<<Statistics`DescriptiveStatistics`

(*    Hermite spline interpolation of the Perfect Spline of degree 2*
    k      *)
(*    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
Print["nn = number of monte-carlo replications"]
nn=100

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)

PS[x_,vec_List, 
    k_]:=(1/Factorial[2*k])* (x^(2*k)+2*
          Sum[(-1)^(j)*(x-vec[[j]])^(2*k)*UnitStep[x-vec[[j]]], {j, 1,
              2*k-4}])
PSp[x_,vec_List,
    k_]:=(2*k)*(1/
        Factorial[2*k])*(x^(2*k-1) +2*
          Sum[(-1)^(j)*(x-vec[[j]])^(2*k-1)*UnitStep[x-vec[[j]]], {j, 1,
              2*k-4}])

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[HermiteInterp[i/1000,coeff,vec,k]-PS[i/1000,vec, k],{i,1,1000}]
Error[k_]:={
    VectauUnif=Sort[Table[Random[Real,{0,1},40],{j,1,2*k-4}]];
    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}]];
    VecPS=
      Join[Table[PS[Vectau[[j]],VectauUnif,k],{j,1,2*k-2}],
        Table[PSp[Vectau[[j]],VectauUnif,k],{j,1,2*k-2}]];
    HermiteCoeff=LinearSolve[MM,VecPS];
    mtl=mxlist[HermiteCoeff,Vectau,k];
    mx=Max[mtl]
    }
Ert=Flatten[Table[Error[k],{j,1,nn}]];

zz=Sort[Ert];
Print["mean, median, and harmonic mean of the maximum error (on grid of 1000 \
pts)"]
LocationReport[Ert]
Print["sample variance and standard deviation of the maximum error"]
DispersionReport[Ert]
Print["95% quantile of maximum error"]
qu=Quantile[zz,.95]
Print["99% quantile of maximum error"]
quu =Quantile[zz,.99]
Print["maximum (across monte-carlo replications) of maximum error"]
ME=Max[Ert]

hh=Table[k,{k,1,nn}]*(1/nn);//N
J1=Table[{zz[[i]],hh[[i]]},{i,1,nn}];
Print["listplot of empirical distribution of maximal error"]
ListPlot[J1,PlotRange\[Rule]All]
Delta[x_]:=0/;x<0
Delta[x_]:=1/;x³0
DE[x_]:=Table[{Delta[zz[[i]]-x]},{i,1,nn}]
EDF[nn_,x_]:=1.-Apply[Plus,DE[x]]*(1/nn)//N
Print["empirical distribution of errors, up to 1.1 times 99% quantile"]
P1=Plot[EDF[nn,x],{x,0,quu*1.1},PlotRange\[Rule]{0,1},
    PlotStyle\[Rule]{Thickness[1/100],RGBColor[0.000,1.000,0.196],
        Dashing[{}]} ]
Print["empirical distribution of errors, up to 1.1 times max"]
P2=Plot[EDF[nn,x],{x,0,ME*1.1},PlotRange\[Rule]{0,1},
    PlotStyle\[Rule]{Thickness[1/100],RGBColor[0.000,1.000,0.196],
        Dashing[{}]} ]




k: (2k-1) degree spline interpolation
4
d = number of digits of accuracy
30
nn = number of monte-carlo replications
100
\!\(\*
  RowBox[{\(General::"spell"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(mxlist\\)\\\" is similar to existing symbols \
\\!\\({mxlistC, mxlistH}\\). \\!\\(\\*ButtonBox[\\\"MoreÉ\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell\\\"]\\)\"\>"}]\)
mean, median, and harmonic mean of the maximum error (on grid of 1000 pts)
\!\({Mean \[Rule] 
      0.000031054765317461009493948807446401337589503244`33.428707449960285, 
    HarmonicMean \[Rule] \
\(-3.21406400020773741513467745497056382401`29.347666535601068*^-18\), 
    Median \[Rule] 
      0.000038254139259446205201743650935257702549709332`29.34564432370003}\)
sample variance and standard deviation of the maximum error
\!\({Variance \[Rule] 
      3.07700451382716139061166255846247023038476908983`32.715622151313326*^-\
10, StandardDeviation \[Rule] 
      0.000017541392515496486063493111380131617139568627`33.01665214697731, 
    SampleRange \[Rule] 
      0.000049572668027195389643902081398753982989826960056913356`35.\
73305716179084, 
    MeanDeviation \[Rule] 
