# CubicHermiteSpline.jl

CubicHermiteSpline.jl is a naive implementation of cubic Hermite spline interpolation for 1D data points in pure Julia. Currently, the 1st order gradient should be given by the user. It is most useful when the gradient happens to be available. When the function to be interpolated is smooth and the accuracy of the gradients is high, the cubic Hermite spline interpolation should perform extremely well. A demonstration of the power of this interpolation can be found here.

## Features

• Univariate cubic Hermite spline interpolation for 1D data points (regular and irregular grids are both supported).
• Gradient (1st order derivative) of the interpolation. (New in version 0.2.0)
• Bivariate cubic Hermite spline interpolation for 2D data points (regular and irregular grids are both supported). (New in version 0.3.0)

## Usage

### 1D Interpolation

Below shows a trivial example when the function is a cubic polynomial. In such case, the function can be exactly interpolated.

First, prepare a set of data points to be interpolated. Note that here we use a cubic polynomial function which can be exactly interpolated by the cubic Hermite spline method.

julia> using CubicHermiteSpline

julia> f(x) = x^3 - 3x^2 + 2x - 5;

julia> df(x) = 3x^2 - 6x + 2;

julia> x = range(0, 2.5, step=0.5)
0.0:0.5:2.5

julia> y = f.(x)
6-element Array{Float64,1}:
-5.0
-4.625
-5.0
-5.375
-5.0
-3.125


The gradients at each data points are also computed which is required by the cubic Hermite spline method.

julia> gradient = df.(x)
6-element Array{Float64,1}:
2.0
-0.25
-1.0
-0.25
2.0
5.75


Then, we construct a interpolation instance by using CubicHermiteSpline package.

julia> spl = UnivariateCHSInterpolation(x, y, gradient);


Perform interpolation for a single input x.

julia> xi = 1.2;

julia> yi = spl(xi)  # Or using interp(spl, xi)
-5.192


Perform interpolation for an array of input x.

julia> xi = [0.5, 1.2];

julia> yi = spl(xi)
2-element Array{Float64,1}:
-4.625
-5.192


The 1st order derivative of the interpolation can be obtained.

julia> xi = 1.2;



Note that 1st order derivatives at each data point should be provided by the user. Please see doc/tutorial_univariate.ipynb for a detailed example of the univariate cubic Hermite spline interpolation.

### 2D Interpolation

Construct a 2D interpolation instance:

spl2d = BivariateCHSInterpolation(rand(100), rand(100), rand(100), rand(100), rand(100))


Perform interpolation for a single input point (x, y):

spl2d(x, y)
# Or use
interp(spl2d, x, y)


Compute the interpolated 1st order derivatives at a single input point (x, y):

grad(spl2d, x, y)


Please see doc/tutorial_bivariate.ipynb for a concrete example of bivariate cubic Hermite spline interpolation.