/
_bspl.pyx
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/
_bspl.pyx
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"""
Routines for evaluating and manipulating B-splines.
"""
import numpy as np
cimport numpy as cnp
cimport cython
cnp.import_array()
cdef extern from "src/__fitpack.h":
void _deBoor_D(const double *t, double x, int k, int ell, int m, double *result) nogil
cdef extern from "numpy/npy_math.h":
double nan "NPY_NAN"
ctypedef double complex double_complex
ctypedef fused double_or_complex:
double
double complex
#------------------------------------------------------------------------------
# B-splines
#------------------------------------------------------------------------------
@cython.wraparound(False)
@cython.boundscheck(False)
cdef inline int find_interval(const double[::1] t,
int k,
double xval,
int prev_l,
bint extrapolate) nogil:
"""
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.
"""
cdef:
int l
int n = t.shape[0] - k - 1
double tb = t[k]
double te = t[n]
if xval != xval:
# nan
return -1
if ((xval < tb) or (xval > te)) and not extrapolate:
return -1
l = prev_l if k < prev_l < n else k
# xval is in support, search for interval s.t. t[interval] <= xval < t[l+1]
while(xval < t[l] and l != k):
l -= 1
l += 1
while(xval >= t[l] and l != n):
l += 1
return l-1
@cython.wraparound(False)
@cython.boundscheck(False)
@cython.cdivision(True)
def evaluate_spline(const double[::1] t,
double_or_complex[:, ::1] c,
int k,
const double[::1] xp,
int nu,
bint extrapolate,
double_or_complex[:, ::1] out):
"""
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.
out : ndarray, shape (s, m)
Computed values of the spline at each of the input points.
This argument is modified in-place.
"""
cdef int ip, jp, n, a
cdef int i, interval
cdef double xval
# shape checks
if out.shape[0] != xp.shape[0]:
raise ValueError("out and xp have incompatible shapes")
if out.shape[1] != c.shape[1]:
raise ValueError("out and c have incompatible shapes")
# check derivative order
if nu < 0:
raise NotImplementedError("Cannot do derivative order %s." % nu)
n = c.shape[0]
cdef double[::1] work = np.empty(2*k+2, dtype=np.float_)
# evaluate
with nogil:
interval = k
for ip in range(xp.shape[0]):
xval = xp[ip]
# Find correct interval
interval = find_interval(t, k, xval, interval, extrapolate)
if interval < 0:
# xval was nan etc
for jp in range(c.shape[1]):
out[ip, jp] = nan
continue
# Evaluate (k+1) b-splines which are non-zero on the interval.
# on return, first k+1 elemets of work are B_{m-k},..., B_{m}
_deBoor_D(&t[0], xval, k, interval, nu, &work[0])
# Form linear combinations
for jp in range(c.shape[1]):
out[ip, jp] = 0.
for a in range(k+1):
out[ip, jp] = out[ip, jp] + c[interval + a - k, jp] * work[a]
def evaluate_all_bspl(const double[::1] t, int k, double xval, int m, int nu=0):
"""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., 2., 2., 3., 4.] # internal knots
>>> a, b = t[0], t[-1] # base interval is [a, b)
>>> t = [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),
... 'r-', lw=5, alpha=0.5)
>>> c = ['b', 'g', 'c', 'k']
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, c[i] + '--', lw=3, label=str(i))
...
>>> plt.grid(True)
>>> plt.legend()
>>> plt.show()
"""
bbb = np.empty(2*k+2, dtype=np.float_)
cdef double[::1] work = bbb
_deBoor_D(&t[0], xval, k, m, nu, &work[0])
return bbb[:k+1]
@cython.wraparound(False)
@cython.boundscheck(False)
def _colloc(const double[::1] x, const double[::1] t, int k, double[::1, :] ab,
int offset=0):
"""Build the B-spline collocation matrix.
The collocation 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: zeroed out.
On exit: B-spline collocation matrix in the band storage with
``ku`` upper diagonals and ``kl`` lower diagonals.