      0.000015153788647937188837478888073361790492745527`32.81607042847744, 
    MedianDeviation \[Rule] 
      9.977725439620334255194352753256747216074766`27.357130578444963*^-6, 
    QuartileDeviation \[Rule] 
      0.000013916989181462147092896504016764671513645846712446628`34.\
98702212866607}\)
95% quantile of maximum error
0.000049340709198597434788285291814
99% quantile of maximum error
0.000049560802373136037292350807822
maximum (across monte-carlo replications) of maximum error
0.0000495726680265714136060696156212068880
listplot of empirical distribution of maximal error
GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 19211.8 0.0087696 0.594549 [
[.21593 -0.00373 -21 -9 ]
[.21593 -0.00373 21 0 ]
[.40805 -0.00373 -21 -9 ]
[.40805 -0.00373 21 0 ]
[.60016 -0.00373 -21 -9 ]
[.60016 -0.00373 21 0 ]
[.79228 -0.00373 -21 -9 ]
[.79228 -0.00373 21 0 ]
[.9844 -0.00373 -21 -9 ]
[.9844 -0.00373 21 0 ]
[.01131 .12768 -18 -4.5 ]
[.01131 .12768 0 4.5 ]
[.01131 .24659 -18 -4.5 ]
[.01131 .24659 0 4.5 ]
[.01131 .3655 -18 -4.5 ]
[.01131 .3655 0 4.5 ]
[.01131 .48441 -18 -4.5 ]
[.01131 .48441 0 4.5 ]
[.01131 .60332 -6 -4.5 ]
[.01131 .60332 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.21593 .00877 m
.21593 .01502 L
s
[(0.00001)] .21593 -0.00373 0 1 Mshowa
.40805 .00877 m
.40805 .01502 L
s
[(0.00002)] .40805 -0.00373 0 1 Mshowa
.60016 .00877 m
.60016 .01502 L
s
[(0.00003)] .60016 -0.00373 0 1 Mshowa
.79228 .00877 m
.79228 .01502 L
s
[(0.00004)] .79228 -0.00373 0 1 Mshowa
.9844 .00877 m
.9844 .01502 L
s
[(0.00005)] .9844 -0.00373 0 1 Mshowa
.125 Mabswid
.06223 .00877 m
.06223 .01252 L
s
.10066 .00877 m
.10066 .01252 L
s
.13908 .00877 m
.13908 .01252 L
s
.1775 .00877 m
.1775 .01252 L
s
.25435 .00877 m
.25435 .01252 L
s
.29277 .00877 m
.29277 .01252 L
s
.3312 .00877 m
.3312 .01252 L
s
.36962 .00877 m
.36962 .01252 L
s
.44647 .00877 m
.44647 .01252 L
s
.48489 .00877 m
.48489 .01252 L
s
.52332 .00877 m
.52332 .01252 L
s
.56174 .00877 m
.56174 .01252 L
s
.63859 .00877 m
.63859 .01252 L
s
.67701 .00877 m
.67701 .01252 L
s
.71543 .00877 m
.71543 .01252 L
s
.75386 .00877 m
.75386 .01252 L
s
.83071 .00877 m
.83071 .01252 L
s
.86913 .00877 m
.86913 .01252 L
s
.90755 .00877 m
.90755 .01252 L
s
.94598 .00877 m
.94598 .01252 L
s
.25 Mabswid
0 .00877 m
1 .00877 L
s
.02381 .12768 m
.03006 .12768 L
s
[(0.2)] .01131 .12768 1 0 Mshowa
.02381 .24659 m
.03006 .24659 L
s
[(0.4)] .01131 .24659 1 0 Mshowa
.02381 .3655 m
.03006 .3655 L
s
[(0.6)] .01131 .3655 1 0 Mshowa
.02381 .48441 m
.03006 .48441 L
s
[(0.8)] .01131 .48441 1 0 Mshowa
.02381 .60332 m
.03006 .60332 L
s
[(1)] .01131 .60332 1 0 Mshowa
.125 Mabswid
.02381 .0385 m
.02756 .0385 L
s
.02381 .06822 m
.02756 .06822 L
s
.02381 .09795 m
.02756 .09795 L
s
.02381 .15741 m
.02756 .15741 L
s
.02381 .18713 m
.02756 .18713 L
s
.02381 .21686 m
.02756 .21686 L
s
.02381 .27632 m
.02756 .27632 L
s
.02381 .30604 m
.02756 .30604 L
s
.02381 .33577 m
.02756 .33577 L
s
.02381 .39523 m
.02756 .39523 L
s
.02381 .42495 m
.02756 .42495 L
s
.02381 .45468 m
.02756 .45468 L
s
.02381 .51414 m
.02756 .51414 L
s
.02381 .54386 m
.02756 .54386 L
s
.02381 .57359 m
.02756 .57359 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
.008 w
.02381 .01472 Mdot
.02381 .02066 Mdot
.02381 .02661 Mdot
.02381 .03255 Mdot
.02381 .0385 Mdot
.02381 .04444 Mdot
.02381 .05039 Mdot
.02383 .05633 Mdot
.02391 .06228 Mdot
.0251 .06822 Mdot
.03504 .07417 Mdot
.04057 .08012 Mdot
.0448 .08606 Mdot
.05538 .09201 Mdot
