Algorithm6_2.m

function [ UO,VO,UI,VI,Uinv,Vinv,S ] = Algorithm6_2( UO,VO,UI,VI,Uinv,Vinv,S,x,j,tol,maxrank )
%Given extended r-LRA UOUISVI^TVO^T compute extended r-LRA
%UOUISVI^TVO^T+xe_j with truncation criterion (truncate largest triple)
rows=length(UO);
m=UI'*(UO'*x);
r=length(m);
p=x-UO*(UI*m);
normp=norm(p);
n=VI'*(VO(j,:))';
q=-VO*(VI*n);
q(j)=q(j)+1;
normq=norm(q);
if ~exist('maxrank','var')
     % maxrank does not exist, so default is sqrt(number of rows)
      maxrank = sqrt(rows);
 end
if normp > tol && normq > tol
   
    zerovector=zeros(r,1);
    K=[S+m*n' m*normq;
        n'*normp normp*normq];
    [L,S,R]=svd(K);
    if r+1<=maxrank
    UO=[UO,p/normp];
    VO=[VO,q/normq];
    
    UI=[UI zerovector;
        zerovector' 1]*L;
    VI=[VI zerovector;
        zerovector' 1]*R;
    Uinv=L'*[Uinv zerovector;
            zerovector' 1];
    Vinv=R'*[Vinv zerovector;
            zerovector' 1];
    else
        [ UO,UI,Uinv] = Algorithm5( [UO,p/normp],UI,Uinv,L );
        [ VO,VI,Vinv] = Algorithm5( [VO,q/normq],VI,Vinv,R );

        
        S=S(1:r,1:r);
    end
elseif normp > tol && normq < tol
    K=[S+m*n';n'*normp];
    [L,S,R]=svd(K);
    UO=[UO,p/normp];
    [UO,UI,Uinv]=Algorithm5( UO,UI,Uinv,L );
    VI=VI*R;
    Vinv=R'*Vinv;
    S=S(1:r,:);
elseif normp < tol && normq > tol
    K=[S+m*n' normq*m];
    [L,S,R]=svd(K);
    VO=[VO,q/normq];
    [VO,VI,Vinv]=Algorithm5( VO,VI,Vinv,R );
    UI=UI*L;
    Uinv=L'*Uinv;
    S=S(:,1:r);
else 
    K=S+m*n';
    [L,S,R]=svd(K);
    [UO,UI,Uinv]=Algorithm5( UO,UI,Uinv,L );
    [VO,VI,Vinv]=Algorithm5( VO,VI,Vinv,R );
    S=S(1:end-1,1:end-1);
end

        



end