factpca<-function(x){
#Performs Principal Components Factor Analysis on x#
#using the correlation matrix.  
#It returns the un-rotated loadings.#
xeig<-eigen(cor(x))
xval<-xeig$values
xvec<-xeig$vectors
for (i in 1:length(xval)){
 xvec[,i]<-xvec[,i]*sqrt(xval[i])}
rownames(xvec)<-colnames(x)
return(xvec)
}


factpf<-function(x){
#Performs Principal Factor Factor Analysis on x#
#using the correlation matrix.  
#It returns the un-rotated loadings.#
n<-ncol(x)
redcormat<-cor(x)
diag(redcormat)<-apply(abs(cor(x)-diag(1,nrow=n,ncol=n)),2,max)
xeig<-eigen(redcormat)
xval<-xeig$values
xvec<-xeig$vectors
for (i in 1:length(xval[xval>0])){
 xvec[,i]<-xvec[,i]*sqrt(xval[i])}
rownames(xvec)<-colnames(x)
return(xvec[,xval>0])
}


factiter<-function(x,niter=10,maxfactors=ncol(x)){
#Performs Iterated Principal Factor Factor Analysis on x#
#using the correlation matrix.#  
#It returns the communalties after each iteration#
#and the final un-rotated loadings.#
n<-ncol(x)
temp<-matrix(0,nrow=n,ncol=n)
comm<-matrix(0,nrow=niter+1,ncol=n)
y<-factpf(x)
m<-ncol(y)
temp[1:n,1:m]<-y
comm[1,]<-apply(temp^2,1,sum)                 
for (i in 2:(niter+1)){
  redcormat<-cor(x)
  diag(redcormat)<-comm[i-1,]
  xeig<-eigen(redcormat)
  m<-min(maxfactors,length(xeig$values[xeig$values>0]))
  for (j in 1:m){
     xeig$vectors[,j]<-xeig$vectors[,j]*sqrt(xeig$values[j])} 
  temp[1:n,1:m]<-xeig$vectors[1:n,1:m]
  comm[i,]<-apply(temp[1:n,1:m]^2,1,sum)
}
f.loadings<-temp[1:n,1:m]
rownames(f.loadings)<-colnames(x)
return(comm,f.loadings)
}


testfit<-function(x,loadings){
#Returns the estimated Cov(error) matrix#
error<-cor(x)-loadings%*%t(loadings)
return(round(error,4))}

fitsummary<-function(x,loadings){
#Returns the estimated Var(error), a summary of #
#the covariances of the errors, and the estimated#
#communalities#
error<-cor(x)-loadings%*%t(loadings)
errorvar<-round(diag(error),4)
errorcov<-round(summary(error[upper.tri(error)]),4)
communality<-round(apply(loadings^2,1,sum),4)
return(errorvar,errorcov,communality)}