<<Statistics`DescriptiveStatistics`

(*    Hermite spline interpolation of the Perfect Spline of degree 2*
    k      *)
(*    on [0,
        1] with 2k-4 interior knots                                     *)
(*    generated as order statistics of 2k-4 i.i.d. uniform(0,
        1)            *)
(*    random variables                                                      *)
\

(*    Restriction:  
    k ³ 3                                                   *)

Print["k: (2k-1) degree spline interpolation"]
k=4
Print["d = number of digits of accuracy"]
d=30
f[x_,j_]:=x^j
g[x_,y_,k_]:=((x-y)*UnitStep[x-y])^(2*k-1)
fp[x_,j_]:=j*x^(j-1)
gp[x_,y_,k_]:=(2*k-1)*((x-y)*UnitStep[x-y])^(2*k-2)

PS[x_,vec_List, 
    k_]:=(1/Factorial[2*k])* (x^(2*k)+2*
          Sum[(-1)^(j)*(x-vec[[j]])^(2*k)*UnitStep[x-vec[[j]]], {j, 1,
              2*k-4}])
PSp[x_,vec_List,
    k_]:=(2*k)*(1/
        Factorial[2*k])*(x^(2*k-1) +2*
          Sum[(-1)^(j)*(x-vec[[j]])^(2*k-1)*UnitStep[x-vec[[j]]], {j, 1,
              2*k-4}])

Basis[x_,vec_List,k_]:=
  Join[{1},Table[f[x,j],{j,1,2*k-1}],Table[g[x,vec[[j]],k],{j,2,2*k-3}]]
HermiteInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k];
mxlist[coeff_List,vec_List, k_]:=
  Table[HermiteInterp[i/1000,coeff,vec,k]-PS[i/1000,vec, k],{i,1,1000}]

Print["Interior knots"]
VectauUnif=Sort[Table[Random[Real,{0,1},40],{j,1,2*k-4}]]
Print["All knots"]
Vectau=Append[Prepend[VectauUnif,0],1];
MM=Join[
      Table[Join[{1},
          Table[f[Vectau[[j]],i],{i,1,2*k-1}],
          Table[g[Vectau[[j]],Vectau[[i]],k],{i,2,2*k-3}]],{j,1,2*k-2}],
      Table[Join[{0,1},
          Table[fp[Vectau[[j]],i],{i,2,2*k-1}],
          Table[gp[Vectau[[j]],Vectau[[i]],k],{i,2,2*k-3}]],{j,1,2*k-2}]];
VecPS=Join[Table[PS[Vectau[[j]],VectauUnif,k],{j,1,2*k-2}],
      Table[PSp[Vectau[[j]],VectauUnif,k],{j,1,2*k-2}]];
HermiteCoeff=LinearSolve[MM,VecPS];
mtl=mxlist[HermiteCoeff,Vectau,k];
Print["Maximal interpolation error"]
mx=Max[mtl]
Print["plot of the perfect spline"] 
P1 = Plot[PS[x,VectauUnif,k],{x,0,1},PlotRange\[Rule]All]
Print["plot of interpolant"]
P2=Plot[HermiteInterp[x,HermiteCoeff,Vectau,k],{x,0,1},PlotRange\[Rule]All]
Print["plot of the perfect spline and its interpolant"]
Show[P1,P2]
Print["Plot of the interpolation error"]
P3=Plot[HermiteInterp[x,HermiteCoeff,Vectau,k]-PS[x,VectauUnif,k],{x,0,1},  
    PlotRange\[Rule]All]

k: (2k-1) degree spline interpolation
4
d = number of digits of accuracy
30
Interior knots
{0.01814868052623412443395642995926938274255,0.\
2920804577425194154029680452301527780015,0.\
2939811492489656920245906969397362858326,0.\
7580923225669031006663108607236009536262}
All knots
Maximal interpolation error
\!\(1.966232848228602952711444481194126654639`30.0365*^-12\)
plot of the perfect spline
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