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OOOiiOOOO|p(NNN)g1g2h1h2offsetlen_tOOOOOOnia1a2b(NNNNN)OOiOiOOiOOOOOidi|iOidipancyinit_pOOOOnOnn|p(NNNN)startrowOOnO|nOOiOipOOOOOOData point is out of boundsOOiOlen(x) = != = len(residuals)OOiO|plen(w) = OnOOOiOO|plen(y) = = mnc = > m = OOOOOiOiOOExpected data points in -D space, got -D points.numpy._core._multiarray_umathnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is NULL pointer_dierckxfpknotfpknot replacementfpbackfpbacpqr_reduceqr_reduce_periodicqr_reduce_augmented_matricesinit_augmented_matricesdata_matrixdata_matrix_periodic_coloc_norm_eq_lsqevaluate_splineevaluate_all_bsplfind_intervalevaluate_ndbspline_coloc_nd%fvector::_M_realloc_appendcannot create std::vector larger than max_size()Cannot do derivative order nu= _ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule was compiled against NumPy C-API version 0x%x (NumPy 1.23) but the running NumPy has C-API version 0x%x. Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem.FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtimenumpy._core.multiarray failed to importbacksubstitution, triangular matrixbacksubstitution for periodic splines, triangular matrixrow-by-row QR triangularizationrow-by-row QR triangularization for periodic splinesrow-by-row QR triangularization of augmented matrices for periodic splinesInitialise augmented matrices for periodic splines(m, k+1) array of non-zero b-splines(m, k+1) array of non-zero for periodic splinesfind_interval: out of bounds with x = Internal error. Please report it to SciPy developers.000102030405060708091011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495293949596979899€ð?@øà?}Ô%­I²Ô;LHÇþÿhøÉþÿHÂÊþÿ¬ËþÿTËþÿ\™Ëþÿ¼úËþÿ^Ìþÿt‚ÌþÿEÍþÿtŠÍþÿÈÑÍþÿ$UÎþÿ€åÎþÿøìÏþÿdÑþÿÈ8Òþÿ0 âÓþÿ� ìÔþÿ( Õþÿ¸ AÕþÿH jÕþÿ
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In the dense representation, each row of the collocation matrix corresponds
to a data point and contains non-zero b-spline basis functions which are
non-zero at this data point.
Parameters
----------
xvals : ndarray, shape(size, ndim)
Data points. ``xvals[j, :]`` gives the ``j``-th data point as an
``ndim``-dimensional array.
t : tuple of 1D arrays, length-ndim
Tuple of knot vectors
k : ndarray, shape (ndim,)
Spline degrees
Returns
-------
csr_data, csr_indices, csr_indptr
The collocation matrix in the CSR array format.
Notes
-----
Algorithm: given `xvals` and the tuple of knots `t`, we construct a tensor
product spline, i.e. a linear combination of
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``.
Since ``B`` functions are localized, for each point `(x1, ..., xN)` we
loop over the dimensions, and
- find the location in the knot array, `t[i] <= x < t[i+1]`,
- compute all non-zero `B` values
- place these values into the relevant row
In the dense representation, the collocation matrix would have had a row per
data point, and each row has the values of the basis elements (i.e., tensor
products of B-splines) evaluated at this data point. Since the matrix is very
sparse (has size = len(x)**ndim, with only (k+1)**ndim non-zero elements per
row), we construct it in the CSR format.
Evaluate an N-dim tensor product spline or its derivative.
Parameters
----------
xi : ndarray, shape(npoints, ndim)
``npoints`` values to evaluate the spline at, each value is
a point in an ``ndim``-dimensional space.
t : ndarray, shape(ndim, max_len_t)
Array of knots for each dimension.
This array packs the tuple of knot arrays per dimension into a single
2D array. The array is ragged (knot lengths may differ), hence
the real knots in dimension ``d`` are ``t[d, :len_t[d]]``.
len_t : ndarray, 1D, shape (ndim,)
Lengths of the knot arrays, per dimension.
k : tuple of ints, len(ndim)
Spline degrees in each dimension.
nu : ndarray of ints, shape(ndim,)
Orders of derivatives to compute, per dimension.
extrapolate : int
Whether to extrapolate out of bounds or return nans.
c1r: ndarray, one-dimensional
Flattened array of coefficients.
