####################
## Gauss Jordan Box1 -- (2020) Dr. W.Lindner, Leichlingen GE
####################

Em(k,i,j) = do( M=unit(n,n), M[j,i]=k, M)    

-- Row interchange
Gi(i,j)=do( Gii=unit(n),
           N= Gii[i],
           M= Gii[j],
           Gii[j]=N,
           Gii[i]=M,
           Gii)

----      GAUSS matrix I, elimination downstairs
Gm(k,A) = do( n   = dim(A,1), -- get the dim
              Gmm = unit(n),  -- k th column
              for(i,k+1,n, Gmm[i,k] = - A[i,k]/A[k,k]),
              Gmm )

----       GAUSS matrix II, elimination upstairs
Gu(k,A) = do( n   = dim(A,1), -- get the dim
              Gmm = unit(n),  -- k th column
              for(i,k,1, Gmm[i,k] = - A[i,k]/A[k,k]),
              Gmm )


----    NORMalization of diagonal
Gn(A) = do( n = dim(A,1), -- get the dim
               Gmm = unit(n),  -- k th column
               for(k,1,n, Gmm[k,k] = 1/A[k,k]),
               Gmm )

----     GAUSS elimination 
Ge(A) =  do( n=dim(A,1),
              X = zero(dim(A,1),dim(A,1),dim(A,2)),
              X[1]=A,
              for( i,1,n-1, X[i+1]=dot(Gm(i,X[i]),X[i]) ), 
              X)

----     JORDAN elimination
Gj(A) =  do( n=dim(A,1),
              X = zero(dim(A,1),dim(A,1),dim(A,2)),
              X[n]=A,
              for( i,n-1,1, X[i]=dot(Gu(i+1,X[i+1]),X[i+1]) ), 
              X)

----     Row Reduced Echelon Form
RREF(A)= do( n = dim(A,1),
             U = Ge(A),              
             V = Gj(U[n]),        
             X=dot(Gn(V[1]),V[1]),
             X)    

backSubst(U) = do( n = dim(U,1),    
                   Z = zero(2,n),          
                   x = Z[1],                       
                   x[n] = U[n,n+1] / U[n,n],  
                   for(i,n-1,1, 
                    x[i] = (U[i,n+1] - sum(j,i+1,n, U[i,j]*x[j])) /U[i,i]),
                   x)

----     syntactic sugar
doGm(j, M) = dot(Gm(j, M), M)  
doGu(j, M) = dot(Gu(j, M), M)
doGn(M) = dot(Gn( M), M)
doGi(i,j,M) = dot(Gi(i,j,M), M)
doGn(M) = dot(Gn(M), M)

################### End of gjBox1 ###################