855 lines
30 KiB
Python
855 lines
30 KiB
Python
__all__ = ['splrep', 'splprep', 'splev', 'splint', 'sproot', 'spalde',
|
|
'bisplrep', 'bisplev', 'insert', 'splder', 'splantider']
|
|
|
|
|
|
import numpy as np
|
|
|
|
# These are in the API for fitpack even if not used in fitpack.py itself.
|
|
from ._fitpack_impl import bisplrep, bisplev, dblint # noqa: F401
|
|
from . import _fitpack_impl as _impl
|
|
from ._bsplines import BSpline
|
|
|
|
|
|
def splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None,
|
|
full_output=0, nest=None, per=0, quiet=1):
|
|
"""
|
|
Find the B-spline representation of an N-D curve.
|
|
|
|
Given a list of N rank-1 arrays, `x`, which represent a curve in
|
|
N-dimensional space parametrized by `u`, find a smooth approximating
|
|
spline curve g(`u`). Uses the FORTRAN routine parcur from FITPACK.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
A list of sample vector arrays representing the curve.
|
|
w : array_like, optional
|
|
Strictly positive rank-1 array of weights the same length as `x[0]`.
|
|
The weights are used in computing the weighted least-squares spline
|
|
fit. If the errors in the `x` values have standard-deviation given by
|
|
the vector d, then `w` should be 1/d. Default is ``ones(len(x[0]))``.
|
|
u : array_like, optional
|
|
An array of parameter values. If not given, these values are
|
|
calculated automatically as ``M = len(x[0])``, where
|
|
|
|
v[0] = 0
|
|
|
|
v[i] = v[i-1] + distance(`x[i]`, `x[i-1]`)
|
|
|
|
u[i] = v[i] / v[M-1]
|
|
|
|
ub, ue : int, optional
|
|
The end-points of the parameters interval. Defaults to
|
|
u[0] and u[-1].
|
|
k : int, optional
|
|
Degree of the spline. Cubic splines are recommended.
|
|
Even values of `k` should be avoided especially with a small s-value.
|
|
``1 <= k <= 5``, default is 3.
|
|
task : int, optional
|
|
If task==0 (default), find t and c for a given smoothing factor, s.
|
|
If task==1, find t and c for another value of the smoothing factor, s.
|
|
There must have been a previous call with task=0 or task=1
|
|
for the same set of data.
|
|
If task=-1 find the weighted least square spline for a given set of
|
|
knots, t.
|
|
s : float, optional
|
|
A smoothing condition. The amount of smoothness is determined by
|
|
satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s``,
|
|
where g(x) is the smoothed interpolation of (x,y). The user can
|
|
use `s` to control the trade-off between closeness and smoothness
|
|
of fit. Larger `s` means more smoothing while smaller values of `s`
|
|
indicate less smoothing. Recommended values of `s` depend on the
|
|
weights, w. If the weights represent the inverse of the
|
|
standard-deviation of y, then a good `s` value should be found in
|
|
the range ``(m-sqrt(2*m),m+sqrt(2*m))``, where m is the number of
|
|
data points in x, y, and w.
|
|
t : array, optional
|
|
The knots needed for ``task=-1``.
|
|
There must be at least ``2*k+2`` knots.
|
|
full_output : int, optional
|
|
If non-zero, then return optional outputs.
|
|
nest : int, optional
|
|
An over-estimate of the total number of knots of the spline to
|
|
help in determining the storage space. By default nest=m/2.
|
|
Always large enough is nest=m+k+1.
|
|
per : int, optional
|
|
If non-zero, data points are considered periodic with period
|
|
``x[m-1] - x[0]`` and a smooth periodic spline approximation is
|
|
returned. Values of ``y[m-1]`` and ``w[m-1]`` are not used.
|
|
quiet : int, optional
|
|
Non-zero to suppress messages.
|
|
|
|
Returns
|
|
-------
|
|
tck : tuple
|
|
A tuple, ``(t,c,k)`` containing the vector of knots, the B-spline
|
|
coefficients, and the degree of the spline.
|
|
u : array
|
|
An array of the values of the parameter.
|
|
fp : float
|
|
The weighted sum of squared residuals of the spline approximation.
|
|
ier : int
|
|
An integer flag about splrep success. Success is indicated
|
|
if ier<=0. If ier in [1,2,3] an error occurred but was not raised.
|
|
Otherwise an error is raised.
|
|
msg : str
|
|
A message corresponding to the integer flag, ier.
