Grid/lib/algorithms/iterative/ImplicitlyRestartedBlockLanczos.h

    /*************************************************************************************

    Grid physics library, www.github.com/paboyle/Grid 

    Source file: ./lib/algorithms/iterative/ImplicitlyRestartedBlockLanczos.h

    Copyright (C) 2015

Author: Peter Boyle <paboyle@ph.ed.ac.uk>
Author: Chulwoo Jung
Author: Yong-Chull Jang <ypj@quark.phy.bnl.gov> 
Author: Guido Cossu

    This program is free software; you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation; either version 2 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License along
    with this program; if not, write to the Free Software Foundation, Inc.,
    51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.

    See the full license in the file "LICENSE" in the top level distribution directory
    *************************************************************************************/
    /*  END LEGAL */
#ifndef GRID_IRBL_H
#define GRID_IRBL_H

#include <string.h> //memset

#define clog std::cout << GridLogMessage 

namespace Grid {

/////////////////////////////////////////////////////////////
// Implicitly restarted block lanczos
/////////////////////////////////////////////////////////////
template<class Field> 
class ImplicitlyRestartedBlockLanczos {

private:       
  
  std::string cname = std::string("ImplicitlyRestartedBlockLanczos");
  int MaxIter;   // Max iterations
  int Nstop;     // Number of evecs checked for convergence
  int Nu;        // Numbeer of vecs in the unit block
  int Nk;        // Number of converged sought
  int Nm;        // total number of vectors
  int Nblock_k;    // Nk/Nu
  int Nblock_m;    // Nm/Nu
  RealD eresid;
  IRLdiagonalisation diagonalisation;
  ////////////////////////////////////
  // Embedded objects
  ////////////////////////////////////
           SortEigen<Field> _sort;
  LinearOperatorBase<Field> &_Linop;
    OperatorFunction<Field> &_poly;

  /////////////////////////
  // Constructor
  /////////////////////////
public:       
 ImplicitlyRestartedBlockLanczos(LinearOperatorBase<Field> &Linop, // op
                                 OperatorFunction<Field> & poly,   // polynomial
                                 int _Nstop, // really sought vecs
                                 int _Nu,    // vecs in the unit block
                                 int _Nk,    // sought vecs
                                 int _Nm,    // total vecs
                                 RealD _eresid, // resid in lmd deficit 
                                 int _MaxIter,  // Max iterations
                                 IRLdiagonalisation _diagonalisation = IRLdiagonaliseWithEigen)
   : _Linop(Linop),    _poly(poly),
      Nstop(_Nstop), Nu(_Nu), Nk(_Nk), Nm(_Nm), 
      Nblock_m(_Nm/_Nu), Nblock_k(_Nk/_Nu),
      eresid(_eresid),  MaxIter(_MaxIter),
      diagonalisation(_diagonalisation)
  { assert( (Nk%Nu==0) && (Nm%Nu==0) ); };

  ////////////////////////////////
  // Helpers
  ////////////////////////////////
  static RealD normalize(Field& v) 
  {
    RealD nn = norm2(v);
    nn = sqrt(nn);
    v = v * (1.0/nn);
    return nn;
  }
  
  void orthogonalize(Field& w, std::vector<Field>& evec, int k)
  {
    typedef typename Field::scalar_type MyComplex;
    MyComplex ip;
    
    for(int j=0; j<k; ++j){
      ip = innerProduct(evec[j],w); 
      w = w - ip * evec[j];
    }
    normalize(w);
  }

