
<<Statistics`DescriptiveStatistics`

(*    Hermite spline interpolation of f_t (x) = (x-t)_+^(k-1)       *)
(*    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)
h[x_,t_,k_]:=((x-t)*UnitStep[x-t])^(k-1)/(k-1)!
hp[x_,t_,k_]:=((x-t)*UnitStep[x-t])^(k-2)/(k-2)!
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,t_, k_]:=
  Table[Abs[HermiteInterp[i/1000,coeff,vec,k]-h[i/1000, t, k]],{i,1,1000}]

Print["Evaluation Point t"]
tt=Random[Real,{0,1},d]
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]
Print["Matrix MM"]
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}]]
Print["h at knot points"]
VecH=Join[Table[h[Vectau[[j]],tt,k],{j,1,2*k-2}],
    Table[hp[Vectau[[j]],tt,k],{j,1,2*k-2}]]

Print["hermite coefficients"]
HermiteCoeff=LinearSolve[MM,VecH]

(*
  Print["interpolation error at i/1000, i=1,..., 1000"]
  *)
mtl=mxlist[HermiteCoeff,Vectau,tt, k];
Print["Maximum interpolation error"]
mx=Max[mtl]
Print["plot of function being interpolated, f_t (x) = (x-t)_+^(k-1)"]
P1 = Plot[h[x,tt,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 interpolation error"]
P3=Plot[HermiteInterp[x,HermiteCoeff,Vectau,k]- h[x,tt,k],{x,0,1}, 
    PlotRange\[Rule]All]