|
|
|
|
See Also
|
|
--------
|
|
splrep, splev, sproot, spalde, splint,
|
|
bisplrep, bisplev
|
|
UnivariateSpline, BivariateSpline
|
|
BSpline
|
|
make_interp_spline
|
|
|
|
Notes
|
|
-----
|
|
See `splev` for evaluation of the spline and its derivatives.
|
|
The number of dimensions N must be smaller than 11.
|
|
|
|
The number of coefficients in the `c` array is ``k+1`` less than the number
|
|
of knots, ``len(t)``. This is in contrast with `splrep`, which zero-pads
|
|
the array of coefficients to have the same length as the array of knots.
|
|
These additional coefficients are ignored by evaluation routines, `splev`
|
|
and `BSpline`.
|
|
|
|
References
|
|
----------
|
|
.. [1] P. Dierckx, "Algorithms for smoothing data with periodic and
|
|
parametric splines, Computer Graphics and Image Processing",
|
|
20 (1982) 171-184.
|
|
.. [2] P. Dierckx, "Algorithms for smoothing data with periodic and
|
|
parametric splines", report tw55, Dept. Computer Science,
|
|
K.U.Leuven, 1981.
|
|
.. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs on
|
|
Numerical Analysis, Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
Generate a discretization of a limacon curve in the polar coordinates:
|
|
|
|
>>> import numpy as np
|
|
>>> phi = np.linspace(0, 2.*np.pi, 40)
|
|
>>> r = 0.5 + np.cos(phi) # polar coords
|
|
>>> x, y = r * np.cos(phi), r * np.sin(phi) # convert to cartesian
|
|
|
|
And interpolate:
|
|
|
|
>>> from scipy.interpolate import splprep, splev
|
|
>>> tck, u = splprep([x, y], s=0)
|
|
>>> new_points = splev(u, tck)
|
|
|
|
Notice that (i) we force interpolation by using `s=0`,
|
|
(ii) the parameterization, ``u``, is generated automatically.
|
|
Now plot the result:
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> fig, ax = plt.subplots()
|
|
>>> ax.plot(x, y, 'ro')
|
|
>>> ax.plot(new_points[0], new_points[1], 'r-')
|
|
>>> plt.show()
|
|
|
|
"""
|
|
|
|
res = _impl.splprep(x, w, u, ub, ue, k, task, s, t, full_output, nest, per,
|
|
quiet)
|
|
return res
|
|
|
|
|
|
def splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None,
|
|
full_output=0, per=0, quiet=1):
|
|
"""
|
|
Find the B-spline representation of a 1-D curve.
|
|
|
|
Given the set of data points ``(x[i], y[i])`` determine a smooth spline
|
|
approximation of degree k on the interval ``xb <= x <= xe``.
|
|
|
|
Parameters
|
|
----------
|
|
x, y : array_like
|
|
The data points defining a curve ``y = f(x)``.
|
|
w : array_like, optional
|
|
Strictly positive rank-1 array of weights the same length as `x` and `y`.
|
|
The weights are used in computing the weighted least-squares spline
|
|
fit. If the errors in the `y` values have standard-deviation given by the
|
|
vector ``d``, then `w` should be ``1/d``. Default is ``ones(len(x))``.
|
|
xb, xe : float, optional
|
|
The interval to fit. If None, these default to ``x[0]`` and ``x[-1]``
|
|
respectively.
|
|
k : int, optional
|
|
The degree of the spline fit. It is recommended to use cubic splines.
|
|
Even values of `k` should be avoided especially with small `s` values.
|
|
``1 <= k <= 5``.
|
|
task : {1, 0, -1}, optional
|
|
If ``task==0``, find ``t`` and ``c`` for a given smoothing factor, `s`.
|
|
|
|
If ``task==1`` find ``t`` and ``c`` for another value of the smoothing factor,
|
|
`s`. There must have been a previous call with ``task=0`` or ``task=1`` for
|
|
the same set of data (``t`` will be stored an used internally)
|
|
|
|
If ``task=-1`` find the weighted least square spline for a given set of
|
|
knots, ``t``. These should be interior knots as knots on the ends will be
|
|
added automatically.
|
|
s : float, optional
|
|
A smoothing condition. The amount of smoothness is determined by
|
|
satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s`` where ``g(x)``
|
|
is the smoothed interpolation of ``(x,y)``. The user can use `s` to control
|
|
the tradeoff between closeness and smoothness of fit. Larger `s` means
|
|
more smoothing while smaller values of `s` indicate less smoothing.
