<<Statistics`DescriptiveStatistics`
<<LinearAlgebra`MatrixManipulation`



(*    Complete and Hermite spline interpolation of Shadrin's null spline phi  \
            *)
(*    on [0,
        1] with 2k-4 interior knots                                           \
        *)
(*    Single computation (no Monte-
        carlo) with plots                                       *)
(*    Part 1 (here):  knots equi-spaced at j/(2k-3), j=1,..., 
  2k-4                         *)
(*    Part 2 (next part: 
          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=3


B1[x_,j_,l_]:=((j!)/(j-l)!)*x^(j-l)
B2[x_,y_,k_,l_]:=((2*k-1)!/(2*k-1-l)!)*((x-y)*UnitStep[x-y])^(2*k-1-l)
Basis[x_,vec_List,k_]:=
  Join[{1},Table[B1[x,j,0],{j,1,2*k-1}],
    Table[B2[x,vec[[j]],k,0],{j,1,2*k-4}]]
BasisNull[x_,vec_List,k_]:=
  Join[{1},Table[B1[x,j,0],{j,1,2*k-2}],
    Table[B2[x,vec[[j]],k,1]/(2*k-1),{j,1,2*k-4}]]
BasisNullDer[x_,l_,vec_List,k_]:=
  Join[{l!},Table[B1[x,j,l],{j,l+1,2*k-2}],
    Table[B2[x,vec[[j]],k,l+1]/(2*k-1),{j,1,2*k-4}]]

CompInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k]
NullSpline[x_,coeff_List,vec_List,k_]:=coeff.BasisNull[x,vec,k]
NullSplDer[x_,l_,k_,vec_List,coeff_List]:=
  coeff[[Table[j,{j,l+1,4*k-5}]]].BasisNullDer[x,l,vec,k]
HermiteInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k]
Vect[k_,l_,vec_List]:= 
  Join[Table[0,{m,1,l}],Table[B1[1,i,l],{i,l,2*k-1}],
    Table[B2[1,vec[[i]],k,l],{i,1,2*k-4}]]
VectNull[k_,l_,vec_List]:=
  Join[Table[0,{m,1,l}],Table[B1[1,i,l],{i,l,2*k-2}],
    Table[B2[1,vec[[i]],k,l+1]/(2*k-1),{i,1,2*k-4}]]

mxlistComp[coeff1_List,coeff2_List,vec_List,k_]:=
  Table[Abs[
      CompInterp[i/100,coeff1,vec,k]-NullSplDer[i/100,k-1,k,vec,coeff2]],{i,1,
      100}]
mxlistHermite[coeff1_List,coeff2_List,vec_List,k_]:=
  Table[Abs[
      HermiteInterp[i/100,coeff1,vec,k]-
        NullSplDer[i/100,k-1,k,vec,coeff2]],{i,1,100}]

Print["interior knots"]  
VectauUnif=Sort[Table[j/(2*k-3),{j,1,2*k-4}]]
Print["all knots"] 
Vectau=Append[Prepend[VectauUnif,0],1]
MM1= AppendRows[DiagonalMatrix[Table[j!,{j,0,(k-1)}]],
    Table[0,{i,1,k},{j,1,3*k-4}]]
MM2= Table[
    Join[{1},Table[B1[VectauUnif[[j]],i,0],{i,1,2*k-1}],
      Table[B2[VectauUnif[[j]],VectauUnif[[i]],k,0],{i,1,2*k-4}]],{j,1,
      2*k-4}]
Vec3= Join[Table[1,{m,1,2*k}], Table[B2[1,VectauUnif[[i]],k,0],{i,1,2*k-4}]]
MM3= Join[{Vec3},Table[Vect[k,l,VectauUnif],{l,1,k-1}]]
 MM= Join[MM1,MM2,MM3]


