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(* ::Title:: *)
(*Algorithms*)


(* ::Section:: *)
(*Introduction*)


(* ::Text:: *)
(*This file contains the algorithms and examples from the manuscript: "Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data" written by K. Sabo, R. Scitovski and I. Vazler, Department of Mathematics, University of Osijek.*)
(*The cells containing the algorithms don't need to be manually executed in order to run the examples, Mathematica should offer to run them automatically the first time a cell is run. A more detailed explanation of the examples and their illustrations can be found in the article.*)


(* ::Section:: *)
(*Algorithms*)


(* ::Subsection:: *)
(*Subroutines necessary for the algorithms *)


(* ::Subsubsection::Closed:: *)
(*QR decomposition*)


(* ::Text:: *)
(*input :*)
(* matrix A*)
(*output:*)
(* orthogonal matrix Q*)
(* upper triangular matrix R*)


QR[A_]:=Module[{m=Length[A],n=Length[A[[1]]],Q,R=N[A],a,gamma,u,H,i,j},
Q=IdentityMatrix[m];
Do[
a=R[[All,i]];
gamma=Norm[a[[i;;m]]];
u=Table[0.,{m}];
u[[i]]=a[[i]]+Sign[Sign[a[[i]]]+.5]*gamma;
Do[u[[j]]=a[[j]],{j,i+1,m}];
u/=Sqrt[2*gamma*(gamma+Abs[a[[i]]])];
H=IdentityMatrix[m]-2*Transpose[{u}].{u};
R=H.R;
Q=Q.H;
,{i,Min[{n,m}]}];
Chop[{Q,R}]
]


(* ::Subsubsection::Closed:: *)
(*QR update - coulumn deletion*)


(* ::Text:: *)
(*input :*)
(* orthogonal matrix Q*)
(* upper triangular matrix R*)
(* integer k*)
(*output:*)
(* orthogonal matrix QQ*)
(* upper triangular matrix RR*)


Izbaci[Q_,R_,k_]:=Module[{m=Length[R],n=Length[R[[1]]],RR=Transpose[Delete[Transpose[R],k]],QQ=Q,a,b,G},
Do[
If[RR[[i,i]]!=0||RR[[i+1,i]]!=0,
a=RR[[i,i]]/Sqrt[RR[[i,i]]^2+RR[[i+1,i]]^2];
b=RR[[i+1,i]]/Sqrt[RR[[i,i]]^2+RR[[i+1,i]]^2];
G=IdentityMatrix[m];
G[[i;;i+1,i;;i+1]]={{a,b},{b,-a}};
RR=Chop[G.RR];
QQ=QQ.G;
]
,{i,k,n-1}];
Chop[N[{QQ,RR}]]
]


(* ::Subsubsection::Closed:: *)
(*QR update - coulumn insertion*)


(* ::Text:: *)
(*input :*)
(* orthogonal matrix Q*)
(* upper triangular matrix R*)
(* vector vekt*)
(*output:*)
(* orthogonal matrix QQ*)
(* upper triangular matrix RR*)


Ubaci[Q_,R_,vekt_]:=Module[{m=Length[R],n=Length[R[[1]]]+1,RR=Transpose[Append[Transpose[R],Transpose[Q].vekt]],QQ=Q,a,b,G},
Do[
If[RR[[i+1,n]]!=0,
a=RR[[i,n]]/Sqrt[RR[[i,n]]^2+RR[[i+1,n]]^2];
b=RR[[i+1,n]]/Sqrt[RR[[i,n]]^2+RR[[i+1,n]]^2];
G=IdentityMatrix[m];
G[[i;;i+1,i;;i+1]]={{a,b},{b,-a}};
RR=Chop[G.RR];
QQ=QQ.G;
]
,{i,m-1,n,-1}];
Chop[N[{QQ,RR}]]
]


(* ::Subsubsection::Closed:: *)
(*Upper triangular matrix inverse*)


(* ::Text:: *)
(*input :*)
(* upper triangular matrix R*)
(*output:*)
(* upper triangular matrix InvR*)


InvGT[R_]:=Module[{InvR,i,j,k,n=Length[R]},
InvR=IdentityMatrix[n];
Do[
Do[
InvR[[i,j]]=(InvR[[i,j]]-Sum[R[[k,j]]*InvR[[i,k]],{k,i,j-1}])/R[[j,j]];
,{j,i,n}];
,{i,n,1,-1}];
InvR
]


(* ::Subsubsection::Closed:: *)
(*QR decomposition w/ pivoting*)


(* ::Text:: *)
(*input :*)
(* matrix A*)
(*output:*)
(* orthogonal matrix Q*)
(* upper triangular matrix R*)
(* permutation vector I*)