|
|
Recommended values of `s` depend on the weights, `w`. If the weights
|
|
represent the inverse of the standard-deviation of `y`, then a good `s`
|
|
value should be found in the range ``(m-sqrt(2*m),m+sqrt(2*m))`` where ``m`` is
|
|
the number of datapoints in `x`, `y`, and `w`. default : ``s=m-sqrt(2*m)`` if
|
|
weights are supplied. ``s = 0.0`` (interpolating) if no weights are
|
|
supplied.
|
|
t : array_like, optional
|
|
The knots needed for ``task=-1``. If given then task is automatically set
|
|
to ``-1``.
|
|
full_output : bool, optional
|
|
If non-zero, then return optional outputs.
|
|
per : bool, optional
|
|
If non-zero, data points are considered periodic with period ``x[m-1]`` -
|
|
``x[0]`` and a smooth periodic spline approximation is returned. Values of
|
|
``y[m-1]`` and ``w[m-1]`` are not used.
|
|
The default is zero, corresponding to boundary condition 'not-a-knot'.
|
|
quiet : bool, optional
|
|
Non-zero to suppress messages.
|
|
|
|
Returns
|
|
-------
|
|
tck : tuple
|
|
A tuple ``(t,c,k)`` containing the vector of knots, the B-spline
|
|
coefficients, and the degree of the spline.
|
|
fp : array, optional
|
|
The weighted sum of squared residuals of the spline approximation.
|
|
ier : int, optional
|
|
An integer flag about splrep success. Success is indicated if ``ier<=0``.
|
|
If ``ier in [1,2,3]``, an error occurred but was not raised. Otherwise an
|
|
error is raised.
|
|
msg : str, optional
|
|
A message corresponding to the integer flag, `ier`.
|
|
|
|
See Also
|
|
--------
|
|
UnivariateSpline, BivariateSpline
|
|
splprep, splev, sproot, spalde, splint
|
|
bisplrep, bisplev
|
|
BSpline
|
|
make_interp_spline
|
|
|
|
Notes
|
|
-----
|
|
See `splev` for evaluation of the spline and its derivatives. Uses the
|
|
FORTRAN routine ``curfit`` from FITPACK.
|
|
|
|
The user is responsible for assuring that the values of `x` are unique.
|
|
Otherwise, `splrep` will not return sensible results.
|
|
|
|
If provided, knots `t` must satisfy the Schoenberg-Whitney conditions,
|
|
i.e., there must be a subset of data points ``x[j]`` such that
|
|
``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``.
|
|
|
|
This routine zero-pads the coefficients array ``c`` to have the same length
|
|
as the array of knots ``t`` (the trailing ``k + 1`` coefficients are ignored
|
|
by the evaluation routines, `splev` and `BSpline`.) This is in contrast with
|
|
`splprep`, which does not zero-pad the coefficients.
|
|
|
|
The default boundary condition is 'not-a-knot', i.e. the first and second
|
|
segment at a curve end are the same polynomial. More boundary conditions are
|
|
available in `CubicSpline`.
|
|
|
|
References
|
|
----------
|
|
Based on algorithms described in [1]_, [2]_, [3]_, and [4]_:
|
|
|
|
.. [1] P. Dierckx, "An algorithm for smoothing, differentiation and
|
|
integration of experimental data using spline functions",
|
|
J.Comp.Appl.Maths 1 (1975) 165-184.
|
|
.. [2] P. Dierckx, "A fast algorithm for smoothing data on a rectangular
|
|
grid while using spline functions", SIAM J.Numer.Anal. 19 (1982)
|
|
1286-1304.
|
|
.. [3] P. Dierckx, "An improved algorithm for curve fitting with spline
|
|
functions", report tw54, Dept. Computer Science,K.U. Leuven, 1981.
|
|
.. [4] P. Dierckx, "Curve and surface fitting with splines", Monographs on
|
|
Numerical Analysis, Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
You can interpolate 1-D points with a B-spline curve.
|
|
Further examples are given in
|
|
:ref:`in the tutorial <tutorial-interpolate_splXXX>`.
