Dierckx.jl

Julia library for 1-d and 2-d 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 Grid.jl, a pure-Julia interpolation package. This package is intended to complement Grid.jl and to serve as a benchmark in cases of overlapping functionality.

Install

julia> Pkg.add("Dierckx")

The Fortran library source code is distributed with the package, so you need a Fortran compiler. On Ubuntu, sudo apt-get install gfortran will do it.

Windows is not yet supported.

Example Usage

using Dierckx

# -------------------------------------------------------
# 1-d splines

x = [0., 1., 2., 3., 4.]
y = [0., 1., 8., 27., 64.]
spl = Spline1D(x, y)
evaluate(spl, [1.5, 2.5])  # result = [3.375, 15.625]
evaluate(spl, 1.5)  # result = 3.375

# -------------------------------------------------------
# 2-d splines

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.]

# Fit a 2-d spline to inputs
spline = Spline2D(x, y, z)

# evaluate at element-wise 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 = evaluate(spline, xi, yi)  # 1-d array of length 7

# 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)  # 2-d array of size (4, 3)

Reference

1-d 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 1-d arrays x and y. The parameter bc 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 minimalizing the discontinuity jumps of the kth derivative of the spline at the knots. The amount of smoothness is determined by the condition that sum((w[i]*(y[i]-spline(x[i])))**2) <= s, with s a given non-negative 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. if s is too large, the spline will be too smooth and signal will be lost ; if s is too small the spline will pick up too much noise. in the extreme cases the program will return an interpolating spline if s=0.0 and the weighted least-squares polynomial of degree k if s 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 kth derivative of the spline at the given knots.

evaluate(spl, x)

Evalute the 1-d spline spl at points given in x, which can be a 1-d array or scalar. If a 1-d array, the values must be monotonically increasing.

2-d 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 2-d spline to the input data. x and y must be 1-d arrays.

If z is also a 1-d array, the inputs are assumed to represent unstructured data, with z[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 2-d 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 that size(z) == (length(x), length(y)).

evaluate(spl, x, y)

Evalute the 2-d 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 2-d spline spl at the grid points spanned by the coordinate arrays x and y. The input arrays must be monotonically increasing.

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:

License

Dierckx.jl is distributed under a 3-clause 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.