The original N-dimensional coefficient array ``c`` has shape
``(n1, ..., nd, ...)`` where each ``ni == len(t[d]) - k[d] - 1``,
and the second '...' represents trailing dimensions of ``c``.
In code, given the C-ordered array ``c``, ``c1r`` is
``c1 = c.reshape(c.shape[:ndim] + (-1,)); c1r = c1.ravel()``
num_c_tr : int
The number of elements of ``c1r``, which correspond to the trailing
dimensions of ``c``. In code, this is
``c1 = c.reshape(c.shape[:ndim] + (-1,)); num_c_tr = c1.shape[-1]``.
strides_c1 : ndarray, one-dimensional
Pre-computed strides of the ``c1`` array.
Note: These are *data* strides, not numpy-style byte strides.
This array is equivalent to
``[stride // s1.dtype.itemsize for stride in s1.strides]``.
indices_k1d : ndarray, shape((k+1)**ndim, ndim)
Pre-computed mapping between indices for iterating over a flattened
array of shape ``[k[d] + 1) for d in range(ndim)`` and
ndim-dimensional indices of the ``(k+1,)*ndim`` dimensional array.
This is essentially a transposed version of
``np.unravel_index(np.arange((k+1)**ndim), (k+1,)*ndim)``.
Returns
-------
out : ndarray, shape (npoints, num_c_tr)
Output values of the b-spline at given ``xi`` points.
Notes
-----
This function is essentially equivalent to the following: given an
N-dimensional vector ``x = (x1, x2, ..., xN)``, iterate over the
dimensions, form linear combinations of products,
B(x1) * B(x2) * ... B(xN) of (k+1)**N b-splines which are non-zero
at ``x``.
Since b-splines are localized, the sum has (k+1)**N non-zero elements.
If ``i = (i1, i2, ..., iN)`` is a vector if intervals of the knot
vectors, ``t[d, id] <= xd < t[d, id+1]``, for ``d=1, 2, ..., N``, then
the core loop of this function is nothing but
```
result = 0
iters = [range(i[d] - self.k[d], i[d] + 1) for d in range(ndim)]
for idx in itertools.product(*iters):
term = self.c[idx] * np.prod([B(x[d], self.k[d], idx[d], self.t[d])
for d in range(ndim)])
result += term
```
For efficiency reasons, we iterate over the flattened versions of the arrays.
Find an interval such that t[interval] <= xval < t[interval+1].
Uses a linear search with locality, see fitpack's splev.
Parameters
----------
t : ndarray, shape (nt,)
Knots
k : int
B-spline degree
xval : double
value to find the interval for
prev_l : int
interval where the previous value was located.
if unknown, use any value < k to start the search.
extrapolate : int
whether to return the last or the first interval if xval
is out of bounds.
Returns
-------
interval : int
Suitable interval or -1 if xval was nan.
Evaluate the ``k+1`` B-splines which are non-zero on interval ``m``.
Parameters
----------
t : ndarray, shape (nt + k + 1,)
sorted 1D array of knots
k : int
spline order
xval: float
argument at which to evaluate the B-splines
m : int
index of the left edge of the evaluation interval, ``t[m] <= x < t[m+1]``
nu : int, optional
Evaluate derivatives order `nu`. Default is zero.
Returns
-------
ndarray, shape (k+1,)
The values of B-splines :math:`[B_{m-k}(xval), ..., B_{m}(xval)]` if
`nu` is zero, otherwise the derivatives of order `nu`.
Examples
--------
A textbook use of this sort of routine is plotting the ``k+1`` polynomial
pieces which make up a B-spline of order `k`.
Consider a cubic spline
>>> k = 3
>>> t = [0., 1., 2., 3., 4.] # internal knots
>>> a, b = t[0], t[-1] # base interval is [a, b)
>>> t = np.array([a]*k + t + [b]*k) # add boundary knots