|
|
|
|
>>> import numpy as np
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> from scipy.interpolate import splev, splrep
|
|
>>> x = np.linspace(0, 10, 10)
|
|
>>> y = np.sin(x)
|
|
>>> spl = splrep(x, y)
|
|
>>> x2 = np.linspace(0, 10, 200)
|
|
>>> y2 = splev(x2, spl)
|
|
>>> plt.plot(x, y, 'o', x2, y2)
|
|
>>> plt.show()
|
|
|
|
"""
|
|
res = _impl.splrep(x, y, w, xb, xe, k, task, s, t, full_output, per, quiet)
|
|
return res
|
|
|
|
|
|
def splev(x, tck, der=0, ext=0):
|
|
"""
|
|
Evaluate a B-spline or its derivatives.
|
|
|
|
Given the knots and coefficients of a B-spline representation, evaluate
|
|
the value of the smoothing polynomial and its derivatives. This is a
|
|
wrapper around the FORTRAN routines splev and splder of FITPACK.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
An array of points at which to return the value of the smoothed
|
|
spline or its derivatives. If `tck` was returned from `splprep`,
|
|
then the parameter values, u should be given.
|
|
tck : BSpline instance or tuple
|
|
If a tuple, then it should be a sequence of length 3 returned by
|
|
`splrep` or `splprep` containing the knots, coefficients, and degree
|
|
of the spline. (Also see Notes.)
|
|
der : int, optional
|
|
The order of derivative of the spline to compute (must be less than
|
|
or equal to k, the degree of the spline).
|
|
ext : int, optional
|
|
Controls the value returned for elements of ``x`` not in the
|
|
interval defined by the knot sequence.
|
|
|
|
* if ext=0, return the extrapolated value.
|
|
* if ext=1, return 0
|
|
* if ext=2, raise a ValueError
|
|
* if ext=3, return the boundary value.
|
|
|
|
The default value is 0.
|
|
|
|
Returns
|
|
-------
|
|
y : ndarray or list of ndarrays
|
|
An array of values representing the spline function evaluated at
|
|
the points in `x`. If `tck` was returned from `splprep`, then this
|
|
is a list of arrays representing the curve in an N-D space.
|
|
|
|
See Also
|
|
--------
|
|
splprep, splrep, sproot, spalde, splint
|
|
bisplrep, bisplev
|
|
BSpline
|
|
|
|
Notes
|
|
-----
|
|
Manipulating the tck-tuples directly is not recommended. In new code,
|
|
prefer using `BSpline` objects.
|
|
|
|
References
|
|
----------
|
|
.. [1] C. de Boor, "On calculating with b-splines", J. Approximation
|
|
Theory, 6, p.50-62, 1972.
|
|
.. [2] M. G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths
|
|
Applics, 10, p.134-149, 1972.
|
|
.. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs
|
|
on Numerical Analysis, Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
Examples are given :ref:`in the tutorial <tutorial-interpolate_splXXX>`.
|
|
|
|
A comparison between `splev`, `splder` and `spalde` to compute the derivatives of a
|
|
B-spline can be found in the `spalde` examples section.
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
if tck.c.ndim > 1:
|
|
mesg = ("Calling splev() with BSpline objects with c.ndim > 1 is "
|
|
"not allowed. Use BSpline.__call__(x) instead.")
|
|
raise ValueError(mesg)
|
|
|
|
# remap the out-of-bounds behavior
|
|
try:
|
|
extrapolate = {0: True, }[ext]
|
|
except KeyError as e:
|
|
raise ValueError("Extrapolation mode %s is not supported "
|
|
"by BSpline." % ext) from e
|
|
|
|
return tck(x, der, extrapolate=extrapolate)
|
|
else:
|
|
return _impl.splev(x, tck, der, ext)
|
|
|
|
|
|
def splint(a, b, tck, full_output=0):
|
|
"""
|
|
Evaluate the definite integral of a B-spline between two given points.
|
|
|
|
Parameters
|
|
----------
|
|
a, b : float
|
|
The end-points of the integration interval.
|
|
tck : tuple or a BSpline instance
|
|
If a tuple, then it should be a sequence of length 3, containing the
|
|
vector of knots, the B-spline coefficients, and the degree of the
|
|
spline (see `splev`).
|
|
full_output : int, optional
|
|
Non-zero to return optional output.
|
|
|
|
Returns
|
|
-------
|
|
integral : float
|
|
The resulting integral.
|
|
wrk : ndarray
|
|
An array containing the integrals of the normalized B-splines
|
|
defined on the set of knots.
