Extracting polynomials from a cubic spline function

Question

I have this set of three data points and I want the natural cubic spline functions for them:

  dat = {{-.0247500, .5}, {.3349375, -.25}, {1.101000, 0}};

I tried:

  f = SplineFit[dat, Cubic]

Using another approach and code, I get the two cubic functions (I am trying to verify the correctness) as:

$$\begin{align*} -0.481347 + 0.753113 x - 0.0338047 x^2 - 0.455282 x^3\\ -0.506486 + 0.978281 x - 0.706073 x^2 + 0.213767 x^3 \end{align*}$$

Is there any way to see these results and the intermediate results to derive the two cubics using Mathematica? (I tried looking at documentation and could not make sense of it.)

Answer 1

I've finally decided to write a "modern" re-implementation of the old SplineFit[] routine from NumericalMath`SplineFit`. In the course of writing it, I was struck by how crude it actually is compared to some of the interpolation methods I've presented in past answers. Thus, except for the purpose of reproducing the output of SplineFit[] from old calculations, I do not recommend its use, as there are now better methods.

splinefit[dat_?MatrixQ] :=
          Module[{n = Length[dat], del = Differences[dat]},
                 del = LinearSolve[SparseArray[
                       {Band[{2, 1}] -> 1, 
                        Band[{1, 1}] -> ArrayPad[ConstantArray[4, n - 2], 1, 2], 
                        Band[{1, 2}] -> 1}], 
                       ListCorrelate[{{3}, {3}}, del, {-{1, 1}, {1, 1}}, {0}]];
                 BSplineFunction[#1, SplineDegree -> 3, SplineKnots -> #2] & @@
                 {Join[{dat[[1]]},
                       Riffle[Most[dat] + Most[del]/3, Rest[dat] - Rest[del]/3],
                       {dat[[-1]]}],
                  ArrayPad[Riffle[#, #] &[Range[0, n - 1]], 2, "Fixed"]}]

For instance, here is the example from the old docs:

sf = splinefit[{{0, 0}, {1, 2}, {-1, 3}, {0, 1}, {3, 0}}];

sf[1.4]
   {0.2651428571428573, 2.7017142857142855}

ParametricPlot[sf[t], {t, 0, 4}, AspectRatio -> 1/GoldenRatio]

---

Starting with

dat = {{-.0247500, .5}, {.3349375, -.25}, {1.101000, 0}};

use the new splinefit[] on it:

f = splinefit[dat];

Unfortunately, as it stands, BSplineFunction[] objects are immune to exposure from PiecewiseExpand[]. The routine could certainly be modified so that the results are entirely in terms of BSplineBasis[]. Instead, I'll demonstrate some undocumented functionality for extracting knots and control points from BSplineFunction[] objects, which we can then use with BSplineBasis[]:

n = First[f["Degree"]];
knots = First[f["Knots"]];
cpts = f["ControlPoints"];

(A complete(?) list of supported methods can be obtained from f["Methods"].)

The explicit form of the piecewise polynomial functions can then be recovered like so:

fun[x_] = FullSimplify[PiecewiseExpand[
                       Table[BSplineBasis[{n, knots}, j - 1, x],
                             {j, Length[cpts]}].cpts]];

I've omitted the result, but you can compare it yourself with the ones obtained by Ray in his answer.

Answer 2

f1[x_] = a1 + b1*x + c1*x^2 + d1*x^3;
f2[x_] = a2 + b2*x + c2*x^2 + d2*x^3;
s = Solve[{f1 @ dat[[1,1]] == dat[[1,2]],
           f1 @ dat[[2,1]] == dat[[2,2]],
           f2 @ dat[[2,1]] == dat[[2,2]],
           f2 @ dat[[3,1]] == dat[[3,2]],
           f1' @ dat[[2,1]] == f2' @ dat[[2,1]],
           f1'' @ dat[[2,1]] == f2'' @ dat[[2,1]],
           f1'' @ dat[[1,1]] == 0,
           f2'' @ dat[[3,1]] == 0},
          {a1, b1, c1, d1, a2, b2, c2, d2}]
(* {{a1 -> 0.438903, b1 -> -2.46492, c1 -> 0.221098, d1 -> 2.97775,  
     a2 -> 0.603324, b2 -> -3.93762, c2 -> 4.61804, d2 -> -1.39813}} *)
f[x_] = Piecewise[{{f1[x], x ≤ dat[[2, 1]]}}, f2[x]] /. s;

Plot[f[x], {x, dat[[1, 1]], dat[[3, 1]]}, Frame -> True, Axes -> None, 
     Prolog -> {PointSize[.02], Point /@ dat}]

EDIT, Re the comment by @george2079 -- SplineFit uses the same algebra that I used, but instead of fitting y to x, which is what I did, it fits x and y separately to their (zero-based) indices, Range[0, Length @ dat - 1].

( spline = SplineFit[dat, Cubic] ) // InputForm

gives the following, which I have formatted, rounded, and annotated:

SplineFunction[Cubic,
{0., 2.},  (* index range *)
{{-0.02475,    0.5 },  (* x[[1]], y[[1]] *)  
 { 0.3349375, -0.25},  (* x[[2]], y[[2]] *)
 { 1.101,      0   }}, (* x[[3]], y[[3]] *)
{{{-0.02475, 0.25809375, 0., 0.10159375},   (* coeffs for fitting x[[{1,2}]] *)
  { 0.5,    -1.,         0., 0.25      }},  (*   "     "     "    y[[{1,2}]] *)
 {{ 0.3349375, 0.562875, 0.30478125, -0.10159375}, (*  "     "    x[[{2,3}]] *)
  {-0.25,     -0.25,     0.75,       -0.25      }} (*  "     "    y[[{2,3}]] *) }]

There are two splines implicit in the above:

X[u_] = Piecewise[{{-.02475 + .25809375 u + .10159375 u^3, 0 ≤ u ≤ 1},
  {.3349375 + .562875 (u-1) + .30478125 (u-1)^2 - .10159375 (u-1)^3, 1 < u ≤ 2}}];

Y[u_] = Piecewise[{{.5 - u + .25 u^3, 0 ≤ u ≤ 1},
  {-.25 - .25 (u-1) + .75 (u-1)^2 - .25 (u-1)^3, 1 < u ≤ 2}}];

Note that u is shifted so that the argument of the polynomial is always in $[0,1]$.

In the following, changing spline[u] to {X[u], Y[u]} does not change the plot:

ParametricPlot[spline[u],{u, 0, 2}, Frame -> True, Axes -> None,  
               AspectRatio -> Automatic, Prolog -> {PointSize[.02],Point /@ dat}]