/* Rudy Arthur's thesis pp.137
------------------------
Require: M > K P = M − K †
Compute the factorization AVM = VM HM + fM eM 
repeat
  Q=I
  for i = 1,...,P do
    QiRi =HM −θiI Q = QQi
    H M = Q †i H M Q i
  end for
  βK =HM(K+1,K) σK =Q(M,K)
  r=vK+1βK +rσK
  VK =VM(1:M)Q(1:M,1:K)
  HK =HM(1:K,1:K)
  →AVK =VKHK +fKe†K † Extend to an M = K + P step factorization AVM = VMHM + fMeM
until convergence
*/
  void calc(std::vector<RealD>& eval,  
            std::vector<Field>& evec, 
            const std::vector<Field>& src, int& Nconv)
  {
    std::string fname = std::string(cname+"::calc()"); 
    GridBase *grid = evec[0]._grid;
    assert(grid == src[0]._grid);
    assert( Nu = src.size() );
    
    clog << std::string(74,'*') << std::endl;
    clog << fname + " starting iteration 0 /  "<< MaxIter<< std::endl;
    clog << std::string(74,'*') << std::endl;
    clog <<" -- seek   Nk    = "<< Nk    <<" vectors"<< std::endl;
    clog <<" -- accept Nstop = "<< Nstop <<" vectors"<< std::endl;
    clog <<" -- total  Nm    = "<< Nm    <<" vectors"<< std::endl;
    clog <<" -- size of eval = "<< eval.size() << std::endl;
    clog <<" -- size of evec = "<< evec.size() << std::endl;
    if ( diagonalisation == IRLdiagonaliseWithDSTEGR ) {
      clog << "Diagonalisation is DSTEGR "<< std::endl;
    } else if ( diagonalisation == IRLdiagonaliseWithQR ) { 
      clog << "Diagonalisation is QR "<< std::endl;
    }  else if ( diagonalisation == IRLdiagonaliseWithEigen ) { 
      clog << "Diagonalisation is Eigen "<< std::endl;
    }
    clog << std::string(74,'*') << std::endl;
    
    assert(Nm == evec.size() && Nm == eval.size());
	
    std::vector<std::vector<ComplexD>> lmd(Nu,std::vector<ComplexD>(Nm,0.0));  
    std::vector<std::vector<ComplexD>> lme(Nu,std::vector<ComplexD>(Nm,0.0));  
    std::vector<std::vector<ComplexD>> lmd2(Nu,std::vector<ComplexD>(Nm,0.0));  
    std::vector<std::vector<ComplexD>> lme2(Nu,std::vector<ComplexD>(Nm,0.0));  
    std::vector<RealD> eval2(Nm);

    Eigen::MatrixXcd    Qt = Eigen::MatrixXcd::Zero(Nm,Nm);

    std::vector<int>   Iconv(Nm);
    std::vector<Field>  B(Nm,grid); // waste of space replicating
    
    std::vector<Field> f(Nu,grid);
    std::vector<Field> f_copy(Nu,grid);
    Field v(grid);
    
    int k1 = 1;
    int k2 = Nk;
    
    Nconv = 0;
    
    RealD beta_k;
  
    // Set initial vector
    for (int i=0; i<Nu; ++i) {
      clog << "norm2(src[" << i << "])= "<< norm2(src[i]) << std::endl;
      evec[i] = src[i];
      orthogonalize(evec[i],evec,i);
      clog << "norm2(evec[" << i << "])= "<< norm2(evec[i]) << std::endl;
    }
    
    // Initial Nblock_k steps
    for(int b=0; b<Nblock_k; ++b) blockwiseStep(lmd,lme,evec,f,f_copy,b);
    
    // Restarting loop begins
    int iter;
    for(iter = 0; iter<MaxIter; ++iter){
      
      clog <<" **********************"<< std::endl;
      clog <<" Restart iteration = "<< iter << std::endl;
      clog <<" **********************"<< std::endl;
      
      for(int b=Nblock_k; b<Nblock_m; ++b) blockwiseStep(lmd,lme,evec,f,f_copy,b);
      
      //f[Nu-1] *= lme[Nm-1]; // ypj[fixme] need to be changed for block method
      
      // getting eigenvalues
      for(int u=0; u<Nu; ++u){
        for(int k=0; k<Nm; ++k){
          lmd2[u][k] = lmd[u][k];
          lme2[u][k] = lme[u][k];
        }
      }
      Qt = Eigen::MatrixXcd::Identity(Nm,Nm);
      diagonalize(eval2,lmd2,lme2,Nu,Nblock_m,Nm,Nm,Qt,grid);