|
|
(Only returned if `full_output` is non-zero)
|
|
|
|
See Also
|
|
--------
|
|
splprep, splrep, sproot, spalde, splev
|
|
bisplrep, bisplev
|
|
BSpline
|
|
|
|
Notes
|
|
-----
|
|
`splint` silently assumes that the spline function is zero outside the data
|
|
interval (`a`, `b`).
|
|
|
|
Manipulating the tck-tuples directly is not recommended. In new code,
|
|
prefer using the `BSpline` objects.
|
|
|
|
References
|
|
----------
|
|
.. [1] P.W. Gaffney, The calculation of indefinite integrals of b-splines",
|
|
J. Inst. Maths Applics, 17, p.37-41, 1976.
|
|
.. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs
|
|
on Numerical Analysis, Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
Examples are given :ref:`in the tutorial <tutorial-interpolate_splXXX>`.
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
if tck.c.ndim > 1:
|
|
mesg = ("Calling splint() with BSpline objects with c.ndim > 1 is "
|
|
"not allowed. Use BSpline.integrate() instead.")
|
|
raise ValueError(mesg)
|
|
|
|
if full_output != 0:
|
|
mesg = ("full_output = %s is not supported. Proceeding as if "
|
|
"full_output = 0" % full_output)
|
|
|
|
return tck.integrate(a, b, extrapolate=False)
|
|
else:
|
|
return _impl.splint(a, b, tck, full_output)
|
|
|
|
|
|
def sproot(tck, mest=10):
|
|
"""
|
|
Find the roots of a cubic B-spline.
|
|
|
|
Given the knots (>=8) and coefficients of a cubic B-spline return the
|
|
roots of the spline.
|
|
|
|
Parameters
|
|
----------
|
|
tck : tuple or a BSpline object
|
|
If a tuple, then it should be a sequence of length 3, containing the
|
|
vector of knots, the B-spline coefficients, and the degree of the
|
|
spline.
|
|
The number of knots must be >= 8, and the degree must be 3.
|
|
The knots must be a montonically increasing sequence.
|
|
mest : int, optional
|
|
An estimate of the number of zeros (Default is 10).
|
|
|
|
Returns
|
|
-------
|
|
zeros : ndarray
|
|
An array giving the roots of the spline.
|
|
|
|
See Also
|
|
--------
|
|
splprep, splrep, splint, spalde, splev
|
|
bisplrep, bisplev
|
|
BSpline
|
|
|
|
Notes
|
|
-----
|
|
Manipulating the tck-tuples directly is not recommended. In new code,
|
|
prefer using the `BSpline` objects.
|
|
|
|
References
|
|
----------
|
|
.. [1] C. de Boor, "On calculating with b-splines", J. Approximation
|
|
Theory, 6, p.50-62, 1972.
|
|
.. [2] M. G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths
|
|
Applics, 10, p.134-149, 1972.
|
|
.. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs
|
|
on Numerical Analysis, Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
|
|
For some data, this method may miss a root. This happens when one of
|
|
the spline knots (which FITPACK places automatically) happens to
|
|
coincide with the true root. A workaround is to convert to `PPoly`,
|
|
which uses a different root-finding algorithm.
|
|
|
|
For example,
|
|
|
|
>>> x = [1.96, 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03, 2.04, 2.05]
|
|
>>> y = [-6.365470e-03, -4.790580e-03, -3.204320e-03, -1.607270e-03,
|
|
... 4.440892e-16, 1.616930e-03, 3.243000e-03, 4.877670e-03,
|
|
... 6.520430e-03, 8.170770e-03]
|
|
>>> from scipy.interpolate import splrep, sproot, PPoly
|
|
>>> tck = splrep(x, y, s=0)
|
|
>>> sproot(tck)
|
|
array([], dtype=float64)
|
|
|
|
Converting to a PPoly object does find the roots at `x=2`:
|
|
|
|
>>> ppoly = PPoly.from_spline(tck)
|
|
>>> ppoly.roots(extrapolate=False)
|
|
array([2.])
|
|
|
|
|
|
Further examples are given :ref:`in the tutorial
|
|
<tutorial-interpolate_splXXX>`.
