EMGLLF_R = function(phiInit,rhoInit,piInit,gamInit,mini,maxi,gamma,lambda,X,Y,tau)
{
  # Matrix dimensions
  n = dim(X)[1]
  p = dim(phiInit)[1]
  m = dim(phiInit)[2]
  k = dim(phiInit)[3]

  # Outputs
  phi = phiInit
  rho = rhoInit
  pi = piInit
  llh = -Inf
  S = array(0, dim=c(p,m,k))

	# Algorithm variables
  gam = gamInit
  Gram2 = array(0, dim=c(p,p,k))
  ps2 = array(0, dim=c(p,m,k))
  b = rep(0, k)
  X2 = array(0, dim=c(n,p,k))
  Y2 = array(0, dim=c(n,m,k))
  EPS = 1e-15

  for (ite in 1:maxi)
	{
		# Remember last pi,rho,phi values for exit condition in the end of loop
    Phi = phi
    Rho = rho
    Pi = pi

    # Calcul associé à Y et X
    for (r in 1:k)
		{
      for (mm in 1:m)
        Y2[,mm,r] = sqrt(gam[,r]) * Y[,mm]
      for (i in 1:n)
        X2[i,,r] = sqrt(gam[i,r]) * X[i,]
      for (mm in 1:m)
        ps2[,mm,r] = crossprod(X2[,,r],Y2[,mm,r])
      for (j in 1:p)
			{
        for (s in 1:p)
          Gram2[j,s,r] = crossprod(X2[,j,r], X2[,s,r])
      }
    }

    ##########
    #Etape M #
    ##########

    # Pour pi
    b = sapply( 1:k, function(r) sum(abs(phi[,,r])) )
    gam2 = colSums(gam)
    a = sum(gam %*% log(pi))

    # Tant que les props sont negatives
    kk = 0
    pi2AllPositive = FALSE
    while (!pi2AllPositive)
		{
      pi2 = pi + 0.1^kk * ((1/n)*gam2 - pi)
      pi2AllPositive = all(pi2 >= 0)
      kk = kk+1
    }

    # t(m) la plus grande valeur dans la grille O.1^k tel que ce soit décroissante ou constante
    while( kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
			-sum(gam2 * log(pi2))/n + lambda * sum(pi2^gamma * b) )
		{
      pi2 = pi + 0.1^kk * (1/n*gam2 - pi)
      kk = kk + 1
    }
    t = 0.1^kk
    pi = (pi + t*(pi2-pi)) / sum(pi + t*(pi2-pi))

    #Pour phi et rho
    for (r in 1:k)
		{
      for (mm in 1:m)
			{
				ps = 0
        for (i in 1:n)
          ps = ps + Y2[i,mm,r] * sum(X2[i,,r] * phi[,mm,r])
        nY2 = sum(Y2[,mm,r]^2)
        rho[mm,mm,r] = (ps+sqrt(ps^2+4*nY2*gam2[r])) / (2*nY2)
			}
    }

    for (r in 1:k)
		{
      for (j in 1:p)
			{
        for (mm in 1:m)
				{
          S[j,mm,r] = -rho[mm,mm,r]*ps2[j,mm,r] + sum(phi[-j,mm,r] * Gram2[j,-j,r])
					if (abs(S[j,mm,r]) <= n*lambda*(pi[r]^gamma))
            phi[j,mm,r]=0
          else if(S[j,mm,r] > n*lambda*(pi[r]^gamma))
            phi[j,mm,r] = (n*lambda*(pi[r]^gamma)-S[j,mm,r]) / Gram2[j,j,r]
          else
            phi[j,mm,r] = -(n*lambda*(pi[r]^gamma)+S[j,mm,r]) / Gram2[j,j,r]
        }
      }
    }

    ##########
    #Etape E #
    ##########

		sumLogLLH2 = 0
    for (i in 1:n)
		{
      # Update gam[,]
      sumLLH1 = 0
			sumGamI = 0
      for (r in 1:k)
			{
				gam[i,r] = pi[r] * exp(-0.5*sum( (Y[i,]%*%rho[,,r]-X[i,]%*%phi[,,r])^2 ))
					* det(rho[,,r])
        sumLLH1 = sumLLH1 + gam[i,r] / (2*base::pi)^(m/2)
				sumGamI = sumGamI + gam[i,r]
      }
      sumLogLLH2 = sumLogLLH2 + log(sumLLH1)
      if(sumGamI > EPS) #else: gam[i,] is already ~=0
        gam[i,] = gam[i,] / sumGamI
    }

    sumPen = sum(pi^gamma * b)
		last_llh = llh
    llh = -sumLogLLH2/n + lambda*sumPen
    dist = ifelse( ite == 1, llh, (llh-last_llh) / (1+abs(llh)) )
    Dist1 = max( (abs(phi-Phi)) / (1+abs(phi)) )
    Dist2 = max( (abs(rho-Rho)) / (1+abs(rho)) )
    Dist3 = max( (abs(pi-Pi)) / (1+abs(Pi)) )
    dist2 = max(Dist1,Dist2,Dist3)

    if (ite>=mini && (dist>= tau || dist2 >= sqrt(tau)))
			break
  }

  affec = apply(gam, 1, which.max)
  list( "phi"=phi, "rho"=rho, "pi"=pi, "llh"=llh, "S"=S, "affec"=affec )
}