QRnez[A_]:=Module[{m=Length[A],n=Length[A[[1]]],Q,R=N[A],a,gamma,u,H,i,j,I,naj},
I=Range[n];
Q=IdentityMatrix[m];
Do[
naj={i,Norm[R[[i;;m,i]]]};
Do[If[naj[[2]]<Norm[R[[i;;m,j]]],naj={j,Norm[R[[i;;m,j]]]}],{j,i+1,n}];
I[[{i,naj[[1]]}]]=I[[{naj[[1]],i}]];
R[[All,{i,naj[[1]]}]]=R[[All,{naj[[1]],i}]];
a=R[[All,i]];
gamma=Norm[a[[i;;m]]];

u=Table[0.,{m}];
u[[i]]=a[[i]]+Sign[Sign[a[[i]]]+.5]*gamma;
Do[u[[j]]=a[[j]],{j,i+1,m}];
u/=Sqrt[2*gamma*(gamma+Abs[a[[i]]])];
H=IdentityMatrix[m]-2*Transpose[{u}].{u};
R=H.R;
Q=Q.H;
,{i,Min[{n,m}]}];
If[Chop[N[R[[Min[{n,m}]]]]]==0,
Print["\n\nNema nezavisnih!\n\n"];];
{Chop[Q],Chop[R],I}
]


(* ::Subsubsection::Closed:: *)
(*QR decomposition w/ pivoting except for the first column*)


(* ::Text:: *)
(*input :*)
(* matrix A*)
(*output:*)
(* orthogonal matrix Q*)
(* upper triangular matrix R*)
(* permutation vector I*)


QRnez2[A_]:=Module[{m=Length[A],n=Length[A[[1]]],Q,R=N[A],a,gamma,u,H,i,j,I,naj},
I=Range[n];
Q=IdentityMatrix[m];
Do[
naj={i,Norm[R[[i;;m,i]]]};
If[i>1,
Do[If[naj[[2]]<Norm[R[[i;;m,j]]],naj={j,Norm[R[[i;;m,j]]]}],{j,i+1,n}];
];
I[[{i,naj[[1]]}]]=I[[{naj[[1]],i}]];
R[[All,{i,naj[[1]]}]]=R[[All,{naj[[1]],i}]];
a=R[[All,i]];
gamma=Norm[a[[i;;m]]];

u=Table[0.,{m}];
u[[i]]=a[[i]]+Sign[Sign[a[[i]]]+.5]*gamma;
Do[u[[j]]=a[[j]],{j,i+1,m}];
u/=Sqrt[2*gamma*(gamma+Abs[a[[i]]])];
H=IdentityMatrix[m]-2*Transpose[{u}].{u};
R=H.R;
Q=Q.H;
,{i,Min[{n,m}]}];
If[Chop[R[[Min[{n,m}]]]]==0,
Print["\n\nNema nezavisnih!\n\n"];];
{Chop[Q],Chop[R],I}
]


(* ::Subsubsection::Closed:: *)
(*Weighted median*)


(* ::Text:: *)
(*input :*)
(* data vector:  podaci*)
(* nonnegative weights:   vector tezine*)
(*output:*)
(* weighted median*)
(* weight of the weighted median*)
(* index of the weighted median in the initial data podaci*)


TMed[podaci_,tezine_]:=Module[{d0,di,ii,midonji=1,migornji=Length[podaci]+1,s,sortpod,sorttezine,G},
s=Ordering[podaci];
sortpod=podaci[[s]];
sorttezine=tezine[[s]];
d0=-Total[sorttezine];
di={d0};
Do[AppendTo[di,di[[-1]]+2*sorttezine[[ii]]],{ii,Length[tezine]}];
Do[
If[di[[ii]]<0,midonji=ii];
If[di[[ii]]>0&&di[[ii]]<di[[migornji]],migornji=ii];
,{ii,Length[tezine]+1}];
If[migornji!=1,migornji-=1];
If[Chop[sortpod[[migornji]]]!=0,
{sortpod[[midonji]],sorttezine[[midonji]],s[[midonji]]},
{sortpod[[migornji]],sorttezine[[migornji]],s[[migornji]]}
]
];


(* ::Subsection::Closed:: *)
(*Algorithm 1*)


(* ::Text:: *)
(*input :*)
(* m*3 dimensional matrix \[CapitalLambda]*)
(*optional input:*)
(* maximum iterations integer maxiter*)
(*output:*)
(* a best LAD-parametars \[Alpha] and \[Beta]  *)


Alg1[\[CapitalLambda]_,maxiter_:20]:=Module[{\[ScriptCapitalL],\[Gamma],Ibez\[Mu],i,m=Length[\[CapitalLambda]],\[Mu],\[Nu],T\[Mu],x\[Mu],y\[Mu],z\[Mu],wi,tezine,podaci,rje,\[Alpha],\[Beta],iter=1},

(*step 1*)
\[ScriptCapitalL]=\[CapitalLambda][[All,1;;2]];
Do[If[\[ScriptCapitalL][[i]]!={0,0},\[Mu]=\[Nu]=i;Break[];],{i,m}];
T\[Mu]={x\[Mu],y\[Mu],z\[Mu]}=\[CapitalLambda][[\[Mu]]];
If[Chop[Norm[\[ScriptCapitalL].{-y\[Mu],x\[Mu]}]]==0,Print["Nisu ispunjeni uvjeti!"];Return[];];
\[Gamma]=0;

(*step 2*)
While[\[Gamma]!=\[Nu]&&iter<=maxiter,
iter++;
\[Gamma]=\[Mu];
\[Mu]=\[Nu];
T\[Mu]={x\[Mu],y\[Mu],z\[Mu]}=\[CapitalLambda][[\[Mu]]];
Ibez\[Mu]={};
wi={};
Do[AppendTo[wi,{-y\[Mu],x\[Mu]}.\[ScriptCapitalL][[i]]];If[Chop[wi[[i]]]!=0,AppendTo[Ibez\[Mu],i]],{i,m}];