MM1Null= AppendRows[DiagonalMatrix[Table[j!,{j,0,(k-2)}]],
    Table[0,{i,1,k-1},{j,1,3*k-4}]]
MM2Null= Table[
    Join[{1},Table[B1[VectauUnif[[j]],i,0],{i,1,2*k-2}],
      Table[B2[VectauUnif[[j]],VectauUnif[[i]],k,1]/(2*k-1),{i,1,2*k-4}]],{j,
      1,2*k-4}]
Vec3Null= 
  Join[Table[1,{m,1,2*k-1}], 
    Table[B2[1,VectauUnif[[i]],k,1]/(2*k-1),{i,1,2*k-4}]]
MM3Null= Join[{Vec3Null},Table[VectNull[k,l,VectauUnif],{l,1,k-1}]]
MMNull= Join[MM1Null,MM2Null,MM3Null]
 VecCNull=Join[Table[0,{l,1,4*k-6}],{(k-1)!}]

NullSplCoeff=LinearSolve[MMNull,VecCNull]
VecPhi=Join[
    Table[NullSplDer[0,l,k,VectauUnif,NullSplCoeff],{l,k-1,
        2*k-3}], {NullSplCoeff[[2*k-1]]*(2*k-2)!},
    Table[NullSplDer[VectauUnif[[j]], k-1,k,VectauUnif, NullSplCoeff],{j,1,
        2*k-4}], 
    Table[NullSplDer[1,l,k,VectauUnif,NullSplCoeff],{l,k-1,2*k-2}]]
 CpltSplineCoeff=LinearSolve[MM,VecPhi]

MMHermite=Join[
      Table[Join[{1},
          Table[B1[Vectau[[j]],i,0],{i,1,2*k-1}],
          Table[B2[Vectau[[j]],Vectau[[i]],k,0],{i,2,2*k-3}]],{j,1,2*k-2}],
      Table[Join[{0,1},
          Table[B1[Vectau[[j]],i,1],{i,2,2*k-1}],
          Table[B2[Vectau[[j]],Vectau[[i]],k,1],{i,2,2*k-3}]],{j,1,2*k-2}]];
VecHermite=
    Join[Table[
        NullSplDer[Vectau[[j]],k-1,k,VectauUnif, NullSplCoeff],{j,1,2*k-2}],
      Table[
        NullSplDer[Vectau[[j]], k,k,VectauUnif, NullSplCoeff],{j,1,2*k-2}]];
HermiteCoeff=LinearSolve[MMHermite,VecHermite];

Print["Shadrin's null spline Sigma"] 
P1=Plot[NullSpline[x,NullSplCoeff,VectauUnif,k],{x,0,1}]
Print["k-1 st derivative of Sigma, Shadrin's Phi"]
P2=Plot[NullSplDer[x,k-1,k,VectauUnif,NullSplCoeff],{x,0,1}]
Print["CompleteSplineInterpolant of Phi, Phi = k-1 th derivative of Sigma"]
P3 =Plot[CompInterp[x,CpltSplineCoeff,VectauUnif,k],{x,0,1}]
Print["Phi and its complete spline interpolant"]
Show[P2,P3]
Print["HermiteSplineInterpolant of Phi, Phi = k-1 th derivative of Sigma"]
P4= Plot[HermiteInterp[x,HermiteCoeff,VectauUnif,k],{x,0,1}]
Print["Phi and its Hermite spline interpolant"]
Show[P2,P4]
Print["supremum norm of the error of the complete spline interpolant"]
mc=Max[mxlistComp[CpltSplineCoeff,NullSplCoeff,VectauUnif,k]]
N[mc]
Print["supremum norm of the error of the Hermite spline interpolant"]
mh=Max[mxlistHermite[HermiteCoeff,NullSplCoeff,VectauUnif,k]]
N[mh]






	



