Splines Module

The Splines module provides cubic, bicubic, and fourier spline interpolation functionality, used for smooth representation of MHD equilibria.

Overview

The module includes:

Cubic spline interpolation for 1D data
Bicubic spline interpolation for 2D data
Fourier spline interpolation for decomposed data
Derivative evaluation capabilities
Support for both real and complex-valued data

API Reference

JPEC.SplinesMod.BicubicSpline — Method

BicubicSpline(xs, ys, fs; bctypex::Union{String, Int}="not-a-knot", bctypey::Union{String, Int}="not-a-knot")

## Arguments:
- `xs`: A vector of Float64 values representing the x-coordinates.
- `ys`: A vector of Float64 values representing the y-coordinates.
- `fs`: A 3D array of Float64 values representing the function values at the (x,y) coordinates.
## Keyword Arguments:
- `bctypex`: An integer specifying the boundary condition type for x (Default is 4, not a knot)
- `bctypey`: An integer specifying the boundary condition type for y  (Default is 4, not a knot)
## Returns:
- A `BicubicSpline` object containing the spline handle, x-coordinates, y-coordinates,
function values, number of x-coordinates, number of y-coordinates, number of quantities,
and boundary condition types.

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JPEC.SplinesMod.CubicSpline — Method

CubicSpline(xs, fs; bctype="not-a-knot")

Arguments:
  - `xs`: A vector of Float64 values representing the x-coordinates.
  - `fs`: A vector or matrix of Float64/ComplexF64 values representing the function values at the x-coordinates.
Keyword Arguments:
  - `bctype`: Boundary condition type for the cubic spline in `x`.
    - 1: Natural spline (default)
    - 2: Periodic spline
    - 3: Extrapolated spline
    - 4: "Not-a-knot" spline

Returns:
  - A `CubicSpline` object containing the spline handle, x-coordinates, function values,
    number of x-coordinates, number of quantities, and index of x position in the spline.

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JPEC.SplinesMod.FourierSpline — Method

FourierSpline(xs::Vector{Float64}, ys::Vector{Float64}, fs::Array{Float64, 3}, mband::Int; bctype::Union{String, Int}="not-a-knot", fitmethod::Int=1, fitflag::Bool=true)

Creates and fits a function of two variables, f(x, y), to a cubic spline
in the x-direction and a Fourier series in the y-direction. The y-direction
is assumed to be periodic.

## Arguments:
- `xs`: Vector of x-coordinates (length `mx`+1).
- `ys`: Vector of y-coordinates (length `my`+1, periodic direction).
- `fs`: 3D array of function values with dimensions (`mx`+1, `my`+1, `nqty`).
- `mband`: Number of Fourier modes (harmonics) to keep, from 0 to `mband`.

## Keyword Arguments:
- `bctype`: Boundary condition type for the cubic spline in `x`.
    - 1: Natural spline (default)
    - 2: Periodic spline
    - 3: Extrapolated spline
    - 4: "Not-a-knot" spline
- `fit_method`: Algorithm for computing Fourier coefficients.
    - 1: Integration method (for non-uniform `y` grids).
    - 2: Fast Fourier Transform (FFT) method (requires `length(ys)-1` to be a power of 2).

## Returns:
- A `FourierSpline` object ready for evaluation.

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JPEC.SplinesMod.ReadOnlyArray — Type

ReadOnlyArray{T,N,A}

A thin wrapper that provides a read-only view.

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JPEC.SplinesMod.bicube_eval — Function

bicube_eval(bicube::BicubicSpline, x, y, derivs::Int=0) ## Arguments: - bicube: A BicubicSpline object. - x: A Float64 value or a vector of Float64 values representing the x-coordinates to evaluate the bicubic spline at. - y: A Float64 value or a vector of Float64 values representing the y-coordinates to evaluate the bicubic spline at. ## Returns: - If x and y are single Float64 values, returns a vector of Float64 values representing the function values at that (x,y) coordinate. - If x and y are vectors of Float64 values, returns a 3D array of Float64 values where each slice corresponds to the function values at the respective (x,y) coordinates in x and y.

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JPEC.SplinesMod.fspline_eval — Function

fspline_eval(fspline::FourierSpline, x::Float64, y::Float64, derivs::Int=0)

Evaluates a fitted Fourier-Spline at given coordinates.

