from sympy.core import S, sympify
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions import Piecewise, piecewise_fold
from sympy.logic.boolalg import And
from sympy.sets.sets import Interval
from functools import lru_cache
def _ivl(cond, x):
"""return the interval corresponding to the condition
Conditions in spline's Piecewise give the range over
which an expression is valid like (lo <= x) & (x <= hi).
This function returns (lo, hi).
"""
if isinstance(cond, And) and len(cond.args) == 2:
a, b = cond.args
if a.lts == x:
a, b = b, a
return a.lts, b.gts
raise TypeError('unexpected cond type: %s' % cond)
def _add_splines(c, b1, d, b2, x):
"""Construct c*b1 + d*b2."""
if S.Zero in (b1, c):
rv = piecewise_fold(d * b2)
elif S.Zero in (b2, d):
rv = piecewise_fold(c * b1)
else:
new_args = []
# Just combining the Piecewise without any fancy optimization
p1 = piecewise_fold(c * b1)
p2 = piecewise_fold(d * b2)
# Search all Piecewise arguments except (0, True)
p2args = list(p2.args[:-1])
# This merging algorithm assumes the conditions in
# p1 and p2 are sorted
for arg in p1.args[:-1]:
expr = arg.expr
cond = arg.cond
lower = _ivl(cond, x)[0]
# Check p2 for matching conditions that can be merged
for i, arg2 in enumerate(p2args):
expr2 = arg2.expr
cond2 = arg2.cond
lower_2, upper_2 = _ivl(cond2, x)
if cond2 == cond:
# Conditions match, join expressions
expr += expr2
# Remove matching element
del p2args[i]
# No need to check the rest
break
elif lower_2 < lower and upper_2 <= lower:
# Check if arg2 condition smaller than arg1,
# add to new_args by itself (no match expected
# in p1)
new_args.append(arg2)
del p2args[i]
break
# Checked all, add expr and cond
new_args.append((expr, cond))
# Add remaining items from p2args
new_args.extend(p2args)
# Add final (0, True)
new_args.append((0, True))
rv = Piecewise(*new_args, evaluate=False)
return rv.expand()
@lru_cache(maxsize=128)
def bspline_basis(d, knots, n, x):
"""
The $n$-th B-spline at $x$ of degree $d$ with knots.
Explanation
===========
B-Splines are piecewise polynomials of degree $d$. They are defined on a
set of knots, which is a sequence of integers or floats.
Examples
========
The 0th degree splines have a value of 1 on a single interval:
>>> from sympy import bspline_basis
>>> from sympy.abc import x
>>> d = 0
>>> knots = tuple(range(5))
>>> bspline_basis(d, knots, 0, x)
Piecewise((1, (x >= 0) & (x <= 1)), (0, True))
For a given ``(d, knots)`` there are ``len(knots)-d-1`` B-splines
defined, that are indexed by ``n`` (starting at 0).
Here is an example of a cubic B-spline:
>>> bspline_basis(3, tuple(range(5)), 0, x)
Piecewise((x**3/6, (x >= 0) & (x <= 1)),
(-x**3/2 + 2*x**2 - 2*x + 2/3,
(x >= 1) & (x <= 2)),
(x**3/2 - 4*x**2 + 10*x - 22/3,
(x >= 2) & (x <= 3)),
(-x**3/6 + 2*x**2 - 8*x + 32/3,
(x >= 3) & (x <= 4)),
(0, True))
By repeating knot points, you can introduce discontinuities in the
B-splines and their derivatives:
>>> d = 1
>>> knots = (0, 0, 2, 3, 4)
>>> bspline_basis(d, knots, 0, x)
Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True))
It is quite time consuming to construct and evaluate B-splines. If
you need to evaluate a B-spline many times, it is best to lambdify them
first:
>>> from sympy import lambdify
>>> d = 3
>>> knots = tuple(range(10))
>>> b0 = bspline_basis(d, knots, 0, x)
>>> f = lambdify(x, b0)
>>> y = f(0.5)
Parameters
==========
d : integer
degree of bspline
knots : list of integer values
list of knots points of bspline
n : integer
$n$-th B-spline
x : symbol
See Also
========
bspline_basis_set
References
==========
.. [1] https://en.wikipedia.org/wiki/B-spline
"""
# make sure x has no assumptions so conditions don't evaluate
xvar = x
x = Dummy()
knots = tuple(sympify(k) for k in knots)
d = int(d)
n = int(n)
n_knots = len(knots)
n_intervals = n_knots - 1
if n + d + 1 > n_intervals:
raise ValueError("n + d + 1 must not exceed len(knots) - 1")
if d == 0:
result = Piecewise(
(S.One, Interval(knots[n], knots[n + 1]).contains(x)), (0, True)
)
elif d > 0:
denom = knots[n + d + 1] - knots[n + 1]
if denom != S.Zero:
B = (knots[n + d + 1] - x) / denom
b2 = bspline_basis(d - 1, knots, n + 1, x)
else:
b2 = B = S.Zero
denom = knots[n + d] - knots[n]
if denom != S.Zero:
A = (x - knots[n]) / denom
b1 = bspline_basis(d - 1, knots, n, x)
else:
b1 = A = S.Zero
result = _add_splines(A, b1, B, b2, x)
else:
raise ValueError("degree must be non-negative: %r" % n)
# return result with user-given x
return result.xreplace({x: xvar})
def bspline_basis_set(d, knots, x):
"""
Return the ``len(knots)-d-1`` B-splines at *x* of degree *d*
with *knots*.
