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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?@?}Ô%I;LHhHT\^tEt$U`G4 ( ' PH y 4q248l8(p((88<H x*;XTX a c fX gl s L l X@ Ȳ (ضx(XxHX( h8hXzRx $FJ w?;*3$"@DuBEF D(D0JM0A(A BBBzPLRx^ P$nBBB B(A0D8J 8A0A(B BBBH $xA8{AM m7.`b D U A E[ ,PKAo . U F A C 0M3 <L+Ao b. S.} H \ D E  8AM a. z B P H a 0AM a.0 H Dd 8hAM _. I I R F ^. $~ (LD0a8F@R8A0L F FPl6BBB B(D0D8J 8A0A(B BBBH Hb4lwAo zY.`_. H E ,G Ao .S E A C G < AM m1. U.0. E T 4x AM s.@W.^ H  P  BNB B(A0A8G, 8A0A(B BBBE $( A<Pp AM s.0P.m H N J . @![ AM s.@P. D  J $J-A @ 2|AM sT.`a J . I d W 8Js wAM s.P. 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J dHP   +EZ<s5EAF// 7U<i{]M!50M M!50M M!50pM Q}XM 0A\} \  WtL!50`L HHL!504L-#a}LJu-}K-"LK     }K X{}KE:LJ    XK1)@f $Ki_oV }J-# Sq}Jqeo" " """""""M""""I""}J5(EJ}Iyp0B0 0000000000 0"0#0(0`IA80Y^?IIA H1'5S rwH  *%  " $"=V$Q  0  *8ID J:Ki-X<Mk-4B         VQV3 _-ZU@2 32[PaPNRnP}PPPdPP*RXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX6 F V f v !!&!6!F!V!f!v!!!!!!!!!""&"6"F"V"f"v""""""Construct the N-D tensor product collocation matrix as a CSR array. 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 >>> import matplotlib.pyplot as plt >>> xx = np.linspace(a, b, 100) >>> plt.plot(xx, BSpline.basis_element(t[k:-k])(xx), ... lw=3, alpha=0.5, label='basis_element') Now we use slide an interval ``t[m]..t[m+1]`` along the base interval ``a..b`` and use `evaluate_all_bspl` to compute the restriction of the B-spline of interest to this interval: >>> for i in range(k+1): ... x1, x2 = t[2*k - i], t[2*k - i + 1] ... xx = np.linspace(x1 - 0.5, x2 + 0.5) ... yy = [evaluate_all_bspl(t, k, x, 2*k - i)[i] for x in xx] ... plt.plot(xx, yy, '--', label=str(i)) >>> plt.grid(True) >>> plt.legend() >>> plt.show() Evaluate a spline in the B-spline basis. Parameters ---------- t : ndarray, shape (n+k+1) knots c : ndarray, shape (n, m) B-spline coefficients xp : ndarray, shape (s,) Points to evaluate the spline at. nu : int Order of derivative to evaluate. extrapolate : int, optional Whether to extrapolate to ouf-of-bounds points, or to return NaNs. Returns ------- out : ndarray, shape (s, m) Computed values of the spline at each of the input points. Construct the normal equations for the B-spline LSQ problem. The observation equations are ``A @ c = y``, and the normal equations are ``A.T @ A @ c = A.T @ y``. This routine fills in the rhs and lhs for the latter. The B-spline collocation matrix is defined as :math:`A_{j,l} = B_l(x_j)`, so that row ``j`` contains all the B-splines which are non-zero at ``x_j``. The normal eq matrix has at most `2k+1` bands and is constructed in the LAPACK symmetrix banded storage: ``A[i, j] == ab[i-j, j]`` with `i >= j`. See the doctsring for `scipy.linalg.cholesky_banded` for more info. Parameters ---------- x : ndarray, shape (n,) sorted 1D array of x values t : ndarray, shape (nt + k + 1,) sorted 1D array of knots k : int spline order y : ndarray, shape (n, s) a 2D array of y values. The second dimension contains all trailing dimensions of the original array of ordinates. w : ndarray, shape(n,) Weights. ab : ndarray, shape (k+1, n), in Fortran order. This parameter is modified in-place. On entry: should be zeroed out. On exit: LHS of the normal equations. rhs : ndarray, shape (n, s), in C order. This parameter is modified in-place. On entry: should be zeroed out. On exit: RHS of the normal equations. Build the B-spline colocation matrix. The colocation matrix is defined as :math:`B_{j,l} = B_l(x_j)`, so that row ``j`` contains all the B-splines which are non-zero at ``x_j``. The matrix is constructed in the LAPACK banded storage. Basically, for an N-by-N matrix A with ku upper diagonals and kl lower diagonals, the shape of the array Ab is (2*kl + ku +1, N), where the last kl+ku+1 rows of Ab contain the diagonals of A, and the first kl rows of Ab are not referenced. For more info see, e.g. the docs for the ``*gbsv`` routine. This routine is not supposed to be called directly, and does no error checking. Parameters ---------- x : ndarray, shape (n,) sorted 1D array of x values t : ndarray, shape (nt + k + 1,) sorted 1D array of knots k : int spline order ab : ndarray, shape (2*kl + ku + 1, nt), F-order This parameter is modified in-place. On exit: B-spline colocation matrix in the band storage with ``ku`` upper diagonals and ``kl`` lower diagonals. 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