.asarray(indices, dtype=np.intp).T.copy()
# 5. pack the knots into a single array:
# ([1, 2, 3, 4], [5, 6], (7, 8, 9)) -->
# array([[1, 2, 3, 4],
# [5, 6, nan, nan],
# [7, 8, 9, nan]])
ndim = len(t_tpl)
len_t = [len(ti) for ti in t_tpl]
_t = np.empty((ndim, max(len_t)), dtype=float)
_t.fill(np.nan)
for d in range(ndim):
_t[d, :len(t_tpl[d])] = t_tpl[d]
len_t = np.asarray(len_t, dtype=np.int32)
return k, _indices_k1d, (_t, len_t)
def _iter_solve(a, b, solver=ssl.gcrotmk, **solver_args):
# work around iterative solvers not accepting multiple r.h.s.
# also work around a.dtype == float64 and b.dtype == complex128
# cf https://github.com/scipy/scipy/issues/19644
if np.issubdtype(b.dtype, np.complexfloating):
real = _iter_solve(a, b.real, solver, **solver_args)
imag = _iter_solve(a, b.imag, solver, **solver_args)
return real + 1j*imag
if b.ndim == 2 and b.shape[1] !=1:
res = np.empty_like(b)
for j in range(b.shape[1]):
res[:, j], info = solver(a, b[:, j], **solver_args)
if info != 0:
raise ValueError(f"{solver = } returns {info =} for column {j}.")
return res
else:
res, info = solver(a, b, **solver_args)
if info != 0:
raise ValueError(f"{solver = } returns {info = }.")
return res
def make_ndbspl(points, values, k=3, *, solver=ssl.gcrotmk, **solver_args):
"""Construct an interpolating NdBspline.
Parameters
----------
points : tuple of ndarrays of float, with shapes (m1,), ... (mN,)
The points defining the regular grid in N dimensions. The points in
each dimension (i.e. every element of the `points` tuple) must be
strictly ascending or descending.
values : ndarray of float, shape (m1, ..., mN, ...)
The data on the regular grid in n dimensions.
k : int, optional
The spline degree. Must be odd. Default is cubic, k=3
solver : a `scipy.sparse.linalg` solver (iterative or direct), optional.
An iterative solver from `scipy.sparse.linalg` or a direct one,
`sparse.sparse.linalg.spsolve`.
Used to solve the sparse linear system
``design_matrix @ coefficients = rhs`` for the coefficients.
Default is `scipy.sparse.linalg.gcrotmk`
solver_args : dict, optional
Additional arguments for the solver. The call signature is
``solver(csr_array, rhs_vector, **solver_args)``
Returns
-------
spl : NdBSpline object
Notes
-----
Boundary conditions are not-a-knot in all dimensions.
"""
ndim = len(points)
xi_shape = tuple(len(x) for x in points)
try:
len(k)
except TypeError:
# make k a tuple
k = (k,)*ndim
for d, point in enumerate(points):
numpts = len(np.atleast_1d(point))
if numpts <= k[d]:
raise ValueError(f"There are {numpts} points in dimension {d},"
f" but order {k[d]} requires at least "
f" {k[d]+1} points per dimension.")
t = tuple(_not_a_knot(np.asarray(points[d], dtype=float), k[d])
for d in range(ndim))
xvals = np.asarray([xv for xv in itertools.product(*points)], dtype=float)
# construct the colocation matrix
matr = NdBSpline.design_matrix(xvals, t, k)
# Solve for the coefficients given `values`.
# Trailing dimensions: first ndim dimensions are data, the rest are batch
# dimensions, so stack `values` into a 2D array for `spsolve` to undestand.
v_shape = values.shape
vals_shape = (prod(v_shape[:ndim]), prod(v_shape[ndim:]))
vals = values.reshape(vals_shape)
if solver != ssl.spsolve:
solver = functools.partial(_iter_solve, solver=solver)
if "atol" not in solver_args:
# avoid a DeprecationWarning, grumble grumble
solver_args["atol"] = 1e-6
coef = solver(matr, vals, **solver_args)
coef = coef.reshape(xi_shape + v_shape[ndim:])
return NdBSpline(t, coef, k)
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