Explanation
===========
This function returns a list of piecewise polynomials that are the
``len(knots)-d-1`` B-splines of degree *d* for the given knots.
This function calls ``bspline_basis(d, knots, n, x)`` for different
values of *n*.
Examples
========
>>> from sympy import bspline_basis_set
>>> from sympy.abc import x
>>> d = 2
>>> knots = range(5)
>>> splines = bspline_basis_set(d, knots, x)
>>> splines
[Piecewise((x**2/2, (x >= 0) & (x <= 1)),
(-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)),
(x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)),
(0, True)),
Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)),
(-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)),
(x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)),
(0, True))]
Parameters
==========
d : integer
degree of bspline
knots : list of integers
list of knots points of bspline
x : symbol
See Also
========
bspline_basis
"""
n_splines = len(knots) - d - 1
return [bspline_basis(d, tuple(knots), i, x) for i in range(n_splines)]
def interpolating_spline(d, x, X, Y):
"""
Return spline of degree *d*, passing through the given *X*
and *Y* values.
Explanation
===========
This function returns a piecewise function such that each part is
a polynomial of degree not greater than *d*. The value of *d*
must be 1 or greater and the values of *X* must be strictly
increasing.
Examples
========
>>> from sympy import interpolating_spline
>>> from sympy.abc import x
>>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
Piecewise((3*x, (x >= 1) & (x <= 2)),
(7 - x/2, (x >= 2) & (x <= 4)),
(2*x/3 + 7/3, (x >= 4) & (x <= 7)))
>>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)),
(10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4)))
Parameters
==========
d : integer
Degree of Bspline strictly greater than equal to one
x : symbol
X : list of strictly increasing real values
list of X coordinates through which the spline passes
Y : list of real values
list of corresponding Y coordinates through which the spline passes
See Also
========
bspline_basis_set, interpolating_poly
"""
from sympy.solvers.solveset import linsolve
from sympy.matrices.dense import Matrix
# Input sanitization
d = sympify(d)
if not (d.is_Integer and d.is_positive):
raise ValueError("Spline degree must be a positive integer, not %s." % d)
if len(X) != len(Y):
raise ValueError("Number of X and Y coordinates must be the same.")
if len(X) < d + 1:
raise ValueError("Degree must be less than the number of control points.")
if not all(a < b for a, b in zip(X, X[1:])):
raise ValueError("The x-coordinates must be strictly increasing.")
X = [sympify(i) for i in X]
# Evaluating knots value
if d.is_odd:
j = (d + 1) // 2
interior_knots = X[j:-j]
else:
j = d // 2
interior_knots = [
(a + b)/2 for a, b in zip(X[j : -j - 1], X[j + 1 : -j])
]
knots = [X[0]] * (d + 1) + list(interior_knots) + [X[-1]] * (d + 1)
basis = bspline_basis_set(d, knots, x)
A = [[b.subs(x, v) for b in basis] for v in X]
coeff = linsolve((Matrix(A), Matrix(Y)), symbols("c0:{}".format(len(X)), cls=Dummy))
coeff = list(coeff)[0]
intervals = {c for b in basis for (e, c) in b.args if c != True}
# Sorting the intervals
# ival contains the end-points of each interval
ival = [_ivl(c, x) for c in intervals]
com = zip(ival, intervals)
com = sorted(com, key=lambda x: x[0])
intervals = [y for x, y in com]
basis_dicts = [{c: e for (e, c) in b.args} for b in basis]
spline = []
for i in intervals:
piece = sum(
[c * d.get(i, S.Zero) for (c, d) in zip(coeff, basis_dicts)], S.Zero
)
spline.append((piece, i))
return Piecewise(*spline)