Dierckx.jl
Julia library for 1d and 2d splines
This is a Julia wrapper for the dierckx Fortran library, the same library underlying the spline classes in scipy.interpolate. Some of the functionality here overlaps with Interpolations.jl, a pureJulia interpolation package. Take a look at it if you have a use case not covered here.
All new development on Dierckx.jl
will be for Julia v1.3 and above.
The master
branch is therefore incompatible with earlier versions
of Julia.
Features
 Implements Bsplines (basis splines).
 Splines from first order to fifth order; default is third order (cubic).
 Fit and evaluate 1d and 2d splines on irregular grids.
 Fit and evaluate 2d splines at unstructured points.
 Fit "smooth" (noninterpolating) splines with adjustable smoothing factor s.
 Derivatives, integrals and roots of 1d splines.
 Parametric Bsplines.
Install (Julia 1.3 and later)
(v1.3) pkg> add Dierckx
(Type ]
to enter package mode.)
Example Usage
using Dierckx
Fit a 1d spline to some input data (points can be unevenly spaced):
x = [0., 1., 2., 3., 4.]
y = [1., 0., 7., 26., 63.] # x.^3  1.
spl = Spline1D(x, y)
Evaluate the spline at some new points:
spl([1.5, 2.5]) # result = [2.375, 14.625]
spl(1.5) # result = 2.375
Equivalent to the above:
evaluate(spl, [1.5, 2.5])
evaluate(spl, 1.5)
Evaluate derivative, integral, or roots:
derivative(spl, 1.5) # derivate at x=1.5; result is 5.75
integrate(spl, 0., 4.) # integrate from x=0 to x=4; result is 60.0
roots(spl) # result is [1.0]
Note that roots()
only works for cubic splines (k=3).
Fit a 2d spline to data on a (possibly irregular) grid:
x = [0.5, 2., 3., 4., 5.5, 8.]
y = [0.5, 2., 3., 4.]
z = [1. 2. 1. 2.; # size is (length(x), length(y))
1. 2. 1. 2.;
1. 2. 3. 2.;
1. 2. 2. 2.;
1. 2. 1. 2.;
1. 2. 3. 1.]
spline = Spline2D(x, y, z)
Note that if you consider z
as a matrix, x
refers to row
coordinates and y
refers to column coordinates.
Evaluate at elementwise points:
xi = [1., 1.5, 2.3, 4.5, 3.3, 3.2, 3.]
yi = [1., 2.3, 5.3, 0.5, 3.3, 1.2, 3.]
zi = spline(xi, yi) # 1d array of length 7
zi = evaluate(spline, xi, yi) # equivalent to previous line
Evaluate at grid spanned by input arrays:
xi = [1., 1.5, 2.3, 4.5]
yi = [1., 2.3, 5.3]
zi = evalgrid(spline, xi, yi) # 2d array of size (4, 3)
Reference
1d Splines
Spline1D(x, y; w=ones(length(x)), k=3, bc="nearest", s=0.0)
Spline1D(x, y, xknots; w=ones(length(x)), k=3, bc="nearest")

Create a spline of degree
k
(1 = linear, 2 = quadratic, 3 = cubic, up to 5) from 1d arraysx
andy
. The parameterbc
specifies the behavior when evaluating the spline outside the support domain, which is(minimum(x), maximum(x))
. The allowed values are"nearest"
,"zero"
,"extrapolate"
,"error"
.In the first form, the number and positions of knots are chosen automatically. The smoothness of the spline is then achieved by minimizing the discontinuity jumps of the
k
th derivative of the spline at the knots. The amount of smoothness is determined by the condition thatsum((w[i]*(y[i]spline(x[i])))**2) <= s
, withs
a given nonnegative constant, called the smoothing factor. The number of knots is increased until the condition is satisfied. By means of this parameter, the user can control the tradeoff between closeness of fit and smoothness of fit of the approximation. ifs
is too large, the spline will be too smooth and signal will be lost ; ifs
is too small the spline will pick up too much noise. in the extreme cases the program will return an interpolating spline ifs=0.0
and the weighted leastsquares polynomial of degreek
ifs
is very large.In the second form, the knots are supplied by the user. There is no smoothing parameter in this form. The program simply minimizes the discontinuity jumps of the
k
th derivative of the spline at the given knots.
evaluate(spl, x)
 Evaluate the 1d spline
spl
at points given inx
, which can be a 1d array or scalar. If a 1d array, the values must be monotonically increasing.
derivative(spl, x; nu=1)
 Evaluate the
nu
th derivative of the spline at points inx
.
integrate(spl, a, b)
 Definite integral of spline between
x=a
andx=b
.
roots(spl; maxn=8)
 For cubic splines (
k=3
) only, find roots. Only up tomaxn
roots are returned. A warning is issued if the spline has more roots than the number returned.
Parametric Splines
These are the Bspline representation of a curve through Ndimensional space.
ParametricSpline(X; s=0.0, ...)
ParametricSpline(u, X; s=0.0, ...)
ParametricSpline(X, knots, ...)
ParametricSpline(u, X, knots, ...)