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
if tck.c.ndim > 1:
|
|
mesg = ("Calling sproot() with BSpline objects with c.ndim > 1 is "
|
|
"not allowed.")
|
|
raise ValueError(mesg)
|
|
|
|
t, c, k = tck.tck
|
|
|
|
# _impl.sproot expects the interpolation axis to be last, so roll it.
|
|
# NB: This transpose is a no-op if c is 1D.
|
|
sh = tuple(range(c.ndim))
|
|
c = c.transpose(sh[1:] + (0,))
|
|
return _impl.sproot((t, c, k), mest)
|
|
else:
|
|
return _impl.sproot(tck, mest)
|
|
|
|
|
|
def spalde(x, tck):
|
|
"""
|
|
Evaluate a B-spline and all its derivatives at one point (or set of points) up
|
|
to order k (the degree of the spline), being 0 the spline itself.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
A point or a set of points at which to evaluate the derivatives.
|
|
Note that ``t(k) <= x <= t(n-k+1)`` must hold for each `x`.
|
|
tck : tuple
|
|
A tuple (t,c,k) containing the vector of knots,
|
|
the B-spline coefficients, and the degree of the spline whose
|
|
derivatives to compute.
|
|
|
|
Returns
|
|
-------
|
|
results : {ndarray, list of ndarrays}
|
|
An array (or a list of arrays) containing all derivatives
|
|
up to order k inclusive for each point `x`, being the first element the
|
|
spline itself.
|
|
|
|
See Also
|
|
--------
|
|
splprep, splrep, splint, sproot, splev, bisplrep, bisplev,
|
|
UnivariateSpline, BivariateSpline
|
|
|
|
References
|
|
----------
|
|
.. [1] de Boor C : On calculating with b-splines, J. Approximation Theory
|
|
6 (1972) 50-62.
|
|
.. [2] Cox M.G. : The numerical evaluation of b-splines, J. Inst. Maths
|
|
applics 10 (1972) 134-149.
|
|
.. [3] Dierckx P. : Curve and surface fitting with splines, Monographs on
|
|
Numerical Analysis, Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
To calculate the derivatives of a B-spline there are several aproaches.
|
|
In this example, we will demonstrate that `spalde` is equivalent to
|
|
calling `splev` and `splder`.
|
|
|
|
>>> import numpy as np
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> from scipy.interpolate import BSpline, spalde, splder, splev
|
|
|
|
>>> # Store characteristic parameters of a B-spline
|
|
>>> tck = ((-2, -2, -2, -2, -1, 0, 1, 2, 2, 2, 2), # knots
|
|
... (0, 0, 0, 6, 0, 0, 0), # coefficients
|
|
... 3) # degree (cubic)
|
|
>>> # Instance a B-spline object
|
|
>>> # `BSpline` objects are prefered, except for spalde()
|
|
>>> bspl = BSpline(tck[0], tck[1], tck[2])
|
|
>>> # Generate extra points to get a smooth curve
|
|
>>> x = np.linspace(min(tck[0]), max(tck[0]), 100)
|
|
|
|
Evaluate the curve and all derivatives
|
|
|
|
>>> # The order of derivative must be less or equal to k, the degree of the spline
|
|
>>> # Method 1: spalde()
|
|
>>> f1_y_bsplin = [spalde(i, tck)[0] for i in x ] # The B-spline itself
|
|
>>> f1_y_deriv1 = [spalde(i, tck)[1] for i in x ] # 1st derivative
|
|
>>> f1_y_deriv2 = [spalde(i, tck)[2] for i in x ] # 2nd derivative
|
|
>>> f1_y_deriv3 = [spalde(i, tck)[3] for i in x ] # 3rd derivative
|
|
>>> # You can reach the same result by using `splev`and `splder`
|
|
>>> f2_y_deriv3 = splev(x, bspl, der=3)
|
|
>>> f3_y_deriv3 = splder(bspl, n=3)(x)
|
|
|
|
>>> # Generate a figure with three axes for graphic comparison
|
|
>>> fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(16, 5))
|
|
>>> suptitle = fig.suptitle(f'Evaluate a B-spline and all derivatives')
|
|
>>> # Plot B-spline and all derivatives using the three methods
|
|
>>> orders = range(4)
|
|
>>> linetypes = ['-', '--', '-.', ':']
|
|
>>> labels = ['B-Spline', '1st deriv.', '2nd deriv.', '3rd deriv.']
|
|
>>> functions = ['splev()', 'splder()', 'spalde()']
|
|
>>> for order, linetype, label in zip(orders, linetypes, labels):
|
|
... ax1.plot(x, splev(x, bspl, der=order), linetype, label=label)
|
|
... ax2.plot(x, splder(bspl, n=order)(x), linetype, label=label)
|
|
... ax3.plot(x, [spalde(i, tck)[order] for i in x], linetype, label=label)
|
|
>>> for ax, function in zip((ax1, ax2, ax3), functions):
|
|
... ax.set_title(function)
|
|
... ax.legend()
|
|
>>> plt.tight_layout()
|
|
>>> plt.show()
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
raise TypeError("spalde does not accept BSpline instances.")
