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mirror of https://github.com/paboyle/Grid.git synced 2025-06-20 08:46:55 +01:00

Merge branch 'develop' into feature/gpu-port

This commit is contained in:
Peter Boyle
2018-12-13 05:11:34 +00:00
647 changed files with 49155 additions and 11160 deletions

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@ -309,7 +309,8 @@ int main (int argc, char ** argv)
// Momentum space prop
std::cout << " Solving by FFT and Feynman rules" <<std::endl;
Ddwf.FreePropagator(src,ref,mass) ;
bool fiveD = false; //calculate 4d free propagator
Ddwf.FreePropagator(src,ref,mass,fiveD) ;
Gamma G5(Gamma::Algebra::Gamma5);

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@ -392,7 +392,6 @@ int main(int argc, char **argv) {
}
random(Foo);
*/
lex_sites(Foo);
Integer mm[4];
mm[0] = 1;

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@ -141,6 +141,7 @@ int main (int argc, char ** argv)
t1=usecond();
std::cout<<GridLogMessage << "Called Ds ASM"<<std::endl;
std::cout<<GridLogMessage << "norm src "<< norm2(src)<<std::endl;
std::cout<<GridLogMessage << "norm result "<< norm2(tmp)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)<<std::endl;
@ -160,7 +161,8 @@ int main (int argc, char ** argv)
localConvert(sresult,tmp);
std::cout<<GridLogMessage << "Called sDs unroll"<<std::endl;
std::cout<<GridLogMessage << "norm result "<< norm2(sresult)<<std::endl;
std::cout<<GridLogMessage << "norm ssrc "<< norm2(ssrc)<<std::endl;
std::cout<<GridLogMessage << "norm sresult "<< norm2(sresult)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)<<std::endl;
@ -181,6 +183,7 @@ int main (int argc, char ** argv)
localConvert(sresult,tmp);
std::cout<<GridLogMessage << "Called sDs asm"<<std::endl;
std::cout<<GridLogMessage << "norm ssrc "<< norm2(ssrc)<<std::endl;
std::cout<<GridLogMessage << "norm result "<< norm2(sresult)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)*extra<<std::endl;