(*step 3*)
tezine=Abs[wi[[Ibez\[Mu]]]];
If[y\[Mu]!=0,
podaci=Table[({-y\[Mu],z\[Mu]}.\[CapitalLambda][[Ibez\[Mu][[i]],{3,2}]])/wi[[Ibez\[Mu][[i]]]],{i,Length[Ibez\[Mu]]}];
rje=TMed[podaci,tezine];
\[Alpha]=rje[[1]];
\[Beta]=(z\[Mu]-x\[Mu]*\[Alpha])/y\[Mu];
,
podaci=Table[({-z\[Mu],x\[Mu]}.\[CapitalLambda][[Ibez\[Mu][[i]],{1,3}]])/wi[[Ibez\[Mu][[i]]]],{i,Length[Ibez\[Mu]]}];
rje=TMed[podaci,tezine];
\[Beta]=rje[[1]];
\[Alpha]=(z\[Mu]-y\[Mu]*\[Beta])/x\[Mu];
];
\[Nu]=Ibez\[Mu][[rje[[3]]]];
];
If[\[Gamma]!=\[Nu],
Print["Maximum iterations reached. Current point {\[Alpha],\[Beta]}=",{\[Alpha],\[Beta]}];
];
Return[{\[Alpha],\[Beta]}];
]


(* ::Subsection::Closed:: *)
(*Algorithm 2*)


(* ::Text:: *)
(*input :*)
(* m*2 dimensional matrix X*)
(* m dimensional vector z*)
(*optional input:*)
(* initial base index vector poc*)
(* maximum iterations integer maxiter*)
(*output:*)
(* all iterations of the algorithm up to a best LAD-solution vector a*)


Alg2[X_,z_,poc_:Null,maxiter_:20]:=Module[{IbezI1,IbezI2,i,j,k,m=Length[X],\[Mu]=0,\[Nu],\[CurlyTheta],\[CurlyTheta]1,\[CurlyTheta]2,a0,a1,d,d1,d2,B,Q,R,BInv,wi,tezine,podaci,rje,iter=1,aaa},

(*step 1*)
If[poc=!=Null && MatrixRank[X[[poc]]]==2,
{\[Mu],\[Nu]}=poc;
,
Do[Do[If[{-X[[i,2]],X[[i,1]]}.X[[j]]!=0,\[Mu]=i;\[Nu]=j;Break[];],{j,i+1,m}];
If[\[Mu]!=0,Break[];];
,{i,m-1}];
];
B={X[[\[Mu]]],X[[\[Nu]]]};
{Q,R}=QR[Transpose[B]];
BInv=Q.Transpose[InvGT[R]];
{d1,d2}=Transpose[BInv];
a0=BInv.z[[{\[Mu],\[Nu]}]];
aaa={{{\[Mu],\[Nu]},a0}};

(*step 2*)
While[iter<=maxiter,
iter++;
IbezI1={};
Do[If[Chop[d1.X[[i]]]!=0,AppendTo[IbezI1,i]],{i,m}];
wi=Table[d1.X[[IbezI1[[i]]]],{i,Length[IbezI1]}];
tezine=Abs[wi];
podaci=Table[(z[[IbezI1[[i]]]]-a0.X[[IbezI1[[i]]]])/(wi[[i]]),{i,Length[IbezI1]}];
rje=TMed[podaci,tezine];
\[CurlyTheta]1=rje[[1]];
If[Chop[\[CurlyTheta]1]!=0,
k=IbezI1[[rje[[3]]]];d=d1;\[CurlyTheta]=\[CurlyTheta]1;
{\[Mu],\[Nu]}={\[Nu],k};
{Q,R}=Izbaci[Q,R,1];
,

(*step 3*)
IbezI2={};
Do[If[Chop[d2.X[[i]]]!=0,AppendTo[IbezI2,i]],{i,m}];
wi=Table[d2.X[[IbezI2[[i]]]],{i,Length[IbezI2]}];
tezine=Abs[wi];
podaci=Table[(z[[IbezI2[[i]]]]-a0.X[[IbezI2[[i]]]])/(wi[[i]]),{i,Length[IbezI2]}];
rje=TMed[podaci,tezine];
\[CurlyTheta]2=rje[[1]];
If[Chop[\[CurlyTheta]2]==0,
Return[aaa];
,
k=IbezI2[[rje[[3]]]];d=d2;\[CurlyTheta]=\[CurlyTheta]2;
{\[Mu],\[Nu]}={\[Mu],k};
{Q,R}=Izbaci[Q,R,2];
];
];

(*step 4*)
a1=a0+\[CurlyTheta]*d;
{Q,R}=Ubaci[Q,R,X[[\[Nu]]]];
B=Transpose[Q.R];

(*step 5*)
BInv=Q.Transpose[InvGT[R]];
{d1,d2}=Transpose[BInv];
a0=a1;
AppendTo[aaa,{{\[Mu],\[Nu]},a0}];
];
aaa
]


(* ::Subsection::Closed:: *)
(*Algorithm 3*)