.05748 .09795 Mdot
.11333 .1039 Mdot
.11405 .10984 Mdot
.11706 .11579 Mdot
.15194 .12173 Mdot
.18249 .12768 Mdot
.26254 .13362 Mdot
.26725 .13957 Mdot
.29548 .14552 Mdot
.32951 .15146 Mdot
.35169 .15741 Mdot
.39109 .16335 Mdot
.3954 .1693 Mdot
.41444 .17524 Mdot
.44553 .18119 Mdot
.46222 .18713 Mdot
.48498 .19308 Mdot
.48771 .19903 Mdot
.49732 .20497 Mdot
.50924 .21092 Mdot
.55568 .21686 Mdot
.58337 .22281 Mdot
.58454 .22875 Mdot
.58824 .2347 Mdot
.59103 .24064 Mdot
.61163 .24659 Mdot
.66088 .25253 Mdot
.6624 .25848 Mdot
.66915 .26443 Mdot
.68747 .27037 Mdot
.68953 .27632 Mdot
.69697 .28226 Mdot
.7022 .28821 Mdot
.74245 .29415 Mdot
.74489 .3001 Mdot
.75119 .30604 Mdot
.76629 .31199 Mdot
.76989 .31794 Mdot
.77666 .32388 Mdot
.78291 .32983 Mdot
.78474 .33577 Mdot
.80352 .34172 Mdot
.81112 .34766 Mdot
.81398 .35361 Mdot
.82226 .35955 Mdot
.82443 .3655 Mdot
.8287 .37144 Mdot
.83393 .37739 Mdot
.84066 .38334 Mdot
.84615 .38928 Mdot
.85074 .39523 Mdot
.85837 .40117 Mdot
.86507 .40712 Mdot
.86949 .41306 Mdot
.87153 .41901 Mdot
.87407 .42495 Mdot
.87611 .4309 Mdot
.89295 .43685 Mdot
.90121 .44279 Mdot
.90371 .44874 Mdot
.90447 .45468 Mdot
.9078 .46063 Mdot
.9218 .46657 Mdot
.92421 .47252 Mdot
.93111 .47846 Mdot
.93144 .48441 Mdot
.93278 .49035 Mdot
.93556 .4963 Mdot
.9424 .50225 Mdot
.945 .50819 Mdot
.94964 .51414 Mdot
.95122 .52008 Mdot
.95178 .52603 Mdot
.95379 .53197 Mdot
.95606 .53792 Mdot
.95837 .54386 Mdot
.96042 .54981 Mdot
.96361 .55575 Mdot
.96775 .5617 Mdot
.97036 .56765 Mdot
.97173 .57359 Mdot
.97308 .57954 Mdot
.97417 .58548 Mdot
.97596 .59143 Mdot
.97596 .59737 Mdot
.97619 .60332 Mdot
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
% End of Graphics
MathPictureEnd
\
\>"]
\[SkeletonIndicator]Graphics\[SkeletonIndicator]
empirical distribution of errors, up to 1.1 times 99% quantile
GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 17469.5 0 0.618034 [
[.1985 -0.0125 -21 -9 ]
[.1985 -0.0125 21 0 ]
[.3732 -0.0125 -21 -9 ]
[.3732 -0.0125 21 0 ]
[.54789 -0.0125 -21 -9 ]
[.54789 -0.0125 21 0 ]
[.72259 -0.0125 -21 -9 ]
[.72259 -0.0125 21 0 ]
[.89728 -0.0125 -21 -9 ]
[.89728 -0.0125 21 0 ]
[.01131 .12361 -18 -4.5 ]
[.01131 .12361 0 4.5 ]
[.01131 .24721 -18 -4.5 ]
[.01131 .24721 0 4.5 ]
[.01131 .37082 -18 -4.5 ]
[.01131 .37082 0 4.5 ]
[.01131 .49443 -18 -4.5 ]
[.01131 .49443 0 4.5 ]
[.01131 .61803 -6 -4.5 ]
[.01131 .61803 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.1985 0 m
.1985 .00625 L
s
[(0.00001)] .1985 -0.0125 0 1 Mshowa
.3732 0 m
.3732 .00625 L
s
[(0.00002)] .3732 -0.0125 0 1 Mshowa
.54789 0 m
.54789 .00625 L
s
[(0.00003)] .54789 -0.0125 0 1 Mshowa
.72259 0 m
.72259 .00625 L
s
[(0.00004)] .72259 -0.0125 0 1 Mshowa
.89728 0 m
.89728 .00625 L
s
[(0.00005)] .89728 -0.0125 0 1 Mshowa
.125 Mabswid
.05875 0 m
.05875 .00375 L
s
.09369 0 m
.09369 .00375 L
s
.12863 0 m
.12863 .00375 L
s
.16357 0 m
.16357 .00375 L
s
.23344 0 m
.23344 .00375 L
s
.26838 0 m
.26838 .00375 L
s
.30332 0 m
.30332 .00375 L
s
.33826 0 m
.33826 .00375 L
s
.40814 0 m
.40814 .00375 L
s
.44308 0 m
.44308 .00375 L
s
.47802 0 m
.47802 .00375 L
s
.51295 0 m
.51295 .00375 L
s
.58283 0 m
.58283 .00375 L
s
.61777 0 m
.61777 .00375 L
s
.65271 0 m
.65271 .00375 L
s
.68765 0 m
.68765 .00375 L
s
.75753 0 m
.75753 .00375 L
s
.79247 0 m
.79247 .00375 L
s
.82741 0 m
.82741 .00375 L
s
.86234 0 m
.86234 .00375 L
s
.93222 0 m
.93222 .00375 L
s
.96716 0 m
.96716 .00375 L
s
.25 Mabswid
0 0 m
1 0 L
s
.02381 .12361 m
.03006 .12361 L