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@ -0,0 +1,196 @@
/*************************************************************************************
Grid physics library, www.github.com/paboyle/Grid
Source file: ./benchmarks/Benchmark_wilson.cc
Copyright (C) 2015
Author: Peter Boyle <paboyle@ph.ed.ac.uk>
Author: paboyle <paboyle@ph.ed.ac.uk>
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 */
#include <Grid/Grid.h>
using namespace std;
using namespace Grid;
using namespace Grid::QCD;
int main (int argc, char ** argv)
{
Grid_init(&argc,&argv);
std::vector<int> latt_size = GridDefaultLatt();
std::vector<int> simd_layout = GridDefaultSimd(Nd,vComplex::Nsimd());
std::vector<int> mpi_layout = GridDefaultMpi();
const int Ls=16;
GridCartesian * UGrid = SpaceTimeGrid::makeFourDimGrid(GridDefaultLatt(), GridDefaultSimd(Nd,vComplexF::Nsimd()),GridDefaultMpi());
GridRedBlackCartesian * UrbGrid = SpaceTimeGrid::makeFourDimRedBlackGrid(UGrid);
GridCartesian * FGrid = SpaceTimeGrid::makeFiveDimGrid(Ls,UGrid);
GridRedBlackCartesian * FrbGrid = SpaceTimeGrid::makeFiveDimRedBlackGrid(Ls,UGrid);
std::cout << GridLogMessage << "Making s innermost grids"<<std::endl;
GridCartesian * sUGrid = SpaceTimeGrid::makeFourDimDWFGrid(GridDefaultLatt(),GridDefaultMpi());
GridRedBlackCartesian * sUrbGrid = SpaceTimeGrid::makeFourDimRedBlackGrid(sUGrid);
GridCartesian * sFGrid = SpaceTimeGrid::makeFiveDimDWFGrid(Ls,UGrid);
GridRedBlackCartesian * sFrbGrid = SpaceTimeGrid::makeFiveDimDWFRedBlackGrid(Ls,UGrid);
int threads = GridThread::GetThreads();
std::cout<<GridLogMessage << "Grid is setup to use "<<threads<<" threads"<<std::endl;
std::vector<int> seeds({1,2,3,4});
GridParallelRNG pRNG4(UGrid);
GridParallelRNG pRNG5(FGrid);
pRNG4.SeedFixedIntegers(seeds);
pRNG5.SeedFixedIntegers(seeds);
typedef typename ImprovedStaggeredFermion5DF::FermionField FermionField;
typedef typename ImprovedStaggeredFermion5DF::ComplexField ComplexField;
typename ImprovedStaggeredFermion5DF::ImplParams params;
FermionField src (FGrid);
random(pRNG5,src);
/*
std::vector<int> site({0,1,2,0,0});
ColourVector cv = zero;
cv()()(0)=1.0;
src = zero;
pokeSite(cv,src,site);
*/
FermionField result(FGrid); result=zero;
FermionField tmp(FGrid); tmp=zero;
FermionField err(FGrid); tmp=zero;
FermionField phi (FGrid); random(pRNG5,phi);
FermionField chi (FGrid); random(pRNG5,chi);
LatticeGaugeFieldF Umu(UGrid);
SU3::HotConfiguration(pRNG4,Umu);
/*
for(int mu=1;mu<4;mu++){
auto tmp = PeekIndex<LorentzIndex>(Umu,mu);
tmp = zero;
PokeIndex<LorentzIndex>(Umu,tmp,mu);
}
*/
double volume=Ls;
for(int mu=0;mu<Nd;mu++){
volume=volume*latt_size[mu];
}
RealD mass=0.1;
RealD c1=9.0/8.0;
RealD c2=-1.0/24.0;
RealD u0=1.0;
ImprovedStaggeredFermion5DF Ds(Umu,Umu,*FGrid,*FrbGrid,*UGrid,*UrbGrid,mass,c1,c2,u0,params);
ImprovedStaggeredFermionVec5dF sDs(Umu,Umu,*sFGrid,*sFrbGrid,*sUGrid,*sUrbGrid,mass,c1,c2,u0,params);
std::cout<<GridLogMessage<<"=========================================================="<<std::endl;
std::cout<<GridLogMessage<<"= Testing Dhop against cshift implementation "<<std::endl;
std::cout<<GridLogMessage<<"=========================================================="<<std::endl;
int ncall=1000;
int ncall1=1000;
double t0(0),t1(0);
double flops=(16*(3*(6+8+8)) + 15*3*2)*volume*ncall; // == 66*16 + == 1146
std::cout<<GridLogMessage << "Calling staggered operator"<<std::endl;
t0=usecond();
for(int i=0;i<ncall1;i++){
Ds.Dhop(src,result,0);
}
t1=usecond();
std::cout<<GridLogMessage << "Called Ds"<<std::endl;
std::cout<<GridLogMessage << "norm result "<< norm2(result)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)<<std::endl;
std::cout<<GridLogMessage << "Calling vectorised staggered operator"<<std::endl;
#ifdef AVX512
QCD::StaggeredKernelsStatic::Opt=QCD::StaggeredKernelsStatic::OptInlineAsm;
#else
QCD::StaggeredKernelsStatic::Opt=QCD::StaggeredKernelsStatic::OptGeneric;
#endif
t0=usecond();
for(int i=0;i<ncall1;i++){
Ds.Dhop(src,tmp,0);
}
t1=usecond();
std::cout<<GridLogMessage << "Called Ds ASM"<<std::endl;
std::cout<<GridLogMessage << "norm src "<< norm2(src)<<std::endl;
std::cout<<GridLogMessage << "norm result "<< norm2(tmp)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)<<std::endl;
err = tmp-result;
std::cout<<GridLogMessage << "norm diff "<< norm2(err)<<std::endl;
FermionField ssrc (sFGrid); localConvert(src,ssrc);
FermionField sresult(sFGrid); sresult=zero;
QCD::StaggeredKernelsStatic::Opt=QCD::StaggeredKernelsStatic::OptHandUnroll;
t0=usecond();
for(int i=0;i<ncall1;i++){
sDs.Dhop(ssrc,sresult,0);
}
t1=usecond();
localConvert(sresult,tmp);
std::cout<<GridLogMessage << "Called sDs unroll"<<std::endl;
std::cout<<GridLogMessage << "norm ssrc "<< norm2(ssrc)<<std::endl;
std::cout<<GridLogMessage << "norm sresult "<< norm2(sresult)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)<<std::endl;
#ifdef AVX512
QCD::StaggeredKernelsStatic::Opt=QCD::StaggeredKernelsStatic::OptInlineAsm;
#else
QCD::StaggeredKernelsStatic::Opt=QCD::StaggeredKernelsStatic::OptGeneric;
#endif
err = tmp-result;
std::cout<<GridLogMessage << "norm diff "<< norm2(err)<<std::endl;
int extra=1;
t0=usecond();
for(int i=0;i<ncall1*extra;i++){
sDs.Dhop(ssrc,sresult,0);
}
t1=usecond();
localConvert(sresult,tmp);
std::cout<<GridLogMessage << "Called sDs asm"<<std::endl;
std::cout<<GridLogMessage << "norm ssrc "<< norm2(ssrc)<<std::endl;
std::cout<<GridLogMessage << "norm result "<< norm2(sresult)<<std::endl;
std::cout<<GridLogMessage << "mflop/s = "<< flops/(t1-t0)*extra<<std::endl;
err = tmp-result;
std::cout<<GridLogMessage << "norm diff "<< norm2(err)<<std::endl;
Grid_finalize();
}