(* ::Text:: *)
(*input :*)
(* m*n dimensional matrix X*)
(* m dimensional vector z*)
(*optional input:*)
(* initial base index vector poc*)
(* maximum iterations integer maxiter*)
(* numerical precision integer prec*)
(*output:*)
(*a best LAD-solution vector a*)


Alg3[X_,z_,poc_:Null,maxiter_:200,prec_:10]:=Module[{i,j,k,m=Length[X],n=Length[X[[1]]],IB,IBpot,I0,I0bezIB,I1,I2,I0bezI2IB,II,IbezI0,B,Bpot,Q,R,BInv,BInvT,a,h,j0,wi,roi,rje,iter=1,zB,ni,theta,X0,ll,uvjeti,step3},

(*step 1*)
If[poc===Null,
IB=QRnez[Transpose[X]][[3,1;;n]];
,
IB=poc;
];
B=X[[IB]];
zB=z[[IB]];
{Q,R}=QR[Transpose[B]];
BInv=Q.Transpose[InvGT[R]];
BInvT=Transpose[BInv];
a=BInv.zB;
I0={};
Do[If[Chop[z[[i]]-a.X[[i]],10.^(-prec)]==0,AppendTo[I0,i]],{i,m}];
IbezI0=Complement[Range[m],I0];
h=Table[0,{n}]+Sum[Sign[z[[IbezI0[[i]]]]-a.X[[IbezI0[[i]]]]]*X[[IbezI0[[i]]]],{i,Length[IbezI0]}];

While[iter<=maxiter,
iter++;

(*step 2*)
j0=0;
I0bezIB=Complement[I0,IB];
Do[
If[Abs[BInvT[[j]].h]>1+Sum[Abs[BInvT[[j]].X[[I0bezIB[[i]]]]],{i,Length[I0bezIB]}],
j0=IB[[j]];k=j;
Break[];
];
,{j,n}];
Label[step3];
If[j0!=0,

(*step 3*)
I1={};
Do[If[Chop[BInvT[[k]].X[[IbezI0[[i]]]],10.^(-prec)]!=0,AppendTo[I1,IbezI0[[i]]]],{i,Length[IbezI0]}];
II=Join[I0,I1];
wi=Table[Chop[Abs[BInvT[[k]].X[[II[[i]]]]],10.^(-prec)],{i,Length[II]}];
roi=Join[Table[0,{i,Length[I0]}],Table[(z[[I1[[i]]]]-a.X[[I1[[i]]]])/(BInvT[[k]].X[[I1[[i]]]]),{i,Length[I1]}]];
rje=TMed[roi,wi];
ni=II[[rje[[3]]]];
theta=rje[[1]];
a+=theta*BInvT[[k]];
,

I2={};
(*step 5*)
X0=Transpose[X[[I0]]];
ll=Table[ToExpression["l"<>ToString[i]],{i,Length[I0]}];
uvjeti=Flatten[Join[{X0.ll==h},Table[-1<=ll[[i]]<=1,{i,Length[ll]}]]];
If[Depth[FindInstance[And@@uvjeti,ll]]>2,
Return[a];
,
I0bezI2IB=Complement[Complement[I0,I2],IB];
While[Length[I0bezI2IB]>0,

(*step 6*)
ni=I0bezI2IB[[1]];
AppendTo[I2,ni];
IBpot=Prepend[IB,ni];
Bpot=X[[IBpot]];
rje=QRnez2[Transpose[Bpot]];
{Q,R}=Izbaci[rje[[1]],rje[[2]],n+1];
IB=IBpot[[rje[[3,1;;n]]]];
B=X[[IB]];
zB=z[[IB]];
BInv=Q.Transpose[InvGT[R]];
BInvT=Transpose[BInv];

j0=0;
I0bezIB=Complement[I0,IB];
Do[
If[Abs[BInvT[[j]].h]>1+Sum[Abs[BInvT[[j]].X[[I0bezIB[[i]]]]],{i,Length[I0bezIB]}],
j0=IB[[j]];k=j;
Break[];
];
,{j,n}];
If[j0!=0,
Goto[step3];
,
I0bezI2IB=Complement[Complement[I0,I2],IB];
];
];
Print["Error!"];
Return[];
];
];

(*step 4*)
{Q,R}=Izbaci[Q,R,k];
{Q,R}=Ubaci[Q,R,X[[ni]]];
IB=Append[Delete[IB,k],ni];
B=Transpose[Q.R];
BInv=Q.Transpose[InvGT[R]];
BInvT=Transpose[BInv];
I0={};
Do[If[Chop[z[[i]]-a.X[[i]],10.^(-prec)]==0,AppendTo[I0,i]],{i,m}];
IbezI0=Complement[Range[m],I0];
h=Table[0,{n}]+Sum[Sign[z[[IbezI0[[i]]]]-a.X[[IbezI0[[i]]]]]*X[[IbezI0[[i]]]],{i,Length[IbezI0]}];
];
Print["Maximum iterations reached. Current point a=",a];
Return[a];
]


(* ::Section:: *)
(*Examples*)


(* ::Subsection::Closed:: *)
(*The minimizing function*)


F[al_,X_,z_]:=Total[Abs[z-X.al]]


(* ::Subsection::Closed:: *)
(*Algorithm 1 examples*)


(* ::Subsubsection:: *)
(*Example 1*)