## Arguments:
- `fspline`: A `FourierSpline` object.
- `x`: A `Float64` x-coordinate.
- `y`: A `Float64` y-coordinate.

## Returns:
- If `x`, `y` are scalars: A tuple containing the function value(s) and any requested derivatives. If nqty=1, the results are scalars, otherwise they are vectors.

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JPEC.SplinesMod.fspline_eval — Function

fspline_eval(fspline::FourierSpline, xs::Vector{Float64}, ys::Vector{Float64}, derivs::Int=0)

Evaluates a fitted Fourier-Spline at given coordinates.

## Arguments:
- `fspline`: A `FourierSpline` object.
- `x`: A `Vector{Float64}` of x-coordinates.
- `y`: A `Vector{Float64}` of y-coordinates.

## Returns:
- If `x`, `y` are vectors: A tuple of 3D arrays for the function values and derivatives on the grid defined by `x` and `y`.

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JPEC.SplinesMod.parse_bctype — Method

parse_bctype(bctype)

Internal helper to parse a boundary condition into a validated integer code.

Arguments:

bctype: The boundary condition as a String or Int. Valid options are:
- "natural" or 1
- "periodic" or 2
- "extrap" or 3
- "not-a-knot" or 4

Returns:

A validated Int code (1-4).

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JPEC.SplinesMod.spline_eval! — Method

spline_eval!(f, spline, x; derivs=0, f1=nothing, f2=nothing, f3=nothing)

In-place evaluation of CubicSpline at a single point x.

Arguments:

f: preallocated vector for the function values (length = spline.nqty).
spline: the cubic spline object.
x: Float64 input point.
derivs: number of derivatives to evaluate (0–3).
f1, f2, f3: optional preallocated vectors for first, second, third derivatives.

Results are written into f (and optionally f1, f2, f3).

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JPEC.SplinesMod.spline_eval — Method

spline_eval(spline::CubicSpline{T}, x, derivs::Int=0) where {T<:Union{Float64, ComplexF64}} ## Arguments: - spline: A Spline object created by CubicSpline. - x: A Float64 value or a vector of Float64 values representing the x-coordinates to evaluate the spline at. ## Returns: - If x is a single Float64 value, returns a vector of Float64 values representing the function values at that x-coordinate. - If x is a vector of Float64 values, returns a matrix of Float64 values where each row corresponds to the function values at the respective x-coordinate in x. - Depending on the derivatives requested, it may return additional vectors for the first, second, or third derivatives.

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JPEC.SplinesMod.spline_integrate! — Method

spline_integrate!(spline::CubicSpline{T}) where {T<:Union{Float64, ComplexF64}}

## Arguments:
- `spline`: A mutable `Spline` object".

## Returns:
- Nothing. Updates `spline._fsi` in place so that
`spline._fsi[i, :]` equals `∫_{xs[1]}^{xs[i]} f(x) dx` for each component.

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Types

CubicSplineType

Represents a cubic spline interpolation object.

BicubicSplineType

Represents a bicubic spline interpolation object for 2D data.

FourierSplineType

Represents a Fourier spline interpolation object for decomposed data.

Functions

CubicSpline

JPEC.SplinesMod.CubicSpline — Type

CubicSpline(xs, fs; bctype="not-a-knot")

Arguments:
  - `xs`: A vector of Float64 values representing the x-coordinates.
  - `fs`: A vector or matrix of Float64/ComplexF64 values representing the function values at the x-coordinates.
Keyword Arguments:
  - `bctype`: Boundary condition type for the cubic spline in `x`.
    - 1: Natural spline (default)
    - 2: Periodic spline
    - 3: Extrapolated spline
    - 4: "Not-a-knot" spline

Returns:
  - A `CubicSpline` object containing the spline handle, x-coordinates, function values,
    number of x-coordinates, number of quantities, and index of x position in the spline.