s
[(0.2)] .01131 .12361 1 0 Mshowa
.02381 .24721 m
.03006 .24721 L
s
[(0.4)] .01131 .24721 1 0 Mshowa
.02381 .37082 m
.03006 .37082 L
s
[(0.6)] .01131 .37082 1 0 Mshowa
.02381 .49443 m
.03006 .49443 L
s
[(0.8)] .01131 .49443 1 0 Mshowa
.02381 .61803 m
.03006 .61803 L
s
[(1)] .01131 .61803 1 0 Mshowa
.125 Mabswid
.02381 .0309 m
.02756 .0309 L
s
.02381 .0618 m
.02756 .0618 L
s
.02381 .09271 m
.02756 .09271 L
s
.02381 .15451 m
.02756 .15451 L
s
.02381 .18541 m
.02756 .18541 L
s
.02381 .21631 m
.02756 .21631 L
s
.02381 .27812 m
.02756 .27812 L
s
.02381 .30902 m
.02756 .30902 L
s
.02381 .33992 m
.02756 .33992 L
s
.02381 .40172 m
.02756 .40172 L
s
.02381 .43262 m
.02756 .43262 L
s
.02381 .46353 m
.02756 .46353 L
s
.02381 .52533 m
.02756 .52533 L
s
.02381 .55623 m
.02756 .55623 L
s
.02381 .58713 m
.02756 .58713 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
0 1 .196 r
.01 w
.02381 .03708 m
.02499 .0618 L
.02605 .0618 L
.02846 .0618 L
.03071 .0618 L
.03279 .0618 L
.03396 .0618 L
.03521 .06798 L
.03646 .06798 L
.03784 .06798 L
.03903 .06798 L
.04031 .07416 L
.04151 .07416 L
.04262 .07416 L
.04384 .08034 L
.04514 .08034 L
.04748 .08034 L
.04891 .08034 L
.05026 .08034 L
.05147 .08034 L
.05278 .08652 L
.05395 .08652 L
.05523 .09271 L
.05587 .09271 L
.05657 .09271 L
.05781 .09271 L
.06244 .09271 L
.08408 .09271 L
.09473 .09271 L
.09941 .09271 L
.1019 .09271 L
.1032 .09271 L
.10392 .09271 L
.10458 .09271 L
.10581 .09889 L
.10652 .10507 L
.10716 .10507 L
.10831 .10507 L
.10953 .11125 L
.11068 .11125 L
.11193 .11125 L
.11419 .11125 L
.12439 .11125 L
.12984 .11125 L
.13282 .11125 L
.1356 .11125 L
.13798 .11125 L
.13866 .11125 L
.13931 .11125 L
.14053 .11743 L
Mistroke
.14118 .11743 L
.14187 .11743 L
.14312 .11743 L
.14594 .11743 L
.15625 .11743 L
.16167 .11743 L
.16463 .11743 L
.16608 .11743 L
.16739 .11743 L
.16864 .12361 L
.16982 .12361 L
.17241 .12361 L
.17791 .12361 L
.18758 .12361 L
.20583 .12361 L
.21622 .12361 L
.22596 .12361 L
.2355 .12361 L
.23811 .12361 L
.23954 .12361 L
.24023 .12361 L
.24086 .12361 L
.242 .12979 L
.24321 .12979 L
.24453 .12979 L
.24577 .13597 L
.24699 .13597 L
.24832 .13597 L
.25103 .13597 L
.25586 .13597 L
.26147 .13597 L
.26423 .13597 L
.26679 .13597 L
.26916 .13597 L
.27035 .13597 L
.27167 .14215 L
.27299 .14215 L
.27441 .14215 L
.27698 .14215 L
.28628 .14215 L
.29159 .14215 L
.29448 .14215 L
.29718 .14215 L
.29949 .14215 L
.30078 .14215 L
.30196 .14833 L
.30324 .14833 L
.30464 .14833 L
.30718 .14833 L
.31174 .14833 L
Mistroke
.31663 .14833 L
.31807 .14833 L
.31944 .14833 L
.32067 .14833 L
.32136 .14833 L
.322 .15451 L
.32328 .15451 L
.32444 .15451 L
.32704 .15451 L
.33799 .15451 L
.34807 .15451 L
.35292 .15451 L
.35528 .15451 L
.35632 .15451 L
.35744 .15451 L
.35861 .16069 L
.35972 .16069 L
.36098 .16069 L
.36216 .16687 L
.36335 .16687 L
.36467 .16687 L
.36733 .16687 L
.3725 .16687 L
.37522 .16687 L
.3767 .16687 L
.37742 .16687 L
.37807 .16687 L
.37929 .17305 L
.38059 .17305 L
.38334 .17305 L
.38816 .17305 L
.39301 .17305 L
.39815 .17305 L
.40253 .17305 L
.40374 .17305 L
.40503 .17305 L
.40624 .17305 L
.40736 .17923 L
.40864 .17923 L
.41001 .17923 L
.41246 .17923 L
.41533 .17923 L
.41803 .17923 L
.42059 .17923 L
.42129 .17923 L
.42203 .17923 L
.42335 .18541 L
.42453 .18541 L
.42564 .18541 L
.42816 .18541 L
Mistroke
.43319 .18541 L
.43584 .18541 L
.43865 .18541 L
.4411 .18541 L
.44177 .18541 L
.4425 .18541 L
.4438 .19159 L
.44508 .19159 L
.44629 .19777 L
.44738 .19777 L