      // sorting
      _sort.push(eval2,Nm);
      break;
    }
#if 0 // working up to here      
      // Implicitly shifted QR transformations
      Qt = Eigen::MatrixXcd::Identity(Nm,Nm);
      for(int ip=k2; ip<Nm; ++ip){ 
	// Eigen replacement for qr_decomp ???
	qr_decomp(eval,lme,Nm,Nm,Qt,eval2[ip],k1,Nm);
      }
    
      for(int i=0; i<(Nk+1); ++i) B[i] = 0.0;
	  
      for(int j=k1-1; j<k2+1; ++j){
	for(int k=0; k<Nm; ++k){
	  B[j].checkerboard = evec[k].checkerboard;
	  B[j] += Qt(j,k) * evec[k];
	}
      }
      for(int j=k1-1; j<k2+1; ++j) evec[j] = B[j];
      
      // Compressed vector f and beta(k2)
      f *= Qt(k2-1,Nm-1);
      f += lme[k2-1] * evec[k2];
      beta_k = norm2(f);
      beta_k = sqrt(beta_k);
      std::cout<< GridLogMessage<<" beta(k) = "<<beta_k<<std::endl;
      
      RealD betar = 1.0/beta_k;
      evec[k2] = betar * f;
      lme[k2-1] = beta_k;
      
      // Convergence test
      for(int k=0; k<Nm; ++k){    
	eval2[k] = eval[k];
	lme2[k] = lme[k];
      }
      Qt = Eigen::MatrixXcd::Identity(Nm,Nm);
      diagonalize(eval2,lme2,Nk,Nm,Qt,grid);
      
      for(int k = 0; k<Nk; ++k) B[k]=0.0;
      
      for(int j = 0; j<Nk; ++j){
	for(int k = 0; k<Nk; ++k){
	  B[j].checkerboard = evec[k].checkerboard;
	  B[j] += Qt(j,k) * evec[k];
	}
      }

      Nconv = 0;
      for(int i=0; i<Nk; ++i){
	
	_Linop.HermOp(B[i],v);
	    
	RealD vnum = real(innerProduct(B[i],v)); // HermOp.
	RealD vden = norm2(B[i]);
	eval2[i] = vnum/vden;
	v -= eval2[i]*B[i];
	RealD vv = norm2(v);
	
	std::cout.precision(13);
	clog << "[" << std::setw(3)<< std::setiosflags(std::ios_base::right) <<i<<"] ";
	std::cout << "eval = "<<std::setw(25)<< std::setiosflags(std::ios_base::left)<< eval2[i];
	std::cout << " |H B[i] - eval[i]B[i]|^2 "<< std::setw(25)<< std::setiosflags(std::ios_base::right)<< vv<< std::endl;
	
	// change the criteria as evals are supposed to be sorted, all evals smaller(larger) than Nstop should have converged
	if((vv<eresid*eresid) && (i == Nconv) ){
	  Iconv[Nconv] = i;
	  ++Nconv;
	}
	
      }  // i-loop end
      
      std::cout<< GridLogMessage <<" #modes converged: "<<Nconv<<std::endl;

      if( Nconv>=Nstop ){
	goto converged;
      }
    } // end of iter loop
    
    clog <<"**************************************************************************"<< std::endl;
    std::cout<< GridLogError    << fname + " NOT converged.";
    clog <<"**************************************************************************"<< std::endl;
    abort();
	
  converged:
    // Sorting
    eval.resize(Nconv);
    evec.resize(Nconv,grid);
    for(int i=0; i<Nconv; ++i){
      eval[i] = eval2[Iconv[i]];
      evec[i] = B[Iconv[i]];
    }
    _sort.push(eval,evec,Nconv);
    