k (2k-1) degree spline interpolation
3
interior knots
\!\({1\/3, 2\/3}\)
all knots
\!\({0, 1\/3, 2\/3, 1}\)
{{1,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,0},{0,0,2,0,0,0,0,0}}
\!\({{1, 1\/3, 1\/9, 1\/27, 1\/81, 1\/243, 0, 0}, {1, 2\/3, 4\/9, 8\/27, 
      16\/81, 32\/243, 1\/243, 0}}\)
\!\({1, 1, 1, 1, 1, 1, 32\/243, 1\/243}\)
\!\({{1, 1, 1, 1, 1, 1, 32\/243, 1\/243}, {0, 1, 2, 3, 4, 5, 80\/81, 
      5\/81}, {0, 0, 2, 6, 12, 20, 160\/27, 20\/27}}\)
\!\({{1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 2, 0, 0, 0, 
      0, 0}, {1, 1\/3, 1\/9, 1\/27, 1\/81, 1\/243, 0, 0}, {1, 2\/3, 4\/9, 
      8\/27, 16\/81, 32\/243, 1\/243, 0}, {1, 1, 1, 1, 1, 1, 32\/243, 
      1\/243}, {0, 1, 2, 3, 4, 5, 80\/81, 5\/81}, {0, 0, 2, 6, 12, 20, 
      160\/27, 20\/27}}\)
{{1,0,0,0,0,0,0},{0,1,0,0,0,0,0}}
\!\({{1, 1\/3, 1\/9, 1\/27, 1\/81, 0, 0}, {1, 2\/3, 4\/9, 8\/27, 16\/81, 
      1\/81, 0}}\)
\!\({1, 1, 1, 1, 1, 16\/81, 1\/81}\)
\!\({{1, 1, 1, 1, 1, 16\/81, 1\/81}, {0, 1, 2, 3, 4, 32\/27, 4\/27}, {0, 0, 
      2, 6, 12, 16\/3, 4\/3}}\)
\!\({{1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0}, {1, 1\/3, 1\/9, 1\/27, 
      1\/81, 0, 0}, {1, 2\/3, 4\/9, 8\/27, 16\/81, 1\/81, 0}, {1, 1, 1, 1, 1, 
      16\/81, 1\/81}, {0, 1, 2, 3, 4, 32\/27, 4\/27}, {0, 0, 2, 6, 12, 16\/3, 
      4\/3}}\)
{0,0,0,0,0,0,2}
\!\({0, 0, 1, \(-\(54\/11\)\), 63\/11, \(-\(108\/11\)\), 108\/11}\)
\!\({2, \(-\(324\/11\)\), 1512\/11, \(-\(2\/11\)\), \(-\(2\/11\)\), 2, 
    324\/11, 1512\/11}\)
\!\({2, \(-\(324\/11\)\), 756\/11, 54432\/407, \(-\(221616\/407\)\), 
    174960\/407, \(-\(174960\/407\)\), \(-\(174960\/407\)\)}\)
Shadrin's null spline Sigma
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k-1 st derivative of Sigma, Shadrin's Phi
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supremum norm of the error of the complete spline interpolant
\!\(270\/407\)
0.663391
supremum norm of the error of the Hermite spline interpolant
\!\(77534496\/107421875\)
0.721776
<<Statistics`DescriptiveStatistics`
<<LinearAlgebra`MatrixManipulation`
(*    Complete and Hermite spline interpolation of Shadrin's null spline phi  \
            *)
(*    on [0,
        1] with 2k-4 interior knots                                           \
         *)
(*    Part 1 (previous part):  knots equi-spaced at j/(2k-3), j=1,..., 
  2k-4                *)
(*    Part 2 (here):    
    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=3
Print["precision of random numbers"]
d=30


B1[x_,j_,l_]:=((j!)/(j-l)!)*x^(j-l)
B2[x_,y_,k_,l_]:=((2*k-1)!/(2*k-1-l)!)*((x-y)*UnitStep[x-y])^(2*k-1-l)
Basis[x_,vec_List,k_]:=
  Join[{1},Table[B1[x,j,0],{j,1,2*k-1}],
    Table[B2[x,vec[[j]],k,0],{j,1,2*k-4}]]
BasisNull[x_,vec_List,k_]:=
  Join[{1},Table[B1[x,j,0],{j,1,2*k-2}],
    Table[B2[x,vec[[j]],k,1]/(2*k-1),{j,1,2*k-4}]]
BasisNullDer[x_,l_,vec_List,k_]:=
  Join[{l!},Table[B1[x,j,l],{j,l+1,2*k-2}],
    Table[B2[x,vec[[j]],k,l+1]/(2*k-1),{j,1,2*k-4}]]

CompInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k]
NullSpline[x_,coeff_List,vec_List,k_]:=coeff.BasisNull[x,vec,k]
NullSplDer[x_,l_,k_,vec_List,coeff_List]:=
  coeff[[Table[j,{j,l+1,4*k-5}]]].BasisNullDer[x,l,vec,k]
HermiteInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k]
Vect[k_,l_,vec_List]:= 
  Join[Table[0,{m,1,l}],Table[B1[1,i,l],{i,l,2*k-1}],
    Table[B2[1,vec[[i]],k,l],{i,1,2*k-4}]]
VectNull[k_,l_,vec_List]:=
  Join[Table[0,{m,1,l}],Table[B1[1,i,l],{i,l,2*k-2}],
    Table[B2[1,vec[[i]],k,l+1]/(2*k-1),{i,1,2*k-4}]]

mxlistComp[coeff1_List,coeff2_List,vec_List,k_]:=
  Table[Abs[
      CompInterp[i/100,coeff1,vec,k]-NullSplDer[i/100,k-1,k,vec,coeff2]],{i,1,
      100}]
mxlistHermite[coeff1_List,coeff2_List,vec_List,k_]:=
  Table[Abs[
      HermiteInterp[i/100,coeff1,vec,k]-
        NullSplDer[i/100,k-1,k,vec,coeff2]],{i,1,100}]

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]
MM1= AppendRows[DiagonalMatrix[Table[j!,{j,0,(k-1)}]],
    Table[0,{i,1,k},{j,1,3*k-4}]]
MM2= Table[
    Join[{1},Table[B1[VectauUnif[[j]],i,0],{i,1,2*k-1}],
      Table[B2[VectauUnif[[j]],VectauUnif[[i]],k,0],{i,1,2*k-4}]],{j,1,
      2*k-4}]
Vec3= Join[Table[1,{m,1,2*k}], Table[B2[1,VectauUnif[[i]],k,0],{i,1,2*k-4}]]
MM3= Join[{Vec3},Table[Vect[k,l,VectauUnif],{l,1,k-1}]]
 MM= Join[MM1,MM2,MM3]


MM1Null= AppendRows[DiagonalMatrix[Table[j!,{j,0,(k-2)}]],
    Table[0,{i,1,k-1},{j,1,3*k-4}]]
MM2Null= Table[
    Join[{1},Table[B1[VectauUnif[[j]],i,0],{i,1,2*k-2}],
      Table[B2[VectauUnif[[j]],VectauUnif[[i]],k,1]/(2*k-1),{i,1,2*k-4}]],{j,
      1,2*k-4}]
Vec3Null= 
  Join[Table[1,{m,1,2*k-1}], 
    Table[B2[1,VectauUnif[[i]],k,1]/(2*k-1),{i,1,2*k-4}]]
MM3Null= Join[{Vec3Null},Table[VectNull[k,l,VectauUnif],{l,1,k-1}]]
MMNull= Join[MM1Null,MM2Null,MM3Null]
 VecCNull=Join[Table[0,{l,1,4*k-6}],{(k-1)!}]

NullSplCoeff=LinearSolve[MMNull,VecCNull]
VecPhi=Join[
    Table[NullSplDer[0,l,k,VectauUnif,NullSplCoeff],{l,k-1,
        2*k-3}], {NullSplCoeff[[2*k-1]]*(2*k-2)!},
    Table[NullSplDer[VectauUnif[[j]], k-1,k,VectauUnif, NullSplCoeff],{j,1,
        2*k-4}], 
    Table[NullSplDer[1,l,k,VectauUnif,NullSplCoeff],{l,k-1,2*k-2}]]
 CpltSplineCoeff=LinearSolve[MM,VecPhi]

MMHermite=Join[
      Table[Join[{1},
          Table[B1[Vectau[[j]],i,0],{i,1,2*k-1}],
          Table[B2[Vectau[[j]],Vectau[[i]],k,0],{i,2,2*k-3}]],{j,1,2*k-2}],
      Table[Join[{0,1},
          Table[B1[Vectau[[j]],i,1],{i,2,2*k-1}],
          Table[B2[Vectau[[j]],Vectau[[i]],k,1],{i,2,2*k-3}]],{j,1,2*k-2}]];
VecHermite=
    Join[Table[
        NullSplDer[Vectau[[j]],k-1,k,VectauUnif, NullSplCoeff],{j,1,2*k-2}],
      Table[
        NullSplDer[Vectau[[j]], k,k,VectauUnif, NullSplCoeff],{j,1,2*k-2}]];
HermiteCoeff=LinearSolve[MMHermite,VecHermite];