.44857 .19777 L
.45097 .19777 L
.45214 .19777 L
.45323 .19777 L
.45448 .20395 L
.45514 .20395 L
.45583 .20395 L
.4582 .20395 L
.46246 .20395 L
.46363 .20395 L
.46488 .20395 L
.46606 .21013 L
.46714 .21013 L
.48783 .21013 L
.49693 .21013 L
.50176 .21013 L
.5043 .21013 L
.50568 .21013 L
.50697 .21013 L
.50813 .21631 L
.50936 .21631 L
.51197 .21631 L
.51655 .21631 L
.52555 .21631 L
.5281 .21631 L
.52944 .21631 L
.53087 .21631 L
.53211 .21631 L
.53282 .22249 L
.53348 .22249 L
.53464 .22867 L
.53589 .22867 L
.53709 .23485 L
.53841 .23485 L
.53965 .24103 L
.5408 .24103 L
.54529 .24103 L
.55038 .24103 L
.55305 .24103 L
.55451 .24103 L
Mistroke
.55588 .24103 L
.55707 .24103 L
.55835 .24721 L
.55908 .24721 L
.55978 .24721 L
.56107 .24721 L
.56584 .24721 L
.58446 .24721 L
.59413 .24721 L
.59901 .24721 L
.60035 .24721 L
.60105 .24721 L
.60179 .24721 L
.60244 .24721 L
.60315 .25339 L
.6044 .25339 L
.60564 .25957 L
.60682 .25957 L
.60902 .25957 L
.61025 .25957 L
.61139 .26575 L
.61263 .26575 L
.61395 .26575 L
.61851 .26575 L
.62283 .26575 L
.62519 .26575 L
.62584 .26575 L
.62653 .26575 L
.62777 .27193 L
.62888 .27193 L
.62992 .27812 L
.63105 .27812 L
.63228 .27812 L
.63362 .27812 L
.63486 .27812 L
.63616 .2843 L
.63684 .2843 L
.63758 .2843 L
.63888 .2843 L
.64012 .2843 L
.64123 .29048 L
.64243 .29048 L
.66076 .29048 L
.6702 .29048 L
.67279 .29048 L
.67421 .29048 L
.67557 .29048 L
.67682 .29048 L
.67754 .29666 L
.6782 .29666 L
Mistroke
.67936 .29666 L
.6806 .30284 L
.68129 .30284 L
.68202 .30284 L
.68333 .30284 L
.68401 .30284 L
.68473 .30284 L
.68553 .30902 L
.68628 .30902 L
.69163 .30902 L
.69432 .30902 L
.69564 .30902 L
.69686 .30902 L
.69797 .30902 L
.69915 .3152 L
.70033 .3152 L
.70163 .3152 L
.70285 .32138 L
.70398 .32138 L
.70533 .32138 L
.70655 .32138 L
.70784 .32138 L
.70905 .32756 L
.71034 .32756 L
.71173 .32756 L
.71294 .32756 L
.71407 .32756 L
.71525 .33374 L
.71654 .33992 L
.71766 .33992 L
.71868 .33992 L
.72098 .33992 L
.7261 .33992 L
.72881 .33992 L
.73029 .33992 L
.73102 .33992 L
.73168 .33992 L
.73233 .33992 L
.73301 .3461 L
.7337 .3461 L
.73445 .3461 L
.73695 .3461 L
.73818 .3461 L
.73948 .3461 L
.74072 .35228 L
.74184 .35228 L
.74302 .35846 L
.74414 .35846 L
.74667 .35846 L
.74793 .35846 L
Mistroke
.74913 .35846 L
.74979 .35846 L
.75052 .36464 L
.75178 .36464 L
.75299 .37082 L
.75409 .37082 L
.75535 .37082 L
.75655 .377 L
.75765 .377 L
.75867 .377 L
.75979 .377 L
.76097 .38318 L
.76226 .38318 L
.76344 .38318 L
.76471 .38318 L
.76538 .38318 L
.76609 .38318 L
.76736 .38936 L
.76854 .38936 L
.76976 .38936 L
.77087 .38936 L
.77215 .39554 L
.77283 .39554 L
.77353 .39554 L
.77473 .39554 L
.77604 .40172 L
.77679 .40172 L
.77747 .40172 L
.77899 .40172 L
.78041 .40172 L
.78111 .40172 L
.78175 .40172 L
.78286 .4079 L
.78405 .4079 L
.78659 .4079 L
.78784 .4079 L
.78856 .4079 L
.78922 .41408 L
.79043 .41408 L
.79171 .41408 L
.79291 .42026 L
.7942 .42026 L
.79542 .42644 L
.79654 .42644 L
.79755 .43262 L
.79867 .43262 L
.79983 .4388 L
.80091 .4388 L
.80607 .4388 L
.81095 .4388 L
Mistroke
.81226 .4388 L
.81297 .4388 L
.81365 .4388 L
.81437 .44498 L
.81517 .44498 L
.81658 .44498 L
.81904 .44498 L
.81972 .44498 L
.82043 .44498 L
.8217 .45116 L
.82294 .45116 L
.82364 .45116 L
.82428 .45735 L
.82546 .46353 L
.82671 .46353 L
.82783 .46971 L
.82888 .46971 L
.83126 .46971 L
.83618 .46971 L
.8376 .46971 L
.83894 .46971 L
.84015 .46971 L
.84147 .47589 L
.84272 .48207 L
.84389 .48207 L
.84609 .48207 L