(* ::Input:: *)
(*SeedRandom[3];*)
(*m=50;*)
(*\[CapitalLambda]=RandomInteger[{0,2m},{m,3}]//N*)
(*X=\[CapitalLambda][[All,{1,2}]];*)
(*z=\[CapitalLambda][[All,3]];*)


(* ::Input:: *)
(*rj=Alg1[\[CapitalLambda]];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 1:",{F[rj,X,z],rj}];*)


(* ::Subsubsection:: *)
(*Example 2 (Sabo)*)


(* ::Input:: *)
(*\[CapitalLambda]={{8,4,4},{2,-1,1},{4,3,0},{-2,6,6}};*)
(*X=\[CapitalLambda][[All,{1,2}]];*)
(*z=\[CapitalLambda][[All,3]];*)


(* ::Input:: *)
(*rj=Alg1[\[CapitalLambda]];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 1:",{F[rj,X,z],rj}];*)


(* ::Subsection::Closed:: *)
(*Algorithm 2 examples*)


(* ::Subsubsection::Closed:: *)
(*Example 1*)


(* ::Input:: *)
(*SeedRandom[3];*)
(*m=50;*)
(*\[CapitalLambda]=RandomInteger[{0,2m},{m,3}]//N;*)
(*X=\[CapitalLambda][[All,{1,2}]]*)
(*z=\[CapitalLambda][[All,3]]*)


(* ::Input:: *)
(*rj=Alg2[X,z][[-1,2]];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 2:",{F[rj,X,z],rj}];*)


(* ::Subsubsection::Closed:: *)
(*Example 2 (Sabo)*)


(* ::Input:: *)
(*X={{8,4},{2,-1},{4,3},{-2,6}};*)
(*z={4,1,0,6};*)


(* ::Text:: *)
(*In this illustration of the problem the numbers near the edges of the image denote the ordinal number of the equation and the numbers on the intersections are the values of the function which is minimized.*)
(*Below the picture is the result of minimization using Mathematica-s function NMinimize.*)


(* ::Input:: *)
(*m=Length[X];*)
(*TT={};*)
(*Do[bla=NSolve[X[[{i,j}]].{x1,x2}==z[[{i,j}]],{x1,x2}];*)
(*If[Depth[bla]==4&&Length[bla[[1]]]==2,AppendTo[TT,{x1,x2}/.bla[[1]]]];*)
(*,{i,m-1},{j,i+1,m}];*)
(*TT=Union[Round[TT,0.000001]];*)
(*pol={TT[[1]]};*)
(*Do[*)
(*If[Chop[F[TT[[i]],X,z]-F[pol[[-1]],X,z]]==0,AppendTo[pol,TT[[i]]]];*)
(*If[Chop[F[TT[[i]],X,z]-F[pol[[-1]],X,z]]<0,pol={TT[[i]]};];*)
(*,{i,Length[TT]}];*)
(*slpol=Graphics[{RGBColor[1,1,0],Opacity[.5],Polygon[pol]}];*)
(*slpod=Show[{*)
(*Table[ContourPlot[X[[i]].{x1,x2}==z[[i]],{x1,-1,2},{x2,-2.5,2},Frame->False,ColorFunction->(Gray&),ContourLabels->({RGBColor[0,0,1],Text[Style[i,Bold],{#1+.1,#2}]}&)],{i,m}],*)
(*slpol,*)
(*Graphics[Table[Text[Round[F[TT[[i]],X,z],.1],TT[[i]],{0,-1}],{i,Length[TT]}]]},ImageSize->700,AspectRatio->1/2,Axes->True,TicksStyle->Directive[Orange,Opacity[.5]],AxesStyle->Directive[Orange,Opacity[.5]]]*)
(*Print["NMinimize:",NMinimize[F[{a1,a2},X,z],{a1,a2}]]*)


(* ::Text:: *)
(*In the table below we illustrate how the algorithm works starting from four different starting points.*)
(*In every iteration we show the current point and the function value in it.*)


(* ::Input:: *)
(*AAA={Alg2[X,z,{2,3},10],Alg2[X,z,{2,4},10],Alg2[X,z,{3,4},10],Alg2[X,z,{2,1},10]};*)
(*TableForm[Table[Table[{{AAA[[i,j,2,1]],AAA[[i,j,2,2]]},F[AAA[[i,j,2]],X,z]},{j,Length[AAA[[i]]]}],{i,Length[AAA]}],TableHeadings->{{"1.","2.","3.","4."},{"Initial","1st iteration","2nd iteration"}},TableDepth->2]*)
(*TableForm[*)
(*Table[*)
(*a=AAA[[k,-1,2]];*)
(*I0={};*)
(*Do[If[Chop[z[[i]]-X[[i]].a]==0,AppendTo[I0,i]],{i,Length[X]}];*)
(*IbezI0=Complement[Range[Length[X]],I0];*)
(*ha=Sum[Sign[z[[IbezI0[[i]]]]-X[[IbezI0[[i]]]].a]*X[[IbezI0[[i]]]],{i,Length[IbezI0]}];*)
(*ll=Table[ToExpression["l"<>ToString[i]],{i,Length[I0]}];*)
(*uvjeti=Flatten[Join[{Transpose[X[[I0]]].ll==ha},Table[-1<=ll[[i]]<=1,{i,Length[ll]}]]];*)
(*FindInstance[And@@uvjeti,ll]*)
(*,{k,4}],TableHeadings->{{"1.","2.","3.","4."},{"\!\(\*SuperscriptBox[\"\[Lambda]\", \"*\"]\)(\!\(\*SubscriptBox[\"I\", \"0\"]\))"}},TableDepth->2]*)


