interpolation - What's inside InterpolatingFunction[{{1., 4.}}, ]?

I'm curious what's inside the InterpolationFunction object?

For example:

InputForm[Interpolation[{1., 2., 3., 4.}]]
(*
  InterpolatingFunction[
    {{1., 4.}}, 

    {4, 7, 0, {4}, {4}, 0, 0, 0, 0, Automatic}, 
    {{1., 2., 3., 4.}},{Developer`PackedArrayForm, {0, 1, 2, 3, 4}, {1., 2., 3., 4.}}, 
    {Automatic}]
*)

What do those arguments mean?

Answer

From inspection, some investigation and ruebenko's help, what I've found so far is that InterpolatingFunction has the following underlying structure:

InterpolatingFunction[
    domain,                    (* or min/max of grid for each dimension          *)

    List[
        version,               (* 3 in Mathematica 7, 4 from 8 onwards           *)
        bitField,              (* 3 for exact/arbitrary precision
                                  7 for machine numbers, 15 for machine complex,
                                  39 for spline, 4259 for FEM elements. These are 
                                  for version 4, and are different for version 3.*)
        dataDerivatives,       (* Max order of derivatives supplied for input    *)
        domainGridSize,        (* or input sample points in each dimension       *)
        interpolationOrder,    (* actually, order + 1; for each dimension        *)
        nthDerivativeOfIntFun, (* Denotes if the current InterpolatingFunction is 

                                  an nth derivative of an existing Int. Func. and
                                  0 otherwise.                                   *)
        periodicInterpolation, (* 0 for False and {1} for True                   *)
        0, 0,                  (* One of the zeros is a permutation flag for 
                                  time-dependent InterpolatingFunction           *)
        Automatic              (* Extrapolation handler                          *)
    ],

    basicInterpolatingUnit,    (* This is setup such that it agrees with the input
                                   values at the input grid points. You might see 

                                  structures with Developer`PackedArrayForm for 
                                  2D Hermite, BSplineFunction for 2D Spline
                                  NDSolve`FEM`ElementMesh for 3D, or nothing.    *)
    Automatic                  (* Unknown                                        *)
]

You can access most of this internal data using the following arguments to any InterpolatingFunction object:

{"Domain", "Coordinates", "Grid", "ValuesOnGrid", "InterpolationOrder", "DerivativeOrder"}

See the contents of the following package for more information on what exactly the above arguments return:

SystemOpen@FindFile["DifferentialEquations`InterpolatingFunctionAnatomy`"]