.84724 .48207 L
.84845 .48207 L
.84909 .48825 L
.84978 .49443 L
.851 .50061 L
.85216 .50061 L
.85321 .50679 L
.85443 .50679 L
.85558 .50679 L
.85762 .50679 L
.85877 .50679 L
.85984 .51297 L
.86109 .51297 L
.86227 .51915 L
.86333 .51915 L
.8645 .51915 L
.86575 .52533 L
.86646 .52533 L
.86709 .52533 L
.86833 .53769 L
.8695 .54387 L
.87061 .54387 L
.87177 .55005 L
.87276 .55005 L
Mistroke
.87385 .55623 L
.87503 .55623 L
.87629 .56241 L
.87748 .56241 L
.87856 .56859 L
.87984 .56859 L
.88054 .56859 L
.8812 .56859 L
.88235 .57477 L
.88363 .57477 L
.88431 .57477 L
.88502 .58095 L
.88629 .58713 L
.88915 .59949 L
.89034 .61803 L
.89147 .61803 L
.89276 .61803 L
.89394 .61803 L
.89666 .61803 L
.89918 .61803 L
.93765 .61803 L
.97619 .61803 L
Mfstroke
% End of Graphics
MathPictureEnd
\
\>"]
\[SkeletonIndicator]Graphics\[SkeletonIndicator]
empirical distribution of errors, up to 1.1 times max
GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 17465.3 0 0.618034 [
[.19846 -0.0125 -21 -9 ]
[.19846 -0.0125 21 0 ]
[.37312 -0.0125 -21 -9 ]
[.37312 -0.0125 21 0 ]
[.54777 -0.0125 -21 -9 ]
[.54777 -0.0125 21 0 ]
[.72242 -0.0125 -21 -9 ]
[.72242 -0.0125 21 0 ]
[.89707 -0.0125 -21 -9 ]
[.89707 -0.0125 21 0 ]
[.01131 .12361 -18 -4.5 ]
[.01131 .12361 0 4.5 ]
[.01131 .24721 -18 -4.5 ]
[.01131 .24721 0 4.5 ]
[.01131 .37082 -18 -4.5 ]
[.01131 .37082 0 4.5 ]
[.01131 .49443 -18 -4.5 ]
[.01131 .49443 0 4.5 ]
[.01131 .61803 -6 -4.5 ]
[.01131 .61803 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.19846 0 m
.19846 .00625 L
s
[(0.00001)] .19846 -0.0125 0 1 Mshowa
.37312 0 m
.37312 .00625 L
s
[(0.00002)] .37312 -0.0125 0 1 Mshowa
.54777 0 m
.54777 .00625 L
s
[(0.00003)] .54777 -0.0125 0 1 Mshowa
.72242 0 m
.72242 .00625 L
s
[(0.00004)] .72242 -0.0125 0 1 Mshowa
.89707 0 m
.89707 .00625 L
s
[(0.00005)] .89707 -0.0125 0 1 Mshowa
.125 Mabswid
.05874 0 m
.05874 .00375 L
s
.09367 0 m
.09367 .00375 L
s
.1286 0 m
.1286 .00375 L
s
.16353 0 m
.16353 .00375 L
s
.23339 0 m
.23339 .00375 L
s
.26832 0 m
.26832 .00375 L
s
.30325 0 m
.30325 .00375 L
s
.33818 0 m
.33818 .00375 L
s
.40805 0 m
.40805 .00375 L
s
.44298 0 m
.44298 .00375 L
s
.47791 0 m
.47791 .00375 L
s
.51284 0 m
.51284 .00375 L
s
.5827 0 m
.5827 .00375 L
s
.61763 0 m
.61763 .00375 L
s
.65256 0 m
.65256 .00375 L
s
.68749 0 m
.68749 .00375 L
s
.75735 0 m
.75735 .00375 L
s
.79228 0 m
.79228 .00375 L
s
.82721 0 m
.82721 .00375 L
s
.86214 0 m
.86214 .00375 L
s
.932 0 m
.932 .00375 L
s
.96694 0 m
.96694 .00375 L
s
.25 Mabswid
0 0 m
1 0 L
s
.02381 .12361 m
.03006 .12361 L
s
[(0.2)] .01131 .12361 1 0 Mshowa
.02381 .24721 m
.03006 .24721 L
s
[(0.4)] .01131 .24721 1 0 Mshowa
.02381 .37082 m
.03006 .37082 L
s
[(0.6)] .01131 .37082 1 0 Mshowa
.02381 .49443 m
.03006 .49443 L
s
[(0.8)] .01131 .49443 1 0 Mshowa
.02381 .61803 m
.03006 .61803 L
s
[(1)] .01131 .61803 1 0 Mshowa
.125 Mabswid
.02381 .0309 m
.02756 .0309 L
s
.02381 .0618 m
.02756 .0618 L
s
.02381 .09271 m
.02756 .09271 L
s
.02381 .15451 m
.02756 .15451 L
s
.02381 .18541 m
.02756 .18541 L
s
.02381 .21631 m
.02756 .21631 L
s
.02381 .27812 m
.02756 .27812 L
s
.02381 .30902 m
.02756 .30902 L
s
.02381 .33992 m
.02756 .33992 L
s
.02381 .40172 m
.02756 .40172 L
s
.02381 .43262 m
.02756 .43262 L
s
.02381 .46353 m
.02756 .46353 L
s
.02381 .52533 m
.02756 .52533 L
s
.02381 .55623 m
.02756 .55623 L
s
.02381 .58713 m
.02756 .58713 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
0 1 .196 r
.01 w
.02381 .03708 m
.02499 .0618 L
.02605 .0618 L
