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


(*    Hermite spline and complete spline interpolation of f(x) = 
    x^(2k)        *)
(*        
  on [0,1] with 2k-4 interior knots                                    *)
(*  \
      generated as order statistics of 2k-4 i.i.d. uniform(0,
        1)            *)
(*        
  random variables                                                      *)
(* \
   with  the knot configuration                                              *)
\
(*    in the form                                                             \
  *)
(*    {hermite error, 
    complete error, {knot_1 , knot_2 , ... , knot_(2k-4)}}   *)
(*        
  Restriction:  k ³ 3                                          nn       *)

Print["k :  (2k-1) degree spline interpolation"]
k=7
Print["d = number of digits of accuracy"]
d=100


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)
hd[x_,k_,l_]:=((2*k)!/(2*k-l)!)*x^(2*k-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}]]

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}]]

HermiteInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k]
CompInterp[x_,coeff_List,vec_List,k_]:=coeff.Basis[x,vec,k]

mxlistH[coeff_List,vec_List, k_]:=
  Table[Abs[HermiteInterp[i/1000,coeff,vec,k]-hd[i/1000, k,0]],{i,1,1000}]
mxlistC[coeff_List,vec_List,k_]:=
  Table[Abs[CompInterp[i/1000,coeff,vec,k]-hd[i/1000, k,0]],{i,1,1000}]


Print["interior knots, tau_1, ..., tau_{2k-4}"]
VectauUnif=Sort[Table[Random[Real,{0,1},d],{j,1,2*k-4}]]
Print["all the knots"]
Vectau=Append[Prepend[VectauUnif,0],1]
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[hd[Vectau[[j]],k,0],{j,1,2*k-2}],
      Table[hd[Vectau[[j]],k,1],{j,1,2*k-2}]];
HermiteCoeff=LinearSolve[MMHermite,VecHermite];
Hermitemxl=mxlistH[HermiteCoeff,VectauUnif, k];
Print["Maximum Hermite Error (on the grid j/1000, j=1,...,1000)"]
Hmx=Max[Hermitemxl]
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}]];
MMComp= Join[MM1,MM2,MM3];
VecC=Join[Table[hd[0,k,l],{l,0,k-1}], 
      Table[hd[VectauUnif[[j]],k,0],{j,1,2*k-4}], 
      Table[hd[1,k,l],{l,0,k-1}]];
CompCoeff=LinearSolve[MMComp,VecC];
Compmxl=mxlistC[CompCoeff,VectauUnif,k];
Print["Maximum Complete Interpolation error (on the grid j/1000, \
j=1,...,1000)"]
Compmx= Max[Compmxl]
Print["Max Hermite Error, Max Complete Error, knots"]
mx={Hmx,Compmx,Vectau}
Print["Monomial f(x) = x^(2k) being interpolated"]
P1=Plot[hd[x,k,0],{x,0,1},PlotRange\[Rule]All]
Print["Hermite Interpolant"]
P2=Plot[HermiteInterp[x, HermiteCoeff,VectauUnif,k],{x,0,1},
    PlotRange\[Rule]All]
Print["Complete Interpolant"]
P3=Plot[CompInterp[x,CompCoeff,VectauUnif,k],{x,0,1},PlotRange\[Rule]All]
Print["Monomial with Hermite and Complete Interpolants"]
Show[P1,P2,P3]






k :  (2k-1) degree spline interpolation
7
d = number of digits of accuracy
100
interior knots, tau_1, ..., tau_{2k-4}
{0.301889585606863929575407021113245118977944220292751027558639262594461608505\
9291287252777354774404564,0.\
558176320549494814773654760104359917555317557463125235721823431410164912676846\
1255422989052227400179,0.\
580108979480201795781059306584367866588623357546633172756642750300795986015472\
9218654761066554111490,0.\
693962280520180387707833475167125928769041716334011897569475783544615899970069\
0884719766834568291792,0.\
723264673426688255782785595567284802807358199562013292614113846544499666653428\
7130957909882877790477,0.\
726059357083499453668509331838379603749038532291756664100241099115713841457093\
1990939725880314194399,0.\
760688790305925345782601413406529754958830862485329733661904958834345378512977\
5691567292487946966698,0.\
793143679321211067322944340410093281448474045883914972207370996990874329081496\
2002994939606156609815,0.\
856888264447652860870487607318551695066184664708077535120242012510854602720578\
2000221246109278918339,0.\
962451391978492863319947024284463709114261445221218991883280698724753701089466\
4335483981462956235189}
all the knots
{0,0.3018895856068639295754070211132451189779442202927510275586392625944616085\
059291287252777354774404564,0.\
558176320549494814773654760104359917555317557463125235721823431410164912676846\
1255422989052227400179,0.\
580108979480201795781059306584367866588623357546633172756642750300795986015472\
9218654761066554111490,0.\
693962280520180387707833475167125928769041716334011897569475783544615899970069\
0884719766834568291792,0.\
723264673426688255782785595567284802807358199562013292614113846544499666653428\
7130957909882877790477,0.\
726059357083499453668509331838379603749038532291756664100241099115713841457093\
1990939725880314194399,0.\
760688790305925345782601413406529754958830862485329733661904958834345378512977\
5691567292487946966698,0.\
793143679321211067322944340410093281448474045883914972207370996990874329081496\
2002994939606156609815,0.\
856888264447652860870487607318551695066184664708077535120242012510854602720578\
2000221246109278918339,0.\
962451391978492863319947024284463709114261445221218991883280698724753701089466\
4335483981462956235189,1}
Maximum Hermite Error (on the grid j/1000, j=1,...,1000)
\!\(3.805973298689647695325229611809273882681169702760921012271316984490827449\
4683046144931879624615527876996924618`94.66802268364482*^-6\)
Maximum Complete Interpolation error (on the grid j/1000, j=1,...,1000)
\!\(1.748933546012058558843454793599473214751521053093271302546940980970785288\
0274721663613880481626029185295018`90.88178998179872*^-9\)
Max Hermite Error, Max Complete Error, knots
\!\({3.80597329868964769532522961180927388268116970276092101227131698449082744\
94683046144931879624615527876996924618`94.66802268364482*^-6, 
    1.748933546012058558843454793599473214751521053093271302546940980970785288\
0274721663613880481626029185295018`90.88178998179872*^-9, {0, 
      0.3018895856068639295754070211132451189779442202927510275586392625944616\
085059291287252777354774404564378363827386901`100. , 
      0.5581763205494948147736547601043599175553175574631252357218234314101649\
126768461255422989052227400178658787531486931`100. , 
      0.5801089794802017957810593065843678665886233575466331727566427503007959\
860154729218654761066554111490296864052234714`100. , 
      0.6939622805201803877078334751671259287690417163340118975694757835446158\
999700690884719766834568291791983082253088895`100. , 
      0.7232646734266882557827855955672848028073581995620132926141138465444996\
666534287130957909882877790476614789736242949`100. , 
      0.7260593570834994536685093318383796037490385322917566641002410991157138\
414570931990939725880314194399206410224085672`100. , 
      0.7606887903059253457826014134065297549588308624853297336619049588343453\
785129775691567292487946966698006422939843724`100. , 
      0.7931436793212110673229443404100932814484740458839149722073709969908743\
290814962002994939606156609815341664202470987`100. , 
      0.8568882644476528608704876073185516950661846647080775351202420125108546\
027205782000221246109278918339275491690823125`100. , 
      0.9624513919784928633199470242844637091142614452212189918832806987247537\
010894664335483981462956235189416556918212521`100. , 1}}\)
Monomial f(x) = x^(2k) being interpolated
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