X
is a 2d array with size(N, m)
:N
is the number of dimensions of the space (must be between 1 and 10) andm
is the number of points.X[:, i]
gives the coordinates of thei
th point. 
u
is a 1d array giving parameter values at each of them
points. If not given, values are calculated automatically. 
knots
is a 1d array giving userspecified knots, if desired.
Keyword arguments common to all constructor methods:
w
: weight applied to each point (lengthm
1d array).k
: Spline order (between 1 and 5; default 3).bc
: Boundary condition (default'nearest'
).periodic
: Treat curve as periodic? (Default isfalse
).
2d Splines
Spline2D(x, y, z; w=ones(length(x)), kx=3, ky=3, s=0.0)
Spline2D(x, y, z; kx=3, ky=3, s=0.0)

Fit a 2d spline to the input data.
x
andy
must be 1d arrays.If
z
is also a 1d array, the inputs are assumed to represent unstructured data, withz[i]
being the function value at point(x[i], y[i])
. In this case, the lengths of all inputs must match.If
z
is a 2d array, the data are assumed to be gridded:z[i, j]
is the function value at(x[i], y[j])
. In this case, it is required thatsize(z) == (length(x), length(y))
. (Note that when interpretingz
as a matrix,x
gives the row coordinates andy
gives the column coordinates.)
evaluate(spl, x, y)
 Evaluate the 2d spline
spl
at points(x[i], y[i])
. Inputs can be Vectors or scalars. Points outside the domain of the spline are set to the values at the boundary.
evalgrid(spl, x, y)
 Evaluate the 2d spline
spl
at the grid points spanned by the coordinate arraysx
andy
. The input arrays must be monotonically increasing. The output is a 2d array with size(length(x), length(y))
:output[i, j]
is the function value at(x[i], y[j])
. In other words, when interpreting the result as a matrix,x
gives the row coordinates andy
gives the column coordinates.
derivative(spl, x, y; nux = 1, nuy = 1)

Evaluate the partial derivative of the 2d spline
spl
at the grid points spanned by the coordinate arraysx
andy
. Note thatx
refers to the row coordinates, andy
to the column ones. The order of the derivatives may be between0
andkx1
forx
and0
andky1
fory
. 
integral of a 2d spline over the domain
[x0, x1]*[y0, y1]
integrate(spl, x0, x1, y0, y1)
Translation from scipy.interpolate
The Spline
classes in scipy.interpolate are also thin wrappers
for the Dierckx Fortran library. The performance of Dierckx.jl should
be similar or better than the scipy.interpolate classes. (Better for
small arrays where Python overhead is more significant.) The
equivalent of a specific classes in scipy.interpolate:
scipy.interpolate class  Dierckx.jl constructor method 

UnivariateSpline  Spline1D(x, y; s=length(x)) 
InterpolatedUnivariateSpline  Spline1D(x, y; s=0.0) 
LSQUnivariateSpline  Spline1D(x, y, xknots) 
SmoothBivariateSpline  Spline2D(x, y, z; s=length(x)) 
LSQBivariateSpline  
RectBivariateSpline  Spline2D(x, y, z; s=0.0) (z = 2d array) 
SmoothSphereBivariateSpline  
LSQSphereBivariateSpline  
RectSphereBivariateSpline 
Parametric splines:
scipy.interpolate function  Dierckx.jl constructor method 

splprep(X) 
ParametricSpline(X) 
splprep(X, u=...) 
ParametricSpline(u, X) 
splprep(X, t=...) 
ParametricSpline(X, t) (t = knots) 
splprep(X, u=..., t=...) 
ParametricSpline(u, X, t) 
License
Dierckx.jl is distributed under a 3clause BSD license. See LICENSE.md for details. The real*8 version of the Dierckx Fortran library as well as some test cases and error messages are copied from the scipy package, which is distributed under this license.
Citation
If you use this package in a pulication and wish to cite it, you may want to cite the original Fortran dierckx library:
Paul Dierckx, Curve and Surface Fitting with Splines, Oxford University Press, 1993
If convenient, you can also include a link to the github repository for this package: https://github.com/kbarbary/Dierckx.jl.