(* ::Text:: *)
(*Pictures below are illustrations of the algorithm iterations above.*)


(* ::Input:: *)
(*slike=GraphicsGrid[Table[aaa=AAA[[2i+j]];*)
(*slpod=Show[Table[ContourPlot[X[[i]].{x1,x2}==z[[i]],{x1,-1,2},{x2,-2.5,2},Frame->False,ColorFunction->(Gray&)],{i,m}]];*)
(*slF=Graphics[Table[Text[F[TT[[i]],X,z],TT[[i]],{0,-1}],{i,Length[TT]}]];*)
(*plave=Graphics[{{RGBColor[0,0,1],Thickness[.003],Arrowheads[.02],Table[Arrow[{aaa[[i,2]],aaa[[i+1,2]]-.015(aaa[[i+1,2]]-aaa[[i,2]])/Norm[aaa[[i+1,2]]-aaa[[i,2]]]}],{i,Length[aaa]-1}]},{PointSize[.015],Point/@aaa[[All,2]]}}];*)
(*Show[{slpod,slpol,plave,slF},ImageSize->400,Axes->True,TicksStyle->Directive[Orange,Opacity[.5]],AxesStyle->Directive[Orange,Opacity[.5]],AspectRatio->1/2],{i,0,1},{j,2}]]*)


(* ::Subsubsection::Closed:: *)
(*Example 4*)


(* ::Text:: *)
(*This example shows the degenerative case.*)


(* ::Input:: *)
(*X={{-2,-1},{-1,0},{0,-3},{0,-1},{1,-2},{1,-1},{1,1},{2,-3},{2,-2},{2,0}};*)
(*z={-5,-2,-9,-1,-2,0,4,-1,2,2};*)


(* ::Text:: *)
(*This shows the result of minimization using Mathematica-s function NMinimize.*)
(*In the table beneath we illustrate how the algorithm works starting from four different starting points.*)
(*In every iteration we show the current point and the function value in it.*)


(* ::Input:: *)
(*Print["NMinimize:",NMinimize[F[{a1,a2},X,z],{a1,a2}]]*)
(*AAA={Alg2[X,z,{9,4},10],Alg2[X,z,{4,10},10],Alg2[X,z,{3,9},10],Alg2[X,z,{3,10},10]};*)
(*TableForm[Table[Table[{{AAA[[i,j,2,1]],AAA[[i,j,2,2]]},F[AAA[[i,j,2]],X,z]},{j,Length[AAA[[i]]]}],{i,Length[AAA]}],TableHeadings->{{"1.","2.","3.","4."},{"Initial","1st iteration","2nd iteration"}},TableDepth->2]*)
(*TableForm[*)
(*Table[*)
(*a=AAA[[k,-1,2]];*)
(*I0={};*)
(*Do[If[Chop[z[[i]]-X[[i]].a]==0,AppendTo[I0,i]],{i,Length[X]}];*)
(*IbezI0=Complement[Range[Length[X]],I0];*)
(*ha=Sum[Sign[z[[IbezI0[[i]]]]-X[[IbezI0[[i]]]].a]*X[[IbezI0[[i]]]],{i,Length[IbezI0]}];*)
(*ll=Table[ToExpression["l"<>ToString[i]],{i,Length[I0]}];*)
(*uvjeti=Flatten[Join[{Transpose[X[[I0]]].ll==ha},Table[-1<=ll[[i]]<=1,{i,Length[ll]}]]];*)
(*FindInstance[And@@uvjeti,ll]*)
(*,{k,4}],TableHeadings->{{"1.","2.","3.","4."},{"\!\(\*SuperscriptBox[\"\[Lambda]\", \"*\"]\)(\!\(\*SubscriptBox[\"I\", \"0\"]\))"}},TableDepth->2]*)


(* ::Text:: *)
(*In the illustration below the first picture depicts the problem.*)
(*The other pictures show the algorithm iterations above (blue arrows) and the optimal paths to the solution (green arrows).*)