|
|
else:
|
|
return _impl.spalde(x, tck)
|
|
|
|
|
|
def insert(x, tck, m=1, per=0):
|
|
"""
|
|
Insert knots into a B-spline.
|
|
|
|
Given the knots and coefficients of a B-spline representation, create a
|
|
new B-spline with a knot inserted `m` times at point `x`.
|
|
This is a wrapper around the FORTRAN routine insert of FITPACK.
|
|
|
|
Parameters
|
|
----------
|
|
x (u) : float
|
|
A knot value at which to insert a new knot. If `tck` was returned
|
|
from ``splprep``, then the parameter values, u should be given.
|
|
tck : a `BSpline` instance or a tuple
|
|
If tuple, then it is expected to be a tuple (t,c,k) containing
|
|
the vector of knots, the B-spline coefficients, and the degree of
|
|
the spline.
|
|
m : int, optional
|
|
The number of times to insert the given knot (its multiplicity).
|
|
Default is 1.
|
|
per : int, optional
|
|
If non-zero, the input spline is considered periodic.
|
|
|
|
Returns
|
|
-------
|
|
BSpline instance or a tuple
|
|
A new B-spline with knots t, coefficients c, and degree k.
|
|
``t(k+1) <= x <= t(n-k)``, where k is the degree of the spline.
|
|
In case of a periodic spline (``per != 0``) there must be
|
|
either at least k interior knots t(j) satisfying ``t(k+1)<t(j)<=x``
|
|
or at least k interior knots t(j) satisfying ``x<=t(j)<t(n-k)``.
|
|
A tuple is returned iff the input argument `tck` is a tuple, otherwise
|
|
a BSpline object is constructed and returned.
|
|
|
|
Notes
|
|
-----
|
|
Based on algorithms from [1]_ and [2]_.
|
|
|
|
Manipulating the tck-tuples directly is not recommended. In new code,
|
|
prefer using the `BSpline` objects, in particular `BSpline.insert_knot`
|
|
method.
|
|
|
|
See Also
|
|
--------
|
|
BSpline.insert_knot
|
|
|
|
References
|
|
----------
|
|
.. [1] W. Boehm, "Inserting new knots into b-spline curves.",
|
|
Computer Aided Design, 12, p.199-201, 1980.
|
|
.. [2] P. Dierckx, "Curve and surface fitting with splines, Monographs on
|
|
Numerical Analysis", Oxford University Press, 1993.
|
|
|
|
Examples
|
|
--------
|
|
You can insert knots into a B-spline.
|
|
|
|
>>> from scipy.interpolate import splrep, insert
|
|
>>> import numpy as np
|
|
>>> x = np.linspace(0, 10, 5)
|
|
>>> y = np.sin(x)
|
|
>>> tck = splrep(x, y)
|
|
>>> tck[0]
|
|
array([ 0., 0., 0., 0., 5., 10., 10., 10., 10.])
|
|
|
|
A knot is inserted:
|
|
|
|
>>> tck_inserted = insert(3, tck)
|
|
>>> tck_inserted[0]
|
|
array([ 0., 0., 0., 0., 3., 5., 10., 10., 10., 10.])
|
|
|
|
Some knots are inserted:
|
|
|
|
>>> tck_inserted2 = insert(8, tck, m=3)
|
|
>>> tck_inserted2[0]
|
|
array([ 0., 0., 0., 0., 5., 8., 8., 8., 10., 10., 10., 10.])
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
|
|
t, c, k = tck.tck
|
|
|
|
# FITPACK expects the interpolation axis to be last, so roll it over
|
|
# NB: if c array is 1D, transposes are no-ops
|
|
sh = tuple(range(c.ndim))
|
|
c = c.transpose(sh[1:] + (0,))
|
|
t_, c_, k_ = _impl.insert(x, (t, c, k), m, per)
|
|
|
|
# and roll the last axis back
|
|
c_ = np.asarray(c_)
|
|
c_ = c_.transpose((sh[-1],) + sh[:-1])
|
|
return BSpline(t_, c_, k_)
|
|
else:
|
|
return _impl.insert(x, tck, m, per)
|
|
|
|
|
|
def splder(tck, n=1):
|
|
"""
|
|
Compute the spline representation of the derivative of a given spline
|
|
|
|
Parameters
|
|
----------
|
|
tck : BSpline instance or tuple
|
|
BSpline instance or a tuple (t,c,k) containing the vector of knots,
|
|
the B-spline coefficients, and the degree of the spline whose
|
|
derivative to compute
|
|
n : int, optional
|
|
Order of derivative to evaluate. Default: 1
|
|
|
|
Returns
|
|
-------
|
|
`BSpline` instance or tuple
|
|
Spline of order k2=k-n representing the derivative
|
|
of the input spline.
