fraa<-function(missD,Ep,L=2,iter=5,verbose=T)
{#Estimate missing values in matrix missD
 #Ep   =initial solution
 #value: estimate
 # or, uncomment last lines at end of this file to get
 #list: estimate = matrix of estimate
 #      U = matrix from final iteration of svd(A)
 #      sing.values=singular values of Ep, final iteration
 #
 #6 November 2003/17 August 2004
##############set-up
   M<-dim(missD)[2]
   N<-dim(missD)[1]
   if (L > M)
        stop("L must be less than or equal to the number of columns of E")
  
  #Check to see if any rows are all missing
      x<-(1:N)[apply(missD,1,function(A){all(is.na(A))})]
       if (length(x) > 0)
         {
          cat("Rows ", x, " are all NA's\n")
          stop("Stop and remove these rows\n")
         }


    missing<-which(is.na(missD))
    m<-length(missing)
    m2<-m*m


###############begin algorithm


   Xp1<-matrix(0,N,M)
   track<-iter


    while (iter > 0)
   { 
     A<-t(Ep)%*%Ep


      #find singular value decomposition


     svd.decomp<-svd(A)
     U<-svd.decomp$u
     S2<-svd.decomp$d


     #Singular values of Ep
      singular <- sqrt(S2)
       partial.sum<-sum(S2[L:M])
      total.sum<-sum(S2)


      fraction<-partial.sum/total.sum
      if (verbose)
      cat("\n Iteration ", track-iter+1, "\n", fraction, "\n") 


################construct Bp
    
     B<-matrix(0,m,m)
    
     i  <-row(missD)[which(is.na(missD))]
     j  <-col(missD)[which(is.na(missD))]
   
      for (s in 1:m)
        {
         for (t in s:m)
           {
           if (i[s]==i[t])
              {
                B[s,t]<-  U[j[s],L:M]%*%U[j[t],L:M]
                B[t,s]<-B[s,t]
               } #end if          
          } #end for t
        }   #end for s


##########construct W
   W<-numeric(m)
   for (t in 1:m)
    {
     K<-matrix(0,N,M)
     K[missing[t]]<-1
     W[t]<-sum(diag(t(U[,L:M])%*%t(Ep)%*%K%*%U[,L:M]))
    
    }
    # QR decomposition
     qrB <-qr(B)


     xp1<- - qr.coef(qrB,W)



    Xp1[missing]<- xp1


    Ep<- Ep+Xp1


   #set counter
     iter <- iter - 1



} #End while
  


#############################################
  if (verbose)
  {
  cat(" Singular values (final iteration):\n", singular)
  cat("\n")
  }


  #If we need the eigenvectors and singular values, uncomment
  #the next 2 lines
  #temp<-list(estimate=Ep,U=U,singular=singular)
  # return(temp)


  return(Ep)


}