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@ -0,0 +1,357 @@
/*************************************************************************************
Grid physics library, www.github.com/paboyle/Grid
Source file: ./benchmarks/Benchmark_wilson.cc
Copyright (C) 2015
Author: Guido Cossu <guido.cossu@ed.ac.uk>
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 */
#include <Grid/Grid.h>
using namespace std;
using namespace Grid;
using namespace Grid::QCD;
int main(int argc, char **argv)
{
Grid_init(&argc, &argv);
std::vector<int> latt_size = GridDefaultLatt();
std::vector<int> simd_layout = GridDefaultSimd(Nd, vComplex::Nsimd());
std::vector<int> mpi_layout = GridDefaultMpi();
GridCartesian Grid(latt_size, simd_layout, mpi_layout);
GridRedBlackCartesian RBGrid(&Grid);
int threads = GridThread::GetThreads();
std::cout << GridLogMessage << "Grid is setup to use " << threads << " threads" << std::endl;
std::cout << GridLogMessage << "Grid floating point word size is REALF" << sizeof(RealF) << std::endl;
std::cout << GridLogMessage << "Grid floating point word size is REALD" << sizeof(RealD) << std::endl;
std::cout << GridLogMessage << "Grid floating point word size is REAL" << sizeof(Real) << std::endl;
std::vector<int> seeds({1, 2, 3, 4});
GridParallelRNG pRNG(&Grid);
pRNG.SeedFixedIntegers(seeds);
// pRNG.SeedFixedIntegers(std::vector<int>({45,12,81,9});
typedef typename WilsonCloverFermionR::FermionField FermionField;
typename WilsonCloverFermionR::ImplParams params;
WilsonAnisotropyCoefficients anis;
FermionField src(&Grid);
random(pRNG, src);
FermionField result(&Grid);
result = zero;
FermionField result2(&Grid);
result2 = zero;
FermionField ref(&Grid);
ref = zero;
FermionField tmp(&Grid);
tmp = zero;
FermionField err(&Grid);
err = zero;
FermionField err2(&Grid);
err2 = zero;
FermionField phi(&Grid);
random(pRNG, phi);
FermionField chi(&Grid);
random(pRNG, chi);
LatticeGaugeField Umu(&Grid);
SU3::HotConfiguration(pRNG, Umu);
std::vector<LatticeColourMatrix> U(4, &Grid);
double volume = 1;
for (int mu = 0; mu < Nd; mu++)
{
volume = volume * latt_size[mu];
}
RealD mass = 0.1;
RealD csw_r = 1.0;
RealD csw_t = 1.0;
WilsonCloverFermionR Dwc(Umu, Grid, RBGrid, mass, csw_r, csw_t, anis, params);
//Dwc.ImportGauge(Umu); // not necessary, included in the constructor
std::cout << GridLogMessage << "==========================================================" << std::endl;
std::cout << GridLogMessage << "= Testing that Deo + Doe = Dunprec " << std::endl;
std::cout << GridLogMessage << "==========================================================" << std::endl;
FermionField src_e(&RBGrid);
FermionField src_o(&RBGrid);
FermionField r_e(&RBGrid);
FermionField r_o(&RBGrid);
FermionField r_eo(&Grid);
pickCheckerboard(Even, src_e, src);
pickCheckerboard(Odd, src_o, src);
Dwc.Meooe(src_e, r_o);
std::cout << GridLogMessage << "Applied Meo" << std::endl;
Dwc.Meooe(src_o, r_e);
std::cout << GridLogMessage << "Applied Moe" << std::endl;
Dwc.Dhop(src, ref, DaggerNo);
setCheckerboard(r_eo, r_o);
setCheckerboard(r_eo, r_e);
err = ref - r_eo;
std::cout << GridLogMessage << "EO norm diff " << norm2(err) << " " << norm2(ref) << " " << norm2(r_eo) << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
std::cout << GridLogMessage << "= Test Ddagger is the dagger of D by requiring " << std::endl;
