# Interpolations

The package Interpolations.jl implements a variety of interpolation schemes for the Julia language. It has the goals of ease-of-use, broad algorithmic support, and exceptional performance.

Currently this package supports B-splines and irregular grids. The API has been designed with intent to support more options. Pull-requests are more than welcome! It should be noted that the API may continue to evolve over time.

There are many other interpolation packages implemented in Julia. For a listing, see Other Interpolation Packages.

Some of these packages support methods that `Interpolations`

does not, so if you can't find what you need here, check one of them or submit a pull request here.

## Installation

Interpolations.jl can be installed via the following invocation since it is a registered Julia package.

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

## Example Usage

Create a grid `xs`

and an array `A`

of values to be interpolated

```
using Interpolations
xs = 1:0.2:5
A = log.(xs)
```

Create linear interpolation object without extrapolation

`julia> interp_linear = linear_interpolation(xs, A);`

`julia> interp_linear(3) # exactly log(3)`

`1.0986122886681098`

`julia> interp_linear(3.1) # approximately log(3.1)`

`1.1308815492368953`

`julia> interp_linear(0.9) # outside grid: error`

`ERROR: BoundsError: attempt to access 21-element extrapolate(scale(interpolate(::Vector{Float64}, BSpline(Linear())), (1.0:0.2:5.0,)), Throw()) with element type Float64 at index [0.9]`

Create linear interpolation object with extrapolation

`julia> interp_linear_extrap = linear_interpolation(xs, A, extrapolation_bc=Line());`

`julia> interp_linear_extrap(0.9) # outside grid: linear extrapolation`

`-0.09116077839697727`

## Performant Example Usage

The above use of `LinearInterpolation`

is actually a short hand for a composition of `interpolate`

, `scale`

, and `extrapolate`

. You may not need all of the the scaling and extrapolation features.

`interp_linear = extrapolate(scale(interpolate(A, BSpline(Linear())), xs))`

If we know we do not need the extrapolation portion, we can use the following.

`scaled_itp = scale(interpolate(A, BSpline(Linear())), xs)`

We can also remove the scaling for further performance if integer valued knots and regular grids are sufficient.

`itp = interpolate(A, BSpline(Linear()))`

Removing the scaling or extrapolation will help accelerate interpolation by removing unneeded operations and branches. This can permit the use of advanced processor Single Instruction/Multiple Data (SIMD) capabilities.

## Regular Grids

Interpolations.jl is optimized for use with regular grids with uniform spacing. The highest performance is achieved when the knots are an `AbstractUnitRange`

such as `2:5`

or `Base.OneTo(9)`

. The default case if no knots are specified is to assign the knots as a `UnitRange`

starting at `1`

.

## Scaling

If the knots are not unit spaced or start at a distinct value other than `1`

, then the `scale`

function can be used. While this increases the flexibility of the interpolation, some performance penalty is acquired. See Scaled BSplines for further information.

## Irregular Grids

If the knots are irregularly spaced, then the ranges between knots will have to be scaled as in the `Gridded`

interpolation type. See Gridded interpolation for additional details.

## Points outside the knots

For points not between knots, extrapolation can be used. This introduces a branch into the code that checks whether the point to be queried is inside or outside of the knots. This branch can inhibit the use of vectorized SIMD computation, resulting in a reduction of performance. See Extrapolation.