    clog <<"**************************************************************************"<< std::endl;
    clog << fname + " CONVERGED ; Summary :\n";
    clog <<"**************************************************************************"<< std::endl;
    clog << " -- Iterations  = "<< iter   << "\n";
    clog << " -- beta(k)     = "<< beta_k << "\n";
    clog << " -- Nconv       = "<< Nconv  << "\n";
    clog <<"**************************************************************************"<< std::endl;
#endif
  }

private:
/* Saad PP. 195
1. Choose an initial vector v1 of 2-norm unity. Set β1 ≡ 0, v0 ≡ 0
2. For k = 1,2,...,m Do:
3. wk:=Avk−βkv_{k−1}      
4. αk:=(wk,vk)       // 
5. wk:=wk−αkvk       // wk orthog vk 
6. βk+1 := ∥wk∥2. If βk+1 = 0 then Stop
7. vk+1 := wk/βk+1
8. EndDo
 */
  void blockwiseStep(std::vector<std::vector<ComplexD>>& lmd,
	             std::vector<std::vector<ComplexD>>& lme, 
	             std::vector<Field>& evec,
	             std::vector<Field>& w, 
	             std::vector<Field>& w_copy, 
                     int b)
  {
    const RealD tiny = 1.0e-20;
    
    int Nu = w.size();
    int Nm = evec.size();
    assert( b < Nm/Nu );
    
    // converts block index to full indicies for an interval [L,R)
    int L = Nu*b;
    int R = Nu*(b+1);

    Real beta;
    
    clog << "A: b = " << b << std::endl;
    // 3. wk:=Avk−βkv_{k−1}
    for (int k=L, u=0; k<R; ++k, ++u) {
      _poly(_Linop,evec[k],w[u]);      
    }
    
    if (b>0) {
      clog << "B: b = " << b << std::endl;
      for (int u=0; u<Nu; ++u) {
        for (int k=L-Nu; k<L; ++k) {
          w[u] = w[u] - evec[k] * conjugate(lme[u][k]);
        }
      }
    }
    
    // 4. αk:=(vk,wk)
    clog << "C: b = " << b << std::endl;
    for (int u=0; u<Nu; ++u) {
      for (int k=L; k<R; ++k) {
        lmd[u][k] = innerProduct(evec[k],w[u]);  // lmd = transpose of alpha
      }
      lmd[u][L+u] = real(lmd[u][L+u]);  // force diagonal to be real
    }
    
    clog << "D: b = " << b << std::endl;
    // 5. wk:=wk−αkvk
    for (int u=0; u<Nu; ++u) {
      for (int k=L; k<R; ++k) {
        w[u] = w[u] - evec[k]*lmd[u][k];
      }
      w_copy[u] = w[u];
    }
    
    // In block version, the steps 6 and 7 in Lanczos construction is
    // replaced by the QR decomposition of new basis block.
    // It results block version beta and orthonormal block basis. 
    // Here, QR decomposition is done by using Gram-Schmidt
    clog << "E: b = " << b << std::endl;
    for (int u=0; u<Nu; ++u) {
      for (int k=L; k<R; ++k) {
        lme[u][k] = 0.0;
      }
    }

    clog << "F: b = " << b << std::endl;
    beta = normalize(w[0]);
    for (int u=1; u<Nu; ++u) {
      //orthogonalize(w[u],w_copy,u);
      orthogonalize(w[u],w,u);
    }
    
    clog << "G: b = " << b << std::endl;
    for (int u=0; u<Nu; ++u) {
      for (int v=0; v<Nu; ++v) {
        lme[u][L+v] = innerProduct(w[u],w_copy[v]);
      }
    }
    lme[0][L] = beta;
    