std::cout << GridLogMessage << "= < phi | Deo | chi > * = < chi | Deo^dag| phi> " << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
FermionField chi_e(&RBGrid);
FermionField chi_o(&RBGrid);
FermionField dchi_e(&RBGrid);
FermionField dchi_o(&RBGrid);
FermionField phi_e(&RBGrid);
FermionField phi_o(&RBGrid);
FermionField dphi_e(&RBGrid);
FermionField dphi_o(&RBGrid);
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
pickCheckerboard(Even, phi_e, phi);
pickCheckerboard(Odd, phi_o, phi);
Dwc.Meooe(chi_e, dchi_o);
Dwc.Meooe(chi_o, dchi_e);
Dwc.MeooeDag(phi_e, dphi_o);
Dwc.MeooeDag(phi_o, dphi_e);
ComplexD pDce = innerProduct(phi_e, dchi_e);
ComplexD pDco = innerProduct(phi_o, dchi_o);
ComplexD cDpe = innerProduct(chi_e, dphi_e);
ComplexD cDpo = innerProduct(chi_o, dphi_o);
std::cout << GridLogMessage << "e " << pDce << " " << cDpe << std::endl;
std::cout << GridLogMessage << "o " << pDco << " " << cDpo << std::endl;
std::cout << GridLogMessage << "pDce - conj(cDpo) " << pDce - conj(cDpo) << std::endl;
std::cout << GridLogMessage << "pDco - conj(cDpe) " << pDco - conj(cDpe) << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
std::cout << GridLogMessage << "= Test MeeInv Mee = 1 (if csw!=0) " << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
Dwc.Mooee(chi_e, src_e);
Dwc.MooeeInv(src_e, phi_e);
Dwc.Mooee(chi_o, src_o);
Dwc.MooeeInv(src_o, phi_o);
setCheckerboard(phi, phi_e);
setCheckerboard(phi, phi_o);
err = phi - chi;
std::cout << GridLogMessage << "norm diff " << norm2(err) << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
std::cout << GridLogMessage << "= Test MeeDag MeeInvDag = 1 (if csw!=0) " << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
Dwc.MooeeDag(chi_e, src_e);
Dwc.MooeeInvDag(src_e, phi_e);
Dwc.MooeeDag(chi_o, src_o);
Dwc.MooeeInvDag(src_o, phi_o);
setCheckerboard(phi, phi_e);
setCheckerboard(phi, phi_o);
err = phi - chi;
std::cout << GridLogMessage << "norm diff " << norm2(err) << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
std::cout << GridLogMessage << "= Test MeeInv MeeDag = 1 (if csw!=0) " << std::endl;
std::cout << GridLogMessage << "==============================================================" << std::endl;
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
Dwc.MooeeDag(chi_e, src_e);
Dwc.MooeeInv(src_e, phi_e);
Dwc.MooeeDag(chi_o, src_o);
Dwc.MooeeInv(src_o, phi_o);
setCheckerboard(phi, phi_e);
setCheckerboard(phi, phi_o);
err = phi - chi;
std::cout << GridLogMessage << "norm diff " << norm2(err) << std::endl;
std::cout << GridLogMessage << "================================================================" << std::endl;
std::cout << GridLogMessage << "= Testing gauge covariance Clover term with EO preconditioning " << std::endl;
std::cout << GridLogMessage << "================================================================" << std::endl;
chi = zero;
phi = zero;
tmp = zero;
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
pickCheckerboard(Even, phi_e, phi);
pickCheckerboard(Odd, phi_o, phi);
Dwc.Mooee(src_e, chi_e);
Dwc.Mooee(src_o, chi_o);
setCheckerboard(chi, chi_e);
setCheckerboard(chi, chi_o);
setCheckerboard(src, src_e);
setCheckerboard(src, src_o);
////////////////////// Gauge Transformation
std::vector<int> seeds2({5, 6, 7, 8});
GridParallelRNG pRNG2(&Grid);
pRNG2.SeedFixedIntegers(seeds2);
LatticeColourMatrix