(* ::Input:: *)
(*m=Length[X];*)
(*TT={};*)
(*Do[bla=NSolve[X[[{i,j}]].{x1,x2}==z[[{i,j}]],{x1,x2}];*)
(*If[Depth[bla]==4&&Length[bla[[1]]]==2,AppendTo[TT,{x1,x2}/.bla[[1]]]];*)
(*,{i,m-1},{j,i+1,m}];*)
(*TT=Union[Round[TT,10.^(-12)]];*)
(*pol={TT[[1]]};*)
(*Do[*)
(*If[Chop[F[TT[[i]],X,z]-F[pol[[-1]],X,z]]==0,AppendTo[pol,TT[[i]]]];*)
(*If[Chop[F[TT[[i]],X,z]-F[pol[[-1]],X,z]]<0,pol={TT[[i]]};];*)
(*,{i,Length[TT]}];*)
(*slpol=Graphics[{RGBColor[1,1,0],Opacity[.5],Polygon[pol]}];*)
(*sl11=Show[{slpol,*)
(*Table[If[i!=2&&i!=10,*)
(*ContourPlot[X[[i]].{x1,x2}==z[[i]],{x1,-.25,5},{x2,-.25,4},Frame->False,ColorFunction->(Gray&),ContourLabels->({RGBColor[0,0,1],Text[Style[i,Bold],{#1+.1,#2}]}&)],*)
(*ContourPlot[X[[i]].{x1,x2}==z[[i]],{x1,-.25,5},{x2,-.25,4},Frame->False,ColorFunction->(Gray&),ContourLabels->({RGBColor[0,0,1],Text[Style[i,Bold],{#1+.1,3.9}],Text[Style[i,Bold],{#1+.1,-.2}]}&)]*)
(*],{i,m}],*)
(*Graphics[Table[Text[Round[F[TT[[i]],X,z],.1],TT[[i]],{0,-1}],{i,Length[TT]}]]},AspectRatio->Automatic,Axes->True,TicksStyle->Directive[Orange,Opacity[.5]],AxesStyle->Directive[Orange,Opacity[.5]]];*)
(*zelene={Graphics[{RGBColor[0,1,0],Thickness[.001],Arrowheads[.02],Arrow[{AAA[[1,1,2]],pol[[2]]}],Arrow[{AAA[[1,1,2]],pol[[3]]}]}],*)
(*Graphics[{RGBColor[0,1,0],Thickness[.001],Arrowheads[.02],Arrow[{AAA[[2,1,2]],pol[[1]]}],Arrow[{AAA[[2,1,2]],pol[[2]]}]}],*)
(*Graphics[{RGBColor[0,1,0],Thickness[.001],Arrowheads[.02],Arrow[{AAA[[3,1,2]],pol[[4]]}],Arrow[{AAA[[3,1,2]],pol[[3]]}]}],*)
(*Graphics[{RGBColor[0,1,0],Thickness[.001],Arrowheads[.02],Arrow[{AAA[[4,1,2]],pol[[1]]}],Arrow[{AAA[[4,1,2]],pol[[4]]}]}]};*)
(*sldod={ContourPlot[{X[[4]].{x1,x2}==z[[4]],X[[9]].{x1,x2}==z[[9]]},{x1,-.25,4.25},{x2,-.25,3.25},Frame->False,ColorFunction->(Yellow&)],ContourPlot[{X[[4]].{x1,x2}==z[[4]],X[[10]].{x1,x2}==z[[10]]},{x1,-.25,4.25},{x2,-.25,3.25},Frame->False,ColorFunction->(Yellow&)],Graphics[],Graphics[]};*)
(*slike=Table[aaa=AAA[[2i+j]];*)
(*slpod=Show[Table[ContourPlot[X[[i]].{x1,x2}==z[[i]],{x1,-.25,5},{x2,-.25,4},Frame->False,ColorFunction->(Gray&)],{i,m}]];*)
(*slF=Graphics[Table[Text[F[TT[[i]],X,z],TT[[i]],{0,-1}],{i,Length[TT]}]];*)
(*plave=Graphics[{{RGBColor[0,0,1],Thickness[.003],Arrowheads[.02],Table[Arrow[{aaa[[i,2]],aaa[[i+1,2]]-.015(aaa[[i+1,2]]-aaa[[i,2]])/Norm[aaa[[i+1,2]]-aaa[[i,2]]]}],{i,Length[aaa]-1}]},{PointSize[.015],Point/@aaa[[All,2]]}}];*)
(*Show[{slpol,slpod,zelene[[2*i+j]],plave,slF},ImageSize->400,Axes->True,TicksStyle->Directive[Orange,Opacity[.5]],AxesStyle->Directive[Orange,Opacity[.5]],AspectRatio->Automatic],{i,0,1},{j,2}];*)
(*slike[[1]]={sl11,aaa=AAA[[1]];*)
(*slpod=Show[Table[ContourPlot[X[[i]].{x1,x2}==z[[i]],{x1,-.25,5},{x2,-.25,4},Frame->False,ColorFunction->(Gray&)],{i,m}]];*)
(*slF=Graphics[Table[Text[F[TT[[i]],X,z],TT[[i]],{0,-1}],{i,Length[TT]}]];*)
(*plave1=Graphics[{PointSize[.015],Point/@aaa[[All,2]]}];*)
(*aaa=AAA[[2]];*)
(*plave2=Graphics[{PointSize[.015],Point/@aaa[[All,2]]}];*)
(*Show[{slpol,slpod,zelene[[1]],zelene[[2]],plave1,plave2,slF},ImageSize->400,Axes->True,TicksStyle->Directive[Orange,Opacity[.5]],AxesStyle->Directive[Orange,Opacity[.5]],AspectRatio->Automatic]};*)
(*GraphicsGrid[slike,ImageSize->800]*)


(* ::Text:: *)
(*In the table below *)
(*\!\(\*SuperscriptBox["\[Lambda]", "*"]\)(Subscript[I, 0]) for each of the characteristic solutions are calculated.*)