|
|
A tuple is returned if the input argument `tck` is a tuple, otherwise
|
|
a BSpline object is constructed and returned.
|
|
|
|
See Also
|
|
--------
|
|
splantider, splev, spalde
|
|
BSpline
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 0.13.0
|
|
|
|
Examples
|
|
--------
|
|
This can be used for finding maxima of a curve:
|
|
|
|
>>> from scipy.interpolate import splrep, splder, sproot
|
|
>>> import numpy as np
|
|
>>> x = np.linspace(0, 10, 70)
|
|
>>> y = np.sin(x)
|
|
>>> spl = splrep(x, y, k=4)
|
|
|
|
Now, differentiate the spline and find the zeros of the
|
|
derivative. (NB: `sproot` only works for order 3 splines, so we
|
|
fit an order 4 spline):
|
|
|
|
>>> dspl = splder(spl)
|
|
>>> sproot(dspl) / np.pi
|
|
array([ 0.50000001, 1.5 , 2.49999998])
|
|
|
|
This agrees well with roots :math:`\\pi/2 + n\\pi` of
|
|
:math:`\\cos(x) = \\sin'(x)`.
|
|
|
|
A comparison between `splev`, `splder` and `spalde` to compute the derivatives of a
|
|
B-spline can be found in the `spalde` examples section.
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
return tck.derivative(n)
|
|
else:
|
|
return _impl.splder(tck, n)
|
|
|
|
|
|
def splantider(tck, n=1):
|
|
"""
|
|
Compute the spline for the antiderivative (integral) of a given spline.
|
|
|
|
Parameters
|
|
----------
|
|
tck : BSpline instance or a tuple of (t, c, k)
|
|
Spline whose antiderivative to compute
|
|
n : int, optional
|
|
Order of antiderivative to evaluate. Default: 1
|
|
|
|
Returns
|
|
-------
|
|
BSpline instance or a tuple of (t2, c2, k2)
|
|
Spline of order k2=k+n representing the antiderivative of the input
|
|
spline.
|
|
A tuple is returned iff the input argument `tck` is a tuple, otherwise
|
|
a BSpline object is constructed and returned.
|
|
|
|
See Also
|
|
--------
|
|
splder, splev, spalde
|
|
BSpline
|
|
|
|
Notes
|
|
-----
|
|
The `splder` function is the inverse operation of this function.
|
|
Namely, ``splder(splantider(tck))`` is identical to `tck`, modulo
|
|
rounding error.
|
|
|
|
.. versionadded:: 0.13.0
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.interpolate import splrep, splder, splantider, splev
|
|
>>> import numpy as np
|
|
>>> x = np.linspace(0, np.pi/2, 70)
|
|
>>> y = 1 / np.sqrt(1 - 0.8*np.sin(x)**2)
|
|
>>> spl = splrep(x, y)
|
|
|
|
The derivative is the inverse operation of the antiderivative,
|
|
although some floating point error accumulates:
|
|
|
|
>>> splev(1.7, spl), splev(1.7, splder(splantider(spl)))
|
|
(array(2.1565429877197317), array(2.1565429877201865))
|
|
|
|
Antiderivative can be used to evaluate definite integrals:
|
|
|
|
>>> ispl = splantider(spl)
|
|
>>> splev(np.pi/2, ispl) - splev(0, ispl)
|
|
2.2572053588768486
|
|
|
|
This is indeed an approximation to the complete elliptic integral
|
|
:math:`K(m) = \\int_0^{\\pi/2} [1 - m\\sin^2 x]^{-1/2} dx`:
|
|
|
|
>>> from scipy.special import ellipk
|
|
>>> ellipk(0.8)
|
|
2.2572053268208538
|
|
|
|
"""
|
|
if isinstance(tck, BSpline):
|
|
return tck.antiderivative(n)
|
|
else:
|
|
return _impl.splantider(tck, n)
|
|
|