Print["Shadrin's null spline Sigma"] 
P1=Plot[NullSpline[x,NullSplCoeff,VectauUnif,k],{x,0,1},PlotRange\[Rule]All]
Print["k-1 st derivative of Sigma, Shadrin's Phi"]
P2=Plot[NullSplDer[x,k-1,k,VectauUnif,NullSplCoeff],{x,0,1},
    PlotRange\[Rule]All]
Print["CompleteSplineInterpolant of Phi, Phi = k-1 th derivative of Sigma"]
P3 =Plot[CompInterp[x,CpltSplineCoeff,VectauUnif,k],{x,0,1},
    PlotRange\[Rule]All]
Print["Phi and its complete spline interpolant"]
Show[P2,P3]
Print["HermiteSplineInterpolant of Phi, Phi = k-1 th derivative of Sigma"]
P4= Plot[HermiteInterp[x,HermiteCoeff,VectauUnif,k],{x,0,1},
    PlotRange\[Rule]All]
Print["Phi and its Hermite spline interpolant"]
Show[P2,P4]
Print["supremum norm of the error of the complete spline interpolant"]
mc=Max[mxlistComp[CpltSplineCoeff,NullSplCoeff,VectauUnif,k]]
N[mc]
Print["supremum norm of the error of the Hermite spline interpolant"]
mh=Max[mxlistHermite[HermiteCoeff,NullSplCoeff,VectauUnif,k]]
N[mh]






	