#if 0
    for (int u=0; u<Nu; ++u) {
      for (int k=L+u; k<R; ++k) {
        if (lme[u][k] < tiny) {
          clog <<" In block "<< b << ","; 
          std::cout <<" beta[" << u << "," << k-L << "] = ";
          std::cout << lme[u][k] << std::endl;
        }
      }
    }
#else
    clog << "H: b = " << b << std::endl;
    for (int u=0; u<Nu; ++u) {
      clog << "norm2(w[" << u << "])= "<< norm2(w[u]) << std::endl;
      for (int k=L+u; k<R; ++k) {
        clog <<" In block "<< b << ","; 
        std::cout <<" beta[" << u << "," << k-L << "] = ";
        std::cout << lme[u][k] << std::endl;
      }
    }
#endif
    
    // re-orthogonalization for numerical stability
    //clog << "I: b = " << b << std::endl;
    //if (b>0) {
    //  for (int u=0; u<Nu; ++u) {
    //    orthogonalize(w[u],evec,R);
    //  }
    //}

    clog << "J: b = " << b << std::endl;
    if (b < Nm/Nu-1) {
      for (int u=0; u<Nu; ++u) {
        evec[R+u] = w[u];
      }
    }

  }
      
  void diagonalize_Eigen(std::vector<RealD>& eval, 
                         std::vector<std::vector<ComplexD>>& lmd,
                         std::vector<std::vector<ComplexD>>& lme, 
			 int Nu, int Nb, int Nk, int Nm,
			 Eigen::MatrixXcd & Qt, // Nm x Nm
			 GridBase *grid)
  {
    assert( Nk%Nu == 0 && Nm%Nu == 0 );
    assert( Nk <= Nm );
    Eigen::MatrixXcd BlockTriDiag = Eigen::MatrixXcd::Zero(Nk,Nk);
    
    for ( int u=0; u<Nu; ++u ) {
      for (int k=0; k<Nk; ++k ) {
        BlockTriDiag(k,u+(k/Nu)*Nu) = lmd[u][k];
      }
    }
    
    for ( int u=0; u<Nu; ++u ) {
      for (int k=Nu; k<Nk; ++k ) {
        BlockTriDiag(u+(k/Nu)*Nu,k-Nu) = lme[u][k-Nu];
        BlockTriDiag(k,u+(k/Nu)*Nu) = conjugate(lme[u][k-Nu]);
      }
    }
    
    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXcd> eigensolver(BlockTriDiag);

    for (int i = 0; i < Nk; i++) {
      eval[Nk-1-i] = eigensolver.eigenvalues()(i);
    }
    for (int i = 0; i < Nk; i++) {
      for (int j = 0; j < Nk; j++) {
	Qt(Nk-1-i,j) = eigensolver.eigenvectors()(j,i);
      }
    }
  }
  ///////////////////////////////////////////////////////////////////////////
  // File could end here if settle on Eigen ???
  ///////////////////////////////////////////////////////////////////////////

  void qr_decomp(std::vector<RealD>& lmd,   // Nm 
		 std::vector<RealD>& lme,   // Nm 
		 int Nk, int Nm,            // Nk, Nm
		 Eigen::MatrixXd& Qt,       // Nm x Nm matrix
		 RealD Dsh, int kmin, int kmax)
  {
    int k = kmin-1;
    RealD x;
    
    RealD Fden = 1.0/hypot(lmd[k]-Dsh,lme[k]);
    RealD c = ( lmd[k] -Dsh) *Fden;
    RealD s = -lme[k] *Fden;
      
    RealD tmpa1 = lmd[k];
    RealD tmpa2 = lmd[k+1];
    RealD tmpb  = lme[k];

    lmd[k]   = c*c*tmpa1 +s*s*tmpa2 -2.0*c*s*tmpb;
    lmd[k+1] = s*s*tmpa1 +c*c*tmpa2 +2.0*c*s*tmpb;
    lme[k]   = c*s*(tmpa1-tmpa2) +(c*c-s*s)*tmpb;
    x        =-s*lme[k+1];
    lme[k+1] = c*lme[k+1];
      
    for(int i=0; i<Nk; ++i){
      RealD Qtmp1 = Qt(k,i);
      RealD Qtmp2 = Qt(k+1,i);
      Qt(k,i)  = c*Qtmp1 - s*Qtmp2;
      Qt(k+1,i)= s*Qtmp1 + c*Qtmp2; 
    }

