Interpolation and Integration

Provide interpolation and integration.

function dataslice(data, axes, indices)

Return a view of data sliced on given axes with given indices. Works like view(data, (:, ..., :, i1:f1, :, ..., in:fn, :, ..., :)). Type unstable unless slice dims are constant.

#Members:

• data: data to be sliced.
• axes: axes to be sliced. accept Int or NTuple{DIM, Int} for single or multiple axes. when omitted, assume all axes.
• indices: indices of slicing. accept UnitRange{Int} or Vector of UnitRange{Int} like 2:8 or [2:8, 3:7]

function differentiate1D(data, xgrid, x; axis=1)

calculate integration of data[i] on xgrid. For 1D data, return a number; for multiple dimension, reduce the given axis.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• x: point to differentiate
• axis: axis to be differentiated in data

function findneighbor(xgrid::T, x; method=:default) where {T}

Find neighbor grid points and related information for extrapolating the value of x on xgrid.

#Members:

• xgrid: grid to be interpolated
• x: value to be interpolated
• method: :default use optimized method, :linear use linear interp.

function integrate1D(::ChebIntegrate, data, xgrid)

calculate integration of data[i] on xgrid works for grids that have integration weight stored

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data

function integrate1D(::NoIntegrate, data, xgrid)

calculate integration of data[i] on xgrid works for grids that do not have integration weight stored

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data

function integrate1D(::WeightIntegrate, data, xgrid)

calculate integration of data[i] on xgrid works for grids that have integration weight stored

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data

function integrate1D(::CompositeIntegrate, data, xgrid)

calculate integration of data[i] on xgrid call integrate1D for each subgrid and return the sum.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data

function integrate1D(data, xgrid, range; axis=1)

calculate integration of data[i] on xgrid. For 1D data, return a number; for multiple dimension, reduce the given axis.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• range: range of integration, [init, fin] within bound of xgrid.
• axis: axis to be integrated in data

function integrate1D(data, xgrid; axis=1)

calculate integration of data[i] on xgrid. For 1D data, return a number; for multiple dimension, reduce the given axis.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• axis: axis to be integrated in data

function interp1D(::ChebInterp, data, xgrid, x)

linear interpolation of data(x), barycheb for BaryCheb grid

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• x: x

function interp1D(::CompositeInterp,data, xgrid, x)

linear interpolation of data(x), first floor on panel to find subgrid, then call interp1D on subgrid

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• x: x

function interp1D(::LinearInterp,data, xgrid, x)

linear interpolation of data(x), use floor and linear1D

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• x: x

function interp1D(data, xgrid, x; axis=1, method=InterpStyle(T))

linear interpolation of data(x) with single or multiple dimension. For 1D data, return a number; for multiple dimension, reduce the given axis.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• x: x
• axis: axis to be interpolated in data
• method: by default use optimized method; use linear interp if Interp.LinearInterp()

function interp1DGrid(::CompositeInterp, data, xgrid, grid)

linear interpolation of data(grid[1:end]), return a Vector grid should be sorted.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• grid: points to be interpolated on

function interp1DGrid(::Union{LinearInterp,ChebInterp}, data, xgrid, grid)

linear interpolation of data(grid[1:end]), return a Vector simply call interp1D on each points

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• grid: points to be interpolated on

function interp1DGrid(data, xgrid, grid; axis=1, method=InterpStyle(T))

For 1D data, do interpolation of data(grid[1:end]), return a Vector. For ND data, do interpolation of data(grid[1:end]) at given axis, return data of same dimension.

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• grid: points to be interpolated on
• axis: axis to be interpolated in data
• method: by default use optimized method; use linear interp if :linear

function interpsliced(neighbor, data; axis=1)

Interpolate with given neighbor and sliced data. Assume data already sliced on given axis.

#Members:

• neighbor: neighbor from findneighbor()
• data: sliced data
• axis: axis sliced and to be interpolated

function linear1D(data, xgrid, x)

linear interpolation of data(x)

#Arguments:

• xgrid: one-dimensional grid of x
• data: one-dimensional array of data
• x: x

linear2D(data, xgrid, ygrid, x, y)

linear interpolation of data(x, y)

#Arguments:

• xgrid: one-dimensional grid of x
• ygrid: one-dimensional grid of y
• data: two-dimensional array of data
• x: x
• y: y

function linearND(data, xgrids, xs)

linear interpolation of data(xs)

#Arguments:

• xgrids: n-dimensional grids, xgrids[i] is a 1D grid
• data: n-dimensional array of data
• xs: list of x, x[i] corresponds to xgrids[i]

function locate(grid, x)

return the index of grid point closest to x. Useful for Monte Carlo algorithm when variable x is continuous while histogram is stored on grid.

#Arguments:

• grid: one-dimensional grid of x
• x: point to locate

function volume(grid, i)

return the volume of grid point i. The volume is defined as the length/area/volume/... of histogram bar represented by grid point i. In 1D grids of this package, it is defined as the length of interval between (grid[i-1]+grid[i])/2 and (grid[i]+grid[i+1])/2, and for edge points one side is replaced by boundary points. When index i is omitted, the length of the whole grid is returned. It is guaranteed that volume(grid)==sum(volume(grid, i) for i in 1:length(grid)).

#Arguments:

• grid: one-dimensional grid
• i: index of grid point