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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-D 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 : int, optional
The knots needed for task=-1.
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
(t,c,k) a tuple 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()
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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.
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.
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()
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j | |||� � S )a�
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 : 3-tuple or a BSpline object
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.
Notes
-----
Manipulating the tck-tuples directly is not recommended. In new code,
prefer using `BSpline` objects.
See Also
--------
splprep, splrep, sproot, spalde, splint
bisplrep, bisplev
BSpline
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>`.
r zeCalling splev() with BSpline objects with c.ndim > 1 is not allowed. Use BSpline.__call__(x) instead.r
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j | |||� � S )a�
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)
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.
See Also
--------
splprep, splrep, sproot, spalde, splev
bisplrep, bisplev
BSpline
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>`.
r zfCalling splint() with BSpline objects with c.ndim > 1 is not allowed. Use BSpline.integrate() instead.r
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c �` � t | t � � r�| j j dk rd}t |� � �| j \ }}}t
t |j � � � � }|� |dd� dz � � }t j
|||f|� � S t j
| |� � S )a�
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.
Notes
-----
Manipulating the tck-tuples directly is not recommended. In new code,
prefer using the `BSpline` objects.
See Also
--------
splprep, splrep, splint, spalde, splev
bisplrep, bisplev
BSpline
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>`.
r zECalling sproot() with BSpline objects with c.ndim > 1 is not allowed.N�r
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Evaluate all derivatives of a B-spline.
Given the knots and coefficients of a cubic B-spline compute all
derivatives up to order k at a point (or set of points).
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 (see `splev`).
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`.
See Also
--------
splprep, splrep, splint, sproot, splev, bisplrep, bisplev,
BSpline
References
----------
.. [1] C. de Boor: On calculating with b-splines, J. Approximation Theory
6 (1972) 50-62.
.. [2] M. G. Cox : The numerical evaluation of b-splines, J. Inst. Maths
applics 10 (1972) 134-149.
.. [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>`.
z)spalde does not accept BSpline instances.)r, r � TypeErrorr r )r r1 s r# r r s8 � �R �#�w� � � $��C�D�D�D��|�A�s�#�#�#r$ c �� � t |t � � r�|j \ }}}t t |j � � � � }|� |dd� dz � � }t j | |||f||� � \ }} }
t j
| � � } | � |d f|dd� z � � } t || |
� � S t j | |||� � S )a�
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) : array_like
A 1-D point at which to insert a new knot(s). 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.
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.])
r Nr< ���) r, r r1 r= r> r. r? r r
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Compute the spline representation of the derivative of a given spline
Parameters
----------
tck : BSpline instance or a tuple of (t, c, k)
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 iff the input argument `tck` is a tuple, otherwise
a BSpline object is constructed and returned.
Notes
-----
.. versionadded:: 0.13.0
See Also
--------
splantider, splev, spalde
BSpline
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)`.
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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
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