    // Givens transformations
    for(int k = kmin; k < kmax-1; ++k){
      
      RealD Fden = 1.0/hypot(x,lme[k-1]);
      RealD c = lme[k-1]*Fden;
      RealD s = - x*Fden;
	
      RealD tmpa1 = lmd[k];
      RealD tmpa2 = lmd[k+1];
      RealD tmpb  = lme[k];

      lmd[k]   = c*c*tmpa1 +s*s*tmpa2 -2.0*c*s*tmpb;
      lmd[k+1] = s*s*tmpa1 +c*c*tmpa2 +2.0*c*s*tmpb;
      lme[k]   = c*s*(tmpa1-tmpa2) +(c*c-s*s)*tmpb;
      lme[k-1] = c*lme[k-1] -s*x;

      if(k != kmax-2){
	x = -s*lme[k+1];
	lme[k+1] = c*lme[k+1];
      }

      for(int i=0; i<Nk; ++i){
	RealD Qtmp1 = Qt(k,i);
	RealD Qtmp2 = Qt(k+1,i);
	Qt(k,i)     = c*Qtmp1 -s*Qtmp2;
	Qt(k+1,i)   = s*Qtmp1 +c*Qtmp2;
      }
    }
  }

  void diagonalize(std::vector<RealD>& eval, 
                   std::vector<std::vector<ComplexD>>& lmd, 
                   std::vector<std::vector<ComplexD>>& lme, 
		   int Nu, int Nb, int Nk, int Nm,   
		   Eigen::MatrixXcd & Qt,
		   GridBase *grid)
  {
    Qt = Eigen::MatrixXcd::Identity(Nm,Nm);
//    if ( diagonalisation == IRLdiagonaliseWithDSTEGR ) {
//      diagonalize_lapack(lmd,lme,Nk,Nm,Qt,grid);
//    } else if ( diagonalisation == IRLdiagonaliseWithQR ) { 
//      diagonalize_QR(lmd,lme,Nk,Nm,Qt,grid);
//    }  else if ( diagonalisation == IRLdiagonaliseWithEigen ) { 
    if ( diagonalisation == IRLdiagonaliseWithEigen ) { 
      diagonalize_Eigen(eval,lmd,lme,Nu,Nb,Nk,Nm,Qt,grid);
    } else { 
      assert(0);
    }
  }

#ifdef USE_LAPACK
void LAPACK_dstegr(char *jobz, char *range, int *n, double *d, double *e,
                   double *vl, double *vu, int *il, int *iu, double *abstol,
                   int *m, double *w, double *z, int *ldz, int *isuppz,
                   double *work, int *lwork, int *iwork, int *liwork,
                   int *info);
#endif