k (2k-1) degree spline interpolation
3
precision of random numbers
30
interior knots
{0.0758677944819060097910530564033,0.626191290233080060925986772156}
all knots
{0,0.0758677944819060097910530564033,0.626191290233080060925986772156,1}
{{1,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,0},{0,0,2,0,0,0,0,0}}
\!\({{1, 0.075867794481906009791053056403256503677011994152`30. , 
      0.005755922239548727988612594925279814963110987641`29.69897000433602, 
      0.000436689125523915067130604043629965875616574003`29.522878745280337, 
      0.000033130640827731644386935050177320084593941875`29.397940008672037, 
      2.513548649372188806741952635123338648162968`29.301029995663978*^-6, 
      0``154.0945627263275, 0}, {1, 
      0.62619129023308006092598677215569658906222731115`30. , 
      0.392115531963769508103370817559485161590915703332`29.69897000433602, 
      0.245539330880823373622056336126077620037235676463`29.522878745280337, 
      0.153754590407229946791975766784296644420096507339`29.397940008672037, 
      0.096279785346362075008065964358780475923420356973`29.301029995663978, 
      0.050476621128449156134528643758816521016388070332`29.195274387294656, 
      0``149.51131490832591}}\)
{1,1,1,1,1,1,0.674016498386207366292682560098,0.\
0072987263538486991350238231183}
{{1,1,1,1,1,1,0.674016498386207366292682560098,
    0.0072987263538486991350238231183},{0,1,2,3,4,5,
    3.64675364824200200071663931407,0.097626488671166286100141195071},{0,0,2,
    6,12,20,15.78455388294808479051096023045,1.04466788622489935820191600171}}
\!\({{1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 2, 0, 0, 0, 
      0, 0}, {1, 0.075867794481906009791053056403256503677011994152`30. , 
      0.005755922239548727988612594925279814963110987641`29.69897000433602, 
      0.000436689125523915067130604043629965875616574003`29.522878745280337, 
      0.000033130640827731644386935050177320084593941875`29.397940008672037, 
      2.513548649372188806741952635123338648162968`29.301029995663978*^-6, 
      0``154.0945627263275, 0}, {1, 
      0.62619129023308006092598677215569658906222731115`30. , 
      0.392115531963769508103370817559485161590915703332`29.69897000433602, 
      0.245539330880823373622056336126077620037235676463`29.522878745280337, 
      0.153754590407229946791975766784296644420096507339`29.397940008672037, 
      0.096279785346362075008065964358780475923420356973`29.301029995663978, 
      0.050476621128449156134528643758816521016388070332`29.195274387294656, 
      0``149.51131490832591}, {1, 1, 1, 1, 1, 1, 
      0.674016498386207366292682560098467449574205712035`30.38670664203417, 
      0.007298726353848699135023823118295216402725892671`29.07697238924313}, \
{0, 1, 2, 3, 4, 5, 
      3.646753648242002000716639314071551658263727975517`30.48361665504223, 
      0.097626488671166286100141195070553887838693884418`29.17388240225119}, \
{0, 0, 2, 6, 12, 20, 
      15.784553882948084790510960230448799359653608129279`30.60855539165053, 
      1.0446678862248993582019160017057619509765900017`29.298821138859488}}\)
{{1,0,0,0,0,0,0},{0,1,0,0,0,0,0}}
\!\({{1, 0.075867794481906009791053056403256503677011994152`30. , 
      0.005755922239548727988612594925279814963110987641`29.69897000433602, 
      0.000436689125523915067130604043629965875616574003`29.522878745280337, 
      0.000033130640827731644386935050177320084593941875`29.397940008672037, 
      0``123.27565018106199, 0}, {1, 
      0.62619129023308006092598677215569658906222731115`30. , 
      0.392115531963769508103370817559485161590915703332`29.69897000433602, 
      0.245539330880823373622056336126077620037235676463`29.522878745280337, 
      0.153754590407229946791975766784296644420096507339`29.397940008672037, 
      0.091721726435739719436663146945546372166537618874`29.292184400302713, 
      0``119.60905192666071}}\)
{1,1,1,1,1,0.729350729648400400143327862814,0.019525297734233257220028239014}
{{1,1,1,1,1,0.729350729648400400143327862814,
    0.019525297734233257220028239014},{0,1,2,3,4,
    3.156910776589616958102192046090,0.20893357724497987164038320034},{0,0,2,
    6,12,10.24824399930884050087807778543,1.67679541797131263501676727898}}
\!\({{1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0}, {1, 
      0.075867794481906009791053056403256503677011994152`30. , 
      0.005755922239548727988612594925279814963110987641`29.69897000433602, 
      0.000436689125523915067130604043629965875616574003`29.522878745280337, 
      0.000033130640827731644386935050177320084593941875`29.397940008672037, 
      0``123.27565018106199, 0}, {1, 
      0.62619129023308006092598677215569658906222731115`30. , 
      0.392115531963769508103370817559485161590915703332`29.69897000433602, 
      0.245539330880823373622056336126077620037235676463`29.522878745280337, 
      0.153754590407229946791975766784296644420096507339`29.397940008672037, 
      0.091721726435739719436663146945546372166537618874`29.292184400302713, 
      0``119.60905192666071}, {1, 1, 1, 1, 1, 
      0.729350729648400400143327862814310331652745595103`30.48361665504223, 
      0.019525297734233257220028239014110777567738776884`29.17388240225119}, \
{0, 1, 2, 3, 4, 
      3.156910776589616958102192046089759871930721625856`30.60855539165053, 
      0.20893357724497987164038320034115239019531800034`29.298821138859488}, \
{0, 0, 2, 6, 12, 
      10.248243999308840500878077785425201691309043992041`30.784646650706215, 
      1.676795417971312635016767278977103801597568591377`29.474912397915173}}\
\)
{0,0,0,0,0,0,2}
{0,0,1.00000000000000000000000000000,-18.510817806765205480956376977,70.\
253715082740988929184684515,-72.488963729057641855424665155,6.\
5034250948299644035767369845}
{2.0000000000000000000000000000,-111.064906840591232885738261864,1686.\
0891619857837343004324284,-1.573750473665987421854429755,-0.\
42157064401990456919312774,2.0000000000000000000000000,25.\
6238795558094696050155982,102.4362347643194754560821523}
{2.0000000000000000000000000,-111.06490684059123288573826,843.\
0445809928918671502162,1982.6337035623478546648857,-33596.\
583232568994125362036,98379.53760557439053593685,-100176.89133539520059028389,\
3193.6758892643668939823629}
Shadrin's null spline Sigma
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supremum norm of the error of the complete spline interpolant
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supremum norm of the error of the Hermite spline interpolant
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8.82131