<<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 isolation of  the knot configurations corresponding to the           *)
\
(*    10 largest Hermite interpolation errors                                 \
  *)
(*    
  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
Print["nn = number of monte-carlo replications"]
nn=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}]


Error[k_]:={
    VectauUnif=Sort[Table[Random[Real,{0,1},d],{j,1,2*k-4}]];
    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];
    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];
    Compmx= Max[Compmxl];
    mx={Hmx,Compmx,VectauUnif}
    
    }

Ert=Table[Error[k][[1]],{j,1,nn}];
Print["Block of hermite error, complete error, knots, 1st MC rep"]
Ert1=Ert[[1]];
Print["Transpose of Ert"]
ErtT = Transpose[Ert];
Print["Hermite errors"]
ErtT1=ErtT[[1]];
Print["sorted Hermite errors"]
Sort[ErtT1];
Print["Ranks of 10 largest Hermite errors"]
BigIO=Ordering[ErtT1,-10]
Print["block of hermite error, complete error, knots of largest 10 hermite \
errors"]
Ert[[BigIO]]

Print["all the hermite errors"]
ErtH=Table[Ert[[i,1]],{i,1,nn}];
Print["all the complete spline errors"]
ErtC=Table[Ert[[i,2]],{i,1,nn}];
Print["Statistics of Hermite Error and Complete Error"]
Print["Mean Hermite Error, Mean Complete Error"]
{Mean[ErtH],Mean[ErtC]}
Print["Median Hermite Error, Median Complete Error"]
{Median[ErtH],Median[ErtC]}
Print["StandardDev Hermite Error, StandardDev Complete Error"]
{StandardDeviation[ErtH],StandardDeviation[ErtC]}
Print["0.95-q Hermite error, 0.95-q Complete error"]
{Quantile[ErtH,0.95],Quantile[ErtC,0.95]}
Print["0.99-q Hermite error,  0.99-q Complete error"]
{Quantile[ErtH,0.99],Quantile[ErtC,0.99]}
Print["Max Hermite Error, Max Complete Error"]
{Max[ErtH],Max[ErtC]}



zzH=Sort[ErtH];
zzC=Sort[ErtC];
hh=Table[k,{k,1,nn}]*(1/nn);//N
JH=Table[{zzH[[i]],hh[[i]]},{i,1,nn}];
JC=Table[{zzC[[i]],hh[[i]]},{i,1,nn}];
Print["listplot, empirical distribution of hermite errors"]
ListPlot[JH,PlotRange\[Rule]All];
Print["listplot, empirical distribution of complete errors"]
ListPlot[JC,PlotRange\[Rule]All];
Delta[x_]:=0/;x<0
Delta[x_]:=1/;x³0
DEH[x_]:=Table[{Delta[zzH[[i]]-x]},{i,1,nn}]
DEC[x_]:=Table[{Delta[zzC[[i]]-x]},{i,1,nn}]
EDFH[nn_,x_]:=1.-Apply[Plus,DEH[x]]*(1/nn)//N
EDFC[nn_,x_]:=1.-Apply[Plus,DEC[x]]*(1/nn)//N

Print["empirical distribution function of Hermite Errors"]
P1=Plot[EDFH[nn,x],{x,0,Quantile[ErtH,0.99]},
    PlotStyle\[Rule]{Thickness[1/100],RGBColor[0.000,1.000,0.196],
        Dashing[{}]},PlotRange \[Rule] All ]
Print["empirical distribution function of Complete Errors"]
P2=Plot[EDFC[nn,x],{x,0,Quantile[ErtH,0.99]},
    PlotStyle\[Rule]{Thickness[1/100],RGBColor[0.50,0.000,0.50],Dashing[{}]},
    PlotRange \[Rule] All ]
Print["empirical distribution functions of both errors compared"]
Show[P1,P2]

k :  (2k-1) degree spline interpolation
7
d = number of digits of accuracy
100
nn = number of monte-carlo replications
100
\!\(\*
  RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(mxlistC\\)\\\" is similar to existing symbol \
\\\"\\!\\(mxlistH\\)\\\". \\!\\(\\*ButtonBox[\\\"MoreÉ\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]\)
\!\(\*
  RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(Compmx\\)\\\" is similar to existing symbol \
\\\"\\!\\(Compmxl\\)\\\". \\!\\(\\*ButtonBox[\\\"MoreÉ\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]\)
Block of hermite error, complete error, knots, 1st MC rep
Transpose of Ert
\!\(\*
  RowBox[{\(General::"spell"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(ErtT\\)\\\" is similar to existing symbols \
\\!\\({Ert, ErtC, ErtH}\\). \\!\\(\\*ButtonBox[\\\"MoreÉ\\\", ButtonStyle->\\\
\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell\\\"]\\)\"\>"}]\)
Hermite errors
\!\(\*
  RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(ErtT1\\)\\\" is similar to existing symbol \
\\\"\\!\\(Ert1\\)\\\". \\!\\(\\*ButtonBox[\\\"MoreÉ\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]\)
sorted Hermite errors
Ranks of 10 largest Hermite errors
{21,25,100,2,83,82,6,19,80,44}
block of hermite error, complete error, knots of largest 10 hermite errors
\!\({{2.6691506153873373493300668096825667355849652962119606244347583532156403\
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48638037376840349572091173010300267124693452577`100. }}}\)
all the hermite errors
all the complete spline errors
Statistics of Hermite Error and Complete Error
Mean Hermite Error, Mean Complete Error
\!\({0.00015682392311105858722948743971601465892032162484300976073092298073422\
1010019340366738285798513`76.5917720810194, 
    1.123153434407697624727732383271322435316194840974536678828376301266066550\
4777964039277185673493496`84.02365832649099*^-8}\)
Median Hermite Error, Median Complete Error
\!\({1.09672495938046659154003337123140237227120156272342685576566809742222081\
218312374028458951`70.79646621470899*^-7, 
    1.856986113709016100294091069090375834437193603123909762706329069298930366\
36553654263025196055261`81.49832867477843*^-9}\)
StandardDev Hermite Error, StandardDev Complete Error
\!\({0.00150058240358915821732512018288073633046194641613655440620706283580662\
8802761399688665982922186`78.08516197758911, 
    3.719801260517986581228234242368709703475613731188021354603030333103548732\
2345004579232094615173335`84.68940863557822*^-8}\)
0.95-q Hermite error, 0.95-q Complete error
\!\({0.00001008381331912856068034748743291854479483919162815121887009256403130\
865832803222996`70.31934495998934, 
    4.849796536986598693393782199495947585327375893965583323191434958480662059\
232473819133619596123918`81.02120742005879*^-8}\)
0.99-q Hermite error,  0.99-q Complete error
\!\({0.00024629520896775433775339168719715842586163601586430944579693165432977\
025517638167085`70.31934495998934, 
    9.040078450256860574978571809755499862994280852342857258920790727375989719\
1340198934959909582302288`81.02120742005879*^-8}\)
Max Hermite Error, Max Complete Error
\!\({0.01500860545622812255002059775644682494444967235737696904271205226902257\
69556350172801382729945561344903641663171669`95.00107683971821, 
    3.427902888377069175959605491151795080730669733411618927418656579852803526\
19998864438125696176072968`88.62207559199226*^-7}\)
listplot, empirical distribution of hermite errors
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