void diagonalize_lapack(std::vector<RealD>& lmd,
			std::vector<RealD>& lme, 
			int Nk, int Nm,  
			Eigen::MatrixXd& Qt,
			GridBase *grid)
{
#ifdef USE_LAPACK
  const int size = Nm;
  int NN = Nk;
  double evals_tmp[NN];
  double evec_tmp[NN][NN];
  memset(evec_tmp[0],0,sizeof(double)*NN*NN);
  double DD[NN];
  double EE[NN];
  for (int i = 0; i< NN; i++) {
    for (int j = i - 1; j <= i + 1; j++) {
      if ( j < NN && j >= 0 ) {
	if (i==j) DD[i] = lmd[i];
	if (i==j) evals_tmp[i] = lmd[i];
	if (j==(i-1)) EE[j] = lme[j];
      }
    }
  }
  int evals_found;
  int lwork = ( (18*NN) > (1+4*NN+NN*NN)? (18*NN):(1+4*NN+NN*NN)) ;
  int liwork =  3+NN*10 ;
  int iwork[liwork];
  double work[lwork];
  int isuppz[2*NN];
  char jobz = 'V'; // calculate evals & evecs
  char range = 'I'; // calculate all evals
  //    char range = 'A'; // calculate all evals
  char uplo = 'U'; // refer to upper half of original matrix
  char compz = 'I'; // Compute eigenvectors of tridiagonal matrix
  int ifail[NN];
  int info;
  int total = grid->_Nprocessors;
  int node  = grid->_processor;
  int interval = (NN/total)+1;
  double vl = 0.0, vu = 0.0;
  int il = interval*node+1 , iu = interval*(node+1);
  if (iu > NN)  iu=NN;
  double tol = 0.0;
  if (1) {
    memset(evals_tmp,0,sizeof(double)*NN);
    if ( il <= NN){
      LAPACK_dstegr(&jobz, &range, &NN,
		    (double*)DD, (double*)EE,
		    &vl, &vu, &il, &iu, // these four are ignored if second parameteris 'A'
		    &tol, // tolerance
		    &evals_found, evals_tmp, (double*)evec_tmp, &NN,
		    isuppz,
		    work, &lwork, iwork, &liwork,
		    &info);
      for (int i = iu-1; i>= il-1; i--){
	evals_tmp[i] = evals_tmp[i - (il-1)];
	if (il>1) evals_tmp[i-(il-1)]=0.;
	for (int j = 0; j< NN; j++){
	  evec_tmp[i][j] = evec_tmp[i - (il-1)][j];
	  if (il>1) evec_tmp[i-(il-1)][j]=0.;
	}
      }
    }
    {
      grid->GlobalSumVector(evals_tmp,NN);
      grid->GlobalSumVector((double*)evec_tmp,NN*NN);
    }
  } 
  // Safer to sort instead of just reversing it, 
  // but the document of the routine says evals are sorted in increasing order. 
  // qr gives evals in decreasing order.
  for(int i=0;i<NN;i++){
    lmd [NN-1-i]=evals_tmp[i];
    for(int j=0;j<NN;j++){
      Qt((NN-1-i),j)=evec_tmp[i][j];
    }
  }
#else 
  assert(0);
#endif
}

  void diagonalize_QR(std::vector<RealD>& lmd, std::vector<RealD>& lme, 
		      int Nk, int Nm,   
		      Eigen::MatrixXd & Qt,
		      GridBase *grid)
  {
    int Niter = 100*Nm;
    int kmin = 1;
    int kmax = Nk;

    // (this should be more sophisticated)
    for(int iter=0; iter<Niter; ++iter){
      
      // determination of 2x2 leading submatrix
      RealD dsub = lmd[kmax-1]-lmd[kmax-2];
      RealD dd = sqrt(dsub*dsub + 4.0*lme[kmax-2]*lme[kmax-2]);
      RealD Dsh = 0.5*(lmd[kmax-2]+lmd[kmax-1] +dd*(dsub/fabs(dsub)));
      // (Dsh: shift)
	
      // transformation
      qr_decomp(lmd,lme,Nk,Nm,Qt,Dsh,kmin,kmax); // Nk, Nm
	
      // Convergence criterion (redef of kmin and kamx)
      for(int j=kmax-1; j>= kmin; --j){
	RealD dds = fabs(lmd[j-1])+fabs(lmd[j]);
	if(fabs(lme[j-1])+dds > dds){
	  kmax = j+1;
	  goto continued;
	}
      }
      Niter = iter;
      return;

    continued:
      for(int j=0; j<kmax-1; ++j){
	RealD dds = fabs(lmd[j])+fabs(lmd[j+1]);
	if(fabs(lme[j])+dds > dds){
	  kmin = j+1;
	  break;
	}
      }
    }
    std::cout << GridLogError << "[QL method] Error - Too many iteration: "<<Niter<<"\n";
    abort();
  }

 };
}

#undef clog
#endif