# 1-D Normal Splines

This package implements the normal splines method for solving following interpolation problem:

## Instalation

Start Julia and run the following commands:

```
julia> using Pkg
julia> Pkg.add("NormalSplines")
```

## Example usage

To use the NormalSplines package, begin your code with

```
using NormalSplines
```

Construct a normal spline to some function values (knots `x`

can be unevenly distributed):

```
using NormalSplines
x = [0.0, 1.0, 2.0]
u = [0.0, 1.0, 4.0]
interpolate(x, u, RK_W3())
```

here `RK_W3()`

is the reproducing kernel of Sobolev space `W_2^3[0,2]`

.

Evaluate the spline, its first and second derivatives at some points:

```
p = [0.0, 0.5, 1.0, 1.5, 2.0]
σ = evaluate(p) # result = [0.0, 0.26, 1.0, 2.24, 4.0]
σ = evaluate(1.5) # result = 2.24
σ' = evaluate(p, 1) # result = [0.08, 0.98, 1.96, 3.0, 4.04]
σ'' = evaluate(p, 2) # result = [1.71, 1.89, 2.02, 2.08, 2.08]
```

Construct a normal spline to some function and its first and second derivatives values:

```
using NormalSplines
x = [0.0, 1.0, 2.0] # Function knots
u = [0.0, 1.0, 4.0] # Function values
s = [2.0] # First derivative knot
v = [4.0] # First derivative value
t = [0.0, 1.0] # Second derivative knots
w = [2.0 ,2.0] # Second derivative values
interpolate(x, u, s, v, t, w, RK_W3())
```

Evaluate the spline, its first and second derivatives at some points:

```
p = [0.0, 0.5, 1.0, 1.5, 2.0]
σ = evaluate(p) # result = [0.0, 0.25, 1.0, 2.25, 4.0]
σ' = evaluate(p, 1) # result = [0.0, 1.0, 2.0, 3.0, 4.0]
σ'' = evaluate(p, 2) # result = [2.0, 2.0, 2.0, 2.0, 2.0]
```

Further examples are given in documentation.

## Documentation

For more information see Documentation.