.02846 .0618 L
.03071 .0618 L
.03279 .0618 L
.03396 .0618 L
.03521 .06798 L
.03646 .06798 L
.03784 .06798 L
.03903 .06798 L
.04031 .07416 L
.04151 .07416 L
.04262 .07416 L
.04384 .08034 L
.04514 .08034 L
.04748 .08034 L
.04891 .08034 L
.05026 .08034 L
.05147 .08034 L
.05278 .08652 L
.05395 .08652 L
.05523 .09271 L
.05587 .09271 L
.05657 .09271 L
.05781 .09271 L
.06244 .09271 L
.08408 .09271 L
.09473 .09271 L
.09941 .09271 L
.1019 .09271 L
.1032 .09271 L
.10392 .09271 L
.10458 .09271 L
.10581 .09889 L
.10652 .10507 L
.10716 .10507 L
.10831 .10507 L
.10953 .11125 L
.11068 .11125 L
.11193 .11125 L
.11419 .11125 L
.12439 .11125 L
.12984 .11125 L
.13282 .11125 L
.1356 .11125 L
.13798 .11125 L
.13866 .11125 L
.13931 .11125 L
.14053 .11743 L
Mistroke
.14118 .11743 L
.14187 .11743 L
.14312 .11743 L
.14594 .11743 L
.15625 .11743 L
.16167 .11743 L
.16463 .11743 L
.16608 .11743 L
.16739 .11743 L
.16864 .12361 L
.16982 .12361 L
.17241 .12361 L
.17791 .12361 L
.18758 .12361 L
.20583 .12361 L
.21622 .12361 L
.22596 .12361 L
.23061 .12361 L
.2355 .12361 L
.23806 .12361 L
.23876 .12361 L
.23951 .12361 L
.24022 .12361 L
.24086 .12979 L
.24207 .12979 L
.24339 .12979 L
.24463 .12979 L
.24577 .13597 L
.24698 .13597 L
.24829 .13597 L
.25064 .13597 L
.25599 .13597 L
.2616 .13597 L
.26407 .13597 L
.26679 .13597 L
.26801 .13597 L
.26933 .13597 L
.27057 .13597 L
.27171 .14215 L
.27293 .14215 L
.27426 .14215 L
.27695 .14215 L
.28801 .14215 L
.29304 .14215 L
.29581 .14215 L
.29842 .14215 L
.2996 .14215 L
.30086 .14215 L
.30151 .14215 L
.30223 .14833 L
Mistroke
.3035 .14833 L
.30812 .14833 L
.31332 .14833 L
.31589 .14833 L
.31827 .14833 L
.32041 .14833 L
.32161 .14833 L
.3227 .15451 L
.3239 .15451 L
.3252 .15451 L
.32754 .15451 L
.3375 .15451 L
.34278 .15451 L
.34843 .15451 L
.3512 .15451 L
.35383 .15451 L
.35619 .15451 L
.35683 .15451 L
.35752 .15451 L
.35875 .16069 L
.35989 .16069 L
.36111 .16069 L
.36227 .16687 L
.36333 .16687 L
.36838 .16687 L
.3737 .16687 L
.37629 .16687 L
.37742 .16687 L
.37865 .16687 L
.37993 .17305 L
.38114 .17305 L
.38381 .17305 L
.38945 .17305 L
.39893 .17305 L
.40152 .17305 L
.40431 .17305 L
.40566 .17305 L
.40694 .17305 L
.40809 .17923 L
.40934 .17923 L
.41463 .17923 L
.41754 .17923 L
.41896 .17923 L
.42027 .17923 L
.42151 .17923 L
.42282 .18541 L
.42414 .18541 L
.42558 .18541 L
.43039 .18541 L
.43565 .18541 L
Mistroke
.43824 .18541 L
.44063 .18541 L
.44175 .18541 L
.4428 .18541 L
.44389 .19159 L
.4451 .19159 L
.44632 .19777 L
.44744 .19777 L
.44995 .19777 L
.45124 .19777 L
.45261 .19777 L
.45391 .19777 L
.45508 .20395 L
.4558 .20395 L
.45647 .20395 L
.45797 .20395 L
.46068 .20395 L
.46305 .20395 L
.4637 .20395 L
.46439 .20395 L
.46564 .21013 L
.46632 .21013 L
.46705 .21013 L
.46839 .21013 L
.47089 .21013 L
.49065 .21013 L
.50063 .21013 L
.50337 .21013 L
.50481 .21013 L
.50559 .21013 L
.50632 .21013 L
.50696 .21013 L
.50765 .21631 L
.50841 .21631 L
.5091 .21631 L
.51166 .21631 L
.52097 .21631 L
.52606 .21631 L
.52855 .21631 L
.53083 .21631 L
.5321 .21631 L
.53331 .22249 L
.53459 .22867 L
.53531 .22867 L
.53598 .22867 L
.53672 .22867 L
.53742 .23485 L
.53873 .23485 L
.53946 .23485 L
.54013 .24103 L
Mistroke
.54087 .24103 L
.54165 .24103 L
.54687 .24103 L
.55163 .24103 L
.55396 .24103 L
.5552 .24103 L
.5565 .24103 L
.55765 .24103 L
.5589 .24721 L
.56005 .24721 L
.56112 .24721 L
.57129 .24721 L
.58964 .24721 L
.59474 .24721 L
.59949 .24721 L
.60072 .24721 L
.60188 .24721 L
.60253 .24721 L
