# `interpolate.fitpack`¶

## Module Contents¶

### Functions¶

 `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-dimensional curve. `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 1-D curve. `splev`(x,tck,der=0,ext=0) Evaluate a B-spline or its derivatives. `splint`(a,b,tck,full_output=0) Evaluate the definite integral of a B-spline between two given points. `sproot`(tck,mest=10) Find the roots of a cubic B-spline. `spalde`(x,tck) Evaluate all derivatives of a B-spline. `insert`(x,tck,m=1,per=0) Insert knots into a B-spline. `splder`(tck,n=1) Compute the spline representation of the derivative of a given spline `splantider`(tck,n=1) Compute the spline for the antiderivative (integral) of a given spline.
`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-dimensional 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.

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. 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))`.
u : array_like, optional

An array of parameter values. If not given, these values are calculated automatically as `M = len(x)`, where

v = 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 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` 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. This parameter is deprecated; use standard Python warning filters instead.
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.

splrep, splev, sproot, spalde, splint, bisplrep, bisplev UnivariateSpline, BivariateSpline BSpline make_interp_spline

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 then 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.

  P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing”, 20 (1982) 171-184.
  P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines”, report tw55, Dept. Computer Science, K.U.Leuven, 1981.
  P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.

Generate a discretization of a limacon curve in the polar coordinates:

```>>> 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, new_points, 'r-')
>>> plt.show()
```
`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 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`.

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 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 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. This parameter is deprecated; use standard Python warning filters instead.
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.

UnivariateSpline, BivariateSpline splprep, splev, sproot, spalde, splint bisplrep, bisplev BSpline make_interp_spline

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.

Based on algorithms described in _, _, _, and :

  P. Dierckx, “An algorithm for smoothing, differentiation and integration of experimental data using spline functions”, J.Comp.Appl.Maths 1 (1975) 165-184.
  P. Dierckx, “A fast algorithm for smoothing data on a rectangular grid while using spline functions”, SIAM J.Numer.Anal. 19 (1982) 1286-1304.
  P. Dierckx, “An improved algorithm for curve fitting with spline functions”, report tw54, Dept. Computer Science,K.U. Leuven, 1981.
  P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.
```>>> 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()
```
`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.

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).
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.

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 N-dimensional space.

Manipulating the tck-tuples directly is not recommended. In new code, prefer using BSpline objects.

splprep, splrep, sproot, spalde, splint bisplrep, bisplev BSpline

  C. de Boor, “On calculating with b-splines”, J. Approximation Theory, 6, p.50-62, 1972.
  M. G. Cox, “The numerical evaluation of b-splines”, J. Inst. Maths Applics, 10, p.134-149, 1972.
  P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.
`splint`(a, b, tck, full_output=0)

Evaluate the definite integral of a B-spline between two given points.

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.
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)

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.

splprep, splrep, sproot, spalde, splev bisplrep, bisplev BSpline

  P.W. Gaffney, The calculation of indefinite integrals of b-splines”, J. Inst. Maths Applics, 17, p.37-41, 1976.
  P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.
`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.

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).
zeros : ndarray
An array giving the roots of the spline.

Manipulating the tck-tuples directly is not recommended. In new code, prefer using the BSpline objects.

splprep, splrep, splint, spalde, splev bisplrep, bisplev BSpline

  C. de Boor, “On calculating with b-splines”, J. Approximation Theory, 6, p.50-62, 1972.
  M. G. Cox, “The numerical evaluation of b-splines”, J. Inst. Maths Applics, 10, p.134-149, 1972.
  P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.
`spalde`(x, tck)

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).

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).
results : {ndarray, list of ndarrays}
An array (or a list of arrays) containing all derivatives up to order k inclusive for each point x.

splprep, splrep, splint, sproot, splev, bisplrep, bisplev, BSpline

  C. de Boor: On calculating with b-splines, J. Approximation Theory 6 (1972) 50-62.
  M. G. Cox : The numerical evaluation of b-splines, J. Inst. Maths applics 10 (1972) 134-149.
  P. Dierckx : Curve and surface fitting with splines, Monographs on Numerical Analysis, Oxford University Press, 1993.
`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.

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.
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.

Based on algorithms from _ and _.

Manipulating the tck-tuples directly is not recommended. In new code, prefer using the BSpline objects.

  W. Boehm, “Inserting new knots into b-spline curves.”, Computer Aided Design, 12, p.199-201, 1980.
  P. Dierckx, “Curve and surface fitting with splines, Monographs on Numerical Analysis”, Oxford University Press, 1993.
`splder`(tck, n=1)

Compute the spline representation of the derivative of a given spline

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
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.

New in version 0.13.0.

splantider, splev, spalde BSpline

This can be used for finding maxima of a curve:

```>>> from scipy.interpolate import splrep, splder, sproot
>>> 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 of .

`splantider`(tck, n=1)

Compute the spline for the antiderivative (integral) of a given spline.

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
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.

splder, splev, spalde BSpline

The splder function is the inverse operation of this function. Namely, `splder(splantider(tck))` is identical to tck, modulo rounding error.

New in version 0.13.0.

```>>> from scipy.interpolate import splrep, splder, splantider, splev
>>> 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 :

```>>> from scipy.special import ellipk
>>> ellipk(0.8)
2.2572053268208538
```