(* ::Input:: *)
(*TableForm[*)
(*Table[{a=pol[[k]];*)
(*a,*)
(*I0={};*)
(*Do[If[Chop[z[[i]]-X[[i]].a]==0,AppendTo[I0,i]],{i,Length[X]}];*)
(*IbezI0=Complement[Range[Length[X]],I0];*)
(*ha=Sum[Sign[z[[IbezI0[[i]]]]-X[[IbezI0[[i]]]].a]*X[[IbezI0[[i]]]],{i,Length[IbezI0]}];*)
(*ll=Table[ToExpression["l"<>ToString[i]],{i,Length[I0]}];*)
(*uvjeti=Flatten[Join[{Transpose[X[[I0]]].ll==ha},Table[-1<=ll[[i]]<=1,{i,Length[ll]}]]];*)
(*Print["Intersecting lines:",I0,"\nConditions:",uvjeti];*)
(*FindInstance[And@@uvjeti,ll]}*)
(*,{k,4}],TableHeadings->{{"1.","2.","3.","4."},{"a","\!\(\*SuperscriptBox[\"\[Lambda]\", \"*\"]\)(\!\(\*SubscriptBox[\"I\", \"0\"]\))"}},TableDepth->2]*)


(* ::Subsection:: *)
(*Algorithm 3 examples*)


(* ::Subsubsection::Closed:: *)
(*Example 1*)


(* ::Input:: *)
(*X={{1,1,1},{3,3,1},{0,0,3},{3,3,3},{2,3,1}};*)
(*z={1,2,0,4,2};*)


(* ::Text:: *)
(*Below is the comparison of Mathematica function NMinimize and our Algorithm 3.*)


(* ::Input:: *)
(*rj=Alg3[X,z];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 3:",{F[rj,X,z],rj}];*)


(* ::Subsubsection::Closed:: *)
(*Example 2 (Sabo)*)


(* ::Input:: *)
(*X={{8,4},{2,-1},{4,3},{-2,6}};*)
(*z={4,1,0,6};*)


(* ::Text:: *)
(*Below is the comparison of Mathematica function NMinimize and our Algorithm 3.*)


(* ::Input:: *)
(*rj=Alg3[X,z];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 3:",{F[rj,X,z],rj}];*)


(* ::Subsubsection::Closed:: *)
(*Example 3 {Coleman}*)


(* ::Text:: *)
(*This example illustrates a LAD - problem of finding a polynomial approximation of the function.*)


(* ::Input:: *)
(*f[x_]:=Exp[x]+If[1.2<= x<= 1.4,5,0]*)
(*slf=Plot[f[x],{x,1,2},PlotRange->{2,9.5},PlotStyle->{RGBColor[0,0,1],Opacity[.3]}];*)
(*nnn=10;m=50; sig=0.5;SeedRandom[25]*)
(*x=y=Table[1+i /m,{i,m}]; *)
(*error=RandomReal[NormalDistribution[0,sig^2],m];*)
(*Do[y[[i]]=f[x[[i]]]+error[[i]],{i,m}];*)
(*tab=Table[{x[[i]],f[x[[i]]],error[[i]],y[[i]]},{i,m}];*)
(*(*Transpose[tab]//TableForm *)*)
(*pod=Table[{x[[i]],y[[i]]},{i,m}];*)
(*slpod=ListPlot[pod,PlotRange->{2,9.5},PlotStyle->{AbsolutePointSize[4]}];*)
(*Show[slpod,slf,ImageSize->300,AspectRatio->.5,Axes->True,TicksStyle->Directive[Orange,Opacity[.5]],AxesStyle->Directive[Orange,Opacity[.5]]]*)
(**)
(*(* Definiranje sustava *)*)
(*X=Table[Table[x[[i]]^(j-1)//N,{j,nnn}],{i,m}];*)
(*(* Print[Transpose[X]//TableForm]*)*)
(*z=y;*)
(**)
(*rj=Alg3[X,z,,200,6];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 3:",{F[rj,X,z],rj}];*)


(* ::Subsubsection::Closed:: *)
(*Example 4*)


(* ::Text:: *)
(*This is the same Example 4 from the Algorithm 2 examples.*)


(* ::Input:: *)
(*Graphics[{{RGBColor[1, 1, 0], Opacity[0.5], Polygon[{{1.666666666667, 1.666666666667}, {1.75, 1.5}, {2., 1.666666666667}, {2., 2.}}]}, GraphicsComplex[CompressedData["*)
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(* ::Input:: *)
(*X={{-2,-1},{-1,0},{0,-3},{0,-1},{1,-2},{1,-1},{1,1},{2,-3},{2,-2},{2,0}};*)
(*z={-5,-2,-9,-1,-2,0,4,-1,2,2};*)


(* ::Text:: *)
(*Algorithm 3 starting from intersection of lines 4 and 10.*)


(* ::Input:: *)
(*rj=Alg3[X,z,{4,10}];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 3:",{F[rj,X,z],rj}];*)


(* ::Text:: *)
(*Algorithm 3 starting from intersection of lines 5 and 10.*)


(* ::Input:: *)
(*rj=Alg3[X,z,{5,10}];*)
(*a=Table[ToExpression["a"<>ToString[i]],{i,Length[X[[1]]]}];*)
(*Print["NMinimize:",NMinimize[F[a,X,z],a]];*)
(*Print["Algoritam 3:",{F[rj,X,z],rj}];*)