<<Statistics`DescriptiveStatistics`

(*    Hermite spline interpolation of f_0 (x) = x^(2*k)            *)
(*                          PART1:                                  *)
(*         on [0,1] with equidistant 2k-4 interior knots            *)
(*    Restriction:  k ³ 3                                           *)

Print["k: (2k-1) degree spline interpolation"]
k=4


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)
f0[x_,k_]:= x^(2*k)
f0p[x_,k_]:=(2*k)*x^(2*k-1)
    
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[Abs[HermiteInterp[i/1000,coeff,vec,k]-f0[i/1000, k]],{i,1,1000}]


Print["Equidistant interior knots "]
VectauEqui=Table[j/(2*k-3),{j,1,2*k-4}]
Print["All knots"]
Vectau=Append[Prepend[VectauEqui,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}]];
VecMS=Join[Table[f0[Vectau[[j]],k],{j,1,2*k-2}],
      Table[f0p[Vectau[[j]],k],{j,1,2*k-2}]];
HermiteCoeff=LinearSolve[MM,VecMS];
mtl=mxlist[HermiteCoeff,Vectau, k];
Print["Maximum interpolation error"]
mx=Max[mtl]
Print["Value given by the Bernoulli numbers formula"]
bernform= (2/(2*k-3)^(2*k))*Abs[(1-2^(-2*k))*BernoulliB[2*k]]
P1 = Plot[f0[x,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 function and interpolant"]
Show[P1,P2]
Print["Plot of the monospline of degree 2*k"]
P3=Plot[HermiteInterp[x,HermiteCoeff,Vectau,k]-f0[x,k],{x,0,1},  
    PlotRange\[Rule]All]




k: (2k-1) degree spline interpolation
4
Equidistant interior knots 
\!\({1\/5, 2\/5, 3\/5, 4\/5}\)
All knots
\!\({0, 1\/5, 2\/5, 3\/5, 4\/5, 1}\)
Maximum interpolation error
\!\(17\/100000000\)
Value given by the Bernoulli numbers formula
\!\(17\/100000000\)
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<<Statistics`DescriptiveStatistics`

(*    Hermite spline interpolation of f_0 (x) = t^(2*k)            *)
(*                          PART2:                                  *)
(*    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)
f0[x_,k_]:= x^(2*k)
f0p[x_,k_]:=(2*k)*x^(2*k-1)
    
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[Abs[HermiteInterp[i/1000,coeff,vec,k]-f0[i/1000, k]],{i,1,1000}]

Print["Interior knots "]
VectauUnif=Sort[Table[Random[Real,{0,1},d],{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}]];
VecMS=Join[Table[f0[Vectau[[j]],k],{j,1,2*k-2}],
      Table[f0p[Vectau[[j]],k],{j,1,2*k-2}]];
HermiteCoeff=LinearSolve[MM,VecMS];
mtl=mxlist[HermiteCoeff,Vectau, k];
Print["Maximum interpolation error"]
mx=Max[mtl]
Print["plot of function being interpolated, f_0 (x) = x^(2*k)"]
P1 = Plot[f0[x,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 function and interpolant"]
Show[P1,P2]
Print["Plot of the monospline (the interpolation error) of degree 2*k"]
P3=Plot[HermiteInterp[x,HermiteCoeff,Vectau,k]-f0[x,k],{x,0,1},  
    PlotRange\[Rule]All]


k: (2k-1) degree spline interpolation
4
d = number of digits of accuracy
30
Interior knots 
{0.142551544847808661549515683647,0.187172737044248655603114795266,0.\
580469768036344784852041833658,0.662918166918844316473713348601}
All knots
{0,0.142551544847808661549515683647,0.187172737044248655603114795266,0.\
580469768036344784852041833658,0.662918166918844316473713348601,1}
Maximum interpolation error
0.00006866843626044915689512
plot of function being interpolated, f_0 (x) = x^(2*k)
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