k (2k-1) degree spline interpolation
4
d = number of digits of accuracy
30
Evaluation Point t
0.897606106257365950568248956730
interior knots
{0.0251072804778450610126005848880,0.673932144862751332975025890657,0.\
782360241751219600228004295561,0.796535594827871630777090041385}
all knots
{0,0.0251072804778450610126005848880,0.673932144862751332975025890657,0.\
782360241751219600228004295561,0.796535594827871630777090041385,1}
Matrix MM
\!\({{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 
      0.025107280477845061012600584887994932197870032716`30. , 
      0.000630375532993179715256795160029290311583884457`29.69897000433602, 
      0.000015827015313230836225249202021141725071588955`29.522878745280337, 
      3.97373312596435407739627478034546744262758`29.397940008672037*^-7, 
      9.9769632137691055864558761382045093721`29.301029995663978*^-9, 
      2.50494413725243384912681760622168565235256383546`29.221848749616356*^-\
10, 6.289233503533047143005015142394108076576667198`29.15490195998574*^-12, 
      0``219.09419231105474, 0, 0, 0}, {1, 
      0.673932144862751332975025890657033222988055040588`30. , 
      0.454184535879308447885774056442191934043499063841`29.69897000433602, 
      0.306089558428635581267488813228341554640447605852`29.522878745280337, 
      0.206283592631902822822327392403810425599338532832`29.397940008672037, 
      0.139021144032412316698354620351005896962535021785`29.301029995663978, 
      0.093690817779037115410205922750533245875666514277`29.221848749616356, 
      0.063141253779771679370360043049749600549194822657`29.15490195998574, 
      0.048405241636510258316014210327842920454081416123`29.12252777427503, 
      0``209.09247682903555, 0, 0}, {1, 
      0.782360241751219600228004295561151891764378379049`30. , 
      0.61208754787302677652094107901523864002821652583`29.69897000433602, 
      0.47887296192685242929609122014705519919753190769`29.522878745280337, 
      0.374651166261214845974215790894486387257623825515`29.397940008672037, 
      0.29311217700850041519287073330158025975462963279`29.301029995663978, 
      0.22931931366459665631318012232791213162574756485`29.221848749616356, 
      0.179410313676857596336652394868462656311710018702`29.15490195998574, 
      0.142786406006601877515542516764216008861451194081`29.127017877060226, 
      1.76194702314013743143654669885293353628689`28.02679521610316*^-7, 
      0``208.6389426248425, 0}, {1, 
      0.796535594827871630777090041384972475149248952585`30. , 
      0.634468953827791279038927632457645765373472679635`29.69897000433602, 
      0.505377105537037147640139781907924917626337872243`29.522878745280337, 
      0.402550853371331941913956924069246401525410093508`29.397940008672037, 
      0.320646083438601222374841004733167243412728735779`29.301029995663978, 
      0.255406018800993583192071747497374236578571580537`29.221848749616356, 
      0.203439985108267988873147962540235203280306729639`29.15490195998574, 
      0.162580713649398005193492570497052724282135493404`29.12751444135239, 
      4.16407782229751885604035144931348101625799`28.075949150002526*^-7, 
      1.15010421500575295183294350746987956744332225`27.108082365460206*^-13, 
      0``208.5843537149747}, {1, 1, 1, 1, 1, 1, 1, 1, 
      0.836946677608863316325529100708010226834803962143`30.744059112901006, 
      0.00039188253340100703105861066106782192551282064`28.83959377507869, 
      0.000023129574519967883415628645776590566416159629`28.59923342259594, 
      0.000014435076894667561143138106878024368102231326`28.562185215171805}, \
{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 
      0.050214560955690122025201169775989864395740065431`30. , 
      0.001891126598979539145770385480087870934751653372`29.69897000433602, 
      0.000063308061252923344900996808084566900286355821`29.522878745280337, 
      1.986866562982177038698137390172733721313788`29.397940008672037*^-6, 
      5.9861779282614633518735256829227056232597`29.301029995663978*^-8, 
      1.753460896076703694388772324355179956647`29.221848749616356*^-9, 
      0``186.94992394088982, 0, 0, 0}, {0, 1, 
      1.347864289725502665950051781314066445976110081176`30.000000000000004, 
      1.362553607637925343657322169326575802130497191522`29.698970004336015, 
      1.224358233714542325069955252913366218561790423409`29.522878745280337, 
      1.031417963159514114111636962019052127996692664161`29.39794000867204, 
      0.834126864194473900190127722106035381775210130712`29.301029995663978, 
      0.655835724453259807871441459253732721129665599937`29.221848749616356, 
      0.522231360194238305666825822905275982245195201715`29.189474563905648, 
      0``178.37702495630194, 0, 0}, {0, 1, 
      1.564720483502439200456008591122303783528756758097`30. , 
      1.836262643619080329562823237045715920084649577491`29.698970004336015, 
      1.915491847707409717184364880588220796790127630762`29.52287874528034, 
      1.873255831306074229871078954472431936288119127575`29.397940008672037, 
      1.75867306205100249115722439980948155852777779674`29.301029995663978, 
      1.605235195652176594192260856295384921380232953952`29.221848749616356, 
      1.319908792915744946871321807763530160165406221218`29.193964666690842, 
      0.000011374938337862397278058856059774842541844787`28.093742005733777, 
      0``177.9882813527079, 0}, {0, 1, 
      1.59307118965574326155418008276994495029849790517`29.999999999999996, 
      1.903406861483373837116782897372937296120418038906`29.69897000433602, 
      2.02150842214814859056055912763169967050535148897`29.522878745280337, 
      2.012754266856659709569784620346232007627050467541`29.397940008672034, 
      1.923876500631607334249046028399003460476372414674`29.301029995663978, 
      1.787842131606955082344502232481619656050001063762`29.221848749616356, 
      1.475269930304116842635621151535775048541343011594`29.194461230983002, 
      0.000023774652968065058647318871558124336525570622`28.14289593963314, 
      5.6793855232434969344756483556628968602`27.17502915509082*^-11, 
      0``177.9414908585355}, {0, 1, 2, 3, 4, 5, 6, 7, 
      6.00950917566976753399328576235149663606967795819`30.81100590253162, 
      0.008412904524588568491678273778519796040165811646`28.906540564709307, 
      0.000743922079966207054210163994034534940666271395`28.666180212226553, 
      0.000496625137832780396707263733940533142327218658`28.629132004802422}}\
\)
h at knot points
{0,0,0,0,0,0.00017892495820107253083133223270,0,0,0,0,0,0.\
0052422547378889161714240928150}
hermite coefficients
\!\({0, 0``56.86464603180599, 
    8.20808025200586596454504432889726447882711`27.108082365460206*^-7, \
\(-0.000100382101596510454877291908677763501601972761`27.108082365460206\), 
    0.004789370391955437971717655675769830296485013814`27.108082365460206, \
\(-0.11680421621142335869468870589984939795`27.108082365460206\), 
    1.55336221646246063719756071841604248873`27.108082365460206, \
\(-8.83945626461949005213135747327767296547`27.10808236546021\), 
    8.83952234914238505549335412694509337599`27.108082365460206, \
\(-0.002667922701212832498791576795202722120507896675`27.108082365460206\), 
    17.75138496010721312870963049578864067728`27.108082365460206, \
\(-16.0039330853757888548905922432791538181`27.10808236546021\)}\)
Maximum interpolation error
0.00001013672605740444941690
plot of function being interpolated, f_t (x) = (x-t)_+^(k-1)
GraphicsData["PostScript", "\<\
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