.60322 .25339 L
.60444 .25957 L
.60509 .25957 L
.60581 .25957 L
.60726 .25957 L
.60859 .25957 L
.60985 .25957 L
.61101 .26575 L
.61226 .26575 L
.61447 .26575 L
.61942 .26575 L
.62167 .26575 L
.62406 .26575 L
.62505 .26575 L
.62611 .26575 L
.62727 .27193 L
.62836 .27193 L
.62962 .27812 L
.63075 .27812 L
.63331 .27812 L
.63444 .27812 L
.63568 .27812 L
.63684 .2843 L
.63791 .2843 L
.63908 .2843 L
.64033 .2843 L
.64105 .29048 L
.6417 .29048 L
.64299 .29048 L
.64837 .29048 L
.65931 .29048 L
.66463 .29048 L
Mistroke
.66946 .29048 L
.67173 .29048 L
.67419 .29048 L
.67532 .29048 L
.67652 .29048 L
.67753 .29666 L
.67865 .29666 L
.67993 .30284 L
.6811 .30284 L
.68235 .30284 L
.68372 .30284 L
.68498 .30284 L
.68615 .30902 L
.68727 .30902 L
.68845 .30902 L
.69319 .30902 L
.69568 .30902 L
.69704 .30902 L
.6983 .30902 L
.69898 .3152 L
.6997 .3152 L
.70049 .3152 L
.70123 .3152 L
.7019 .3152 L
.70263 .32138 L
.70391 .32138 L
.70524 .32138 L
.70665 .32138 L
.70785 .32138 L
.70918 .32756 L
.71047 .32756 L
.71167 .32756 L
.71299 .32756 L
.71373 .32756 L
.71439 .33374 L
.71562 .33992 L
.71631 .33992 L
.71694 .33992 L
.71934 .33992 L
.72415 .33992 L
.72863 .33992 L
.73097 .33992 L
.73229 .33992 L
.73348 .3461 L
.73479 .3461 L
.73621 .3461 L
.73755 .3461 L
.7388 .3461 L
.74008 .35228 L
.74077 .35228 L
Mistroke
.7415 .35228 L
.74227 .35846 L
.743 .35846 L
.74437 .35846 L
.74702 .35846 L
.74833 .35846 L
.74952 .35846 L
.7507 .36464 L
.75193 .37082 L
.75318 .37082 L
.75389 .37082 L
.75454 .37082 L
.75568 .377 L
.7569 .377 L
.75796 .377 L
.7591 .377 L
.7604 .38318 L
.76161 .38318 L
.76302 .38318 L
.76434 .38318 L
.76548 .38318 L
.7667 .38936 L
.76736 .38936 L
.76806 .38936 L
.76931 .38936 L
.77046 .38936 L
.77151 .39554 L
.77271 .39554 L
.77386 .39554 L
.77491 .39554 L
.77588 .40172 L
.77695 .40172 L
.77809 .40172 L
.77922 .40172 L
.78045 .40172 L
.7816 .40172 L
.78266 .4079 L
.78389 .4079 L
.78524 .4079 L
.78643 .4079 L
.78769 .4079 L
.78833 .4079 L
.78902 .41408 L
.79025 .41408 L
.79094 .41408 L
.79168 .41408 L
.79238 .41408 L
.79301 .42026 L
.79425 .42026 L
.79494 .42644 L
Mistroke
.79559 .42644 L
.79677 .42644 L
.79802 .43262 L
.79913 .4388 L
.80032 .4388 L
.80248 .4388 L
.80739 .4388 L
.81015 .4388 L
.8115 .4388 L
.81272 .4388 L
.81402 .44498 L
.81473 .44498 L
.8154 .44498 L
.81833 .44498 L
.81955 .44498 L
.82084 .44498 L
.82194 .45116 L
.82315 .45116 L
.82445 .46353 L
.82518 .46353 L
.82584 .46353 L
.82714 .46353 L
.82836 .46971 L
.82962 .46971 L
.83097 .46971 L
.83342 .46971 L
.8356 .46971 L
.83799 .46971 L
.83864 .46971 L
.83934 .46971 L
.84059 .47589 L
.84133 .47589 L
.84202 .47589 L
.84333 .48207 L
.84455 .48207 L
.84568 .48207 L
.84691 .48207 L
.84823 .48207 L
.84938 .49443 L
.85043 .50061 L
.85148 .50061 L
.85248 .50061 L
.85364 .50679 L
.85472 .50679 L
.85598 .50679 L
.85716 .50679 L
.85832 .50679 L
.85957 .51297 L
.8607 .51297 L
.86175 .51915 L
Mistroke
.86288 .51915 L
.86413 .51915 L
.86542 .51915 L
.86664 .52533 L
.86906 .53769 L
.87039 .54387 L
.87164 .55005 L
.87229 .55005 L
.87299 .55005 L
.87376 .55623 L
.87447 .55623 L
.87574 .56241 L
.87638 .56241 L
.87709 .56241 L
.87838 .56859 L
.87957 .56859 L
.88022 .56859 L
.88093 .56859 L
.88221 .57477 L
.88699 .59331 L
.88815 .59949 L
.88941 .61185 L
.89059 .61803 L
.89167 .61803 L
.89404 .61803 L
.89664 .61803 L
.90554 .61803 L
.9453 .61803 L
.97619 .61803 L
Mfstroke
% End of Graphics
MathPictureEnd
\
\>"]
\[SkeletonIndicator]Graphics\[SkeletonIndicator]