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spline_eval

JPEC.SplinesMod.spline_eval — Function

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BicubicSpline

JPEC.SplinesMod.BicubicSpline — Type

BicubicSpline(xs, ys, fs; bctypex::Union{String, Int}="not-a-knot", bctypey::Union{String, Int}="not-a-knot")

## Arguments:
- `xs`: A vector of Float64 values representing the x-coordinates.
- `ys`: A vector of Float64 values representing the y-coordinates.
- `fs`: A 3D array of Float64 values representing the function values at the (x,y) coordinates.
## Keyword Arguments:
- `bctypex`: An integer specifying the boundary condition type for x (Default is 4, not a knot)
- `bctypey`: An integer specifying the boundary condition type for y  (Default is 4, not a knot)
## Returns:
- A `BicubicSpline` object containing the spline handle, x-coordinates, y-coordinates,
function values, number of x-coordinates, number of y-coordinates, number of quantities,
and boundary condition types.

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bicube_eval

JPEC.SplinesMod.bicube_eval — Function

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FourierSpline

JPEC.SplinesMod.FourierSpline — Type

FourierSpline(xs::Vector{Float64}, ys::Vector{Float64}, fs::Array{Float64, 3}, mband::Int; bctype::Union{String, Int}="not-a-knot", fitmethod::Int=1, fitflag::Bool=true)

Creates and fits a function of two variables, f(x, y), to a cubic spline
in the x-direction and a Fourier series in the y-direction. The y-direction
is assumed to be periodic.

## Arguments:
- `xs`: Vector of x-coordinates (length `mx`+1).
- `ys`: Vector of y-coordinates (length `my`+1, periodic direction).
- `fs`: 3D array of function values with dimensions (`mx`+1, `my`+1, `nqty`).
- `mband`: Number of Fourier modes (harmonics) to keep, from 0 to `mband`.

## Keyword Arguments:
- `bctype`: Boundary condition type for the cubic spline in `x`.
    - 1: Natural spline (default)
    - 2: Periodic spline
    - 3: Extrapolated spline
    - 4: "Not-a-knot" spline
- `fit_method`: Algorithm for computing Fourier coefficients.
    - 1: Integration method (for non-uniform `y` grids).
    - 2: Fast Fourier Transform (FFT) method (requires `length(ys)-1` to be a power of 2).

## Returns:
- A `FourierSpline` object ready for evaluation.

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fspline_eval

JPEC.SplinesMod.fspline_eval — Function

fspline_eval(fspline::FourierSpline, x::Float64, y::Float64, derivs::Int=0)

Evaluates a fitted Fourier-Spline at given coordinates.

## Arguments:
- `fspline`: A `FourierSpline` object.
- `x`: A `Float64` x-coordinate.
- `y`: A `Float64` y-coordinate.

## Returns:
- If `x`, `y` are scalars: A tuple containing the function value(s) and any requested derivatives. If nqty=1, the results are scalars, otherwise they are vectors.

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fspline_eval(fspline::FourierSpline, xs::Vector{Float64}, ys::Vector{Float64}, derivs::Int=0)

Evaluates a fitted Fourier-Spline at given coordinates.

## Arguments:
- `fspline`: A `FourierSpline` object.
- `x`: A `Vector{Float64}` of x-coordinates.
- `y`: A `Vector{Float64}` of y-coordinates.

## Returns:
- If `x`, `y` are vectors: A tuple of 3D arrays for the function values and derivatives on the grid defined by `x` and `y`.

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Example Usage

1D Cubic Spline

using JPEC

# Create data points
xs = collect(range(0.0, stop=2π, length=21))
fs = sin.(xs)

# Set up spline (1 quantity)
spline = JPEC.SplinesMod.CubicSpline(xs, hcat(fs), 1)

# Evaluate at new points
xs_fine = collect(range(0.0, stop=2π, length=100))
fs_fine = JPEC.SplinesMod.spline_eval(spline, xs_fine)

2D Bicubic Spline

# Create 2D grid
xs = collect(range(0.0, stop=2π, length=20))
ys = collect(range(0.0, stop=2π, length=20))

# Create 2D function data
fs = zeros(20, 20, 1)
for i in 1:20, j in 1:20
    fs[i, j, 1] = sin(xs[i]) * cos(ys[j])
end

# Set up bicubic spline
bcspline = JPEC.SplinesMod.BicubicSpline(xs, ys, fs, 1, 1)

# Evaluate with derivatives
x_eval, y_eval = π/2, π/4
f, fx, fy = JPEC.SplinesMod.bicube_eval(bcspline, x_eval, y_eval, 1)