Here ``kl = ku = k``.
offset : int, optional
skip this many rows
"""
cdef int nt = t.shape[0] - k - 1
cdef int left, j, a, kl, ku, clmn
cdef double xval
kl = ku = k
cdef double[::1] wrk = np.empty(2*k + 2, dtype=np.float_)
# collocation matrix
with nogil:
left = k
for j in range(x.shape[0]):
xval = x[j]
# find interval
left = find_interval(t, k, xval, left, extrapolate=False)
# fill a row
_deBoor_D(&t[0], xval, k, left, 0, &wrk[0])
# for a full matrix it would be ``A[j + offset, left-k:left+1] = bb``
# in the banded storage, need to spread the row over
for a in range(k+1):
clmn = left - k + a
ab[kl + ku + j + offset - clmn, clmn] = wrk[a]
@cython.wraparound(False)
@cython.boundscheck(False)
def _handle_lhs_derivatives(const double[::1]t, int k, double xval,
double[::1, :] ab,
int kl, int ku,
const cnp.int_t[::1] deriv_ords,
int offset=0):
""" Fill in the entries of the collocation matrix corresponding to known
derivatives at xval.
The collocation matrix is in the banded storage, as prepared by _colloc.
No error checking.
Parameters
----------
t : ndarray, shape (nt + k + 1,)
knots
k : integer
B-spline order
xval : float
The value at which to evaluate the derivatives at.
ab : ndarray, shape(2*kl + ku + 1, nt), Fortran order
B-spline collocation matrix.
This argument is modified *in-place*.
kl : integer
Number of lower diagonals of ab.
ku : integer
Number of upper diagonals of ab.
deriv_ords : 1D ndarray
Orders of derivatives known at xval
offset : integer, optional
Skip this many rows of the matrix ab.
"""
cdef:
int left, nu, a, clmn, row
double[::1] wrk = np.empty(2*k+2, dtype=np.float_)
# derivatives @ xval
with nogil:
left = find_interval(t, k, xval, k, extrapolate=False)
for row in range(deriv_ords.shape[0]):
nu = deriv_ords[row]
_deBoor_D(&t[0], xval, k, left, nu, &wrk[0])
# if A were a full matrix, it would be just
# ``A[row + offset, left-k:left+1] = bb``.
for a in range(k+1):
clmn = left - k + a
ab[kl + ku + offset + row - clmn, clmn] = wrk[a]
@cython.wraparound(False)
@cython.boundscheck(False)
def _norm_eq_lsq(const double[::1] x,
const double[::1] t,
int k,
double_or_complex[:, ::1] y,
const double[::1] w,
double[::1, :] ab,
double_or_complex[::1, :] rhs):
"""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.
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
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 Fortran order.
This parameter is modified in-place.
On entry: should be zeroed out.
On exit: RHS of the normal equations.
"""
cdef:
int j, r, s, row, clmn, left, ci
double xval, wval
double[::1] wrk = np.empty(2*k + 2, dtype=np.float_)
with nogil:
left = k
for j in range(x.shape[0]):
xval = x[j]
wval = w[j] * w[j]
# find interval
left = find_interval(t, k, xval, left, extrapolate=False)
# non-zero B-splines at xval
_deBoor_D(&t[0], xval, k, left, 0, &wrk[0])
# non-zero values of A.T @ A: banded storage w/ lower=True
# The colloq matrix in full storage would be
# A[j, left-k:left+1] = wrk,
# Here we work out A.T @ A *in the banded storage* w/lower=True
# see the docstring of `scipy.linalg.cholesky_banded`.
for r in range(k+1):
row = left - k + r
for s in range(r+1):
clmn = left - k + s
ab[r-s, clmn] += wrk[r] * wrk[s] * wval
# ... and A.T @ y
for ci in range(rhs.shape[1]):
rhs[row, ci] = rhs[row, ci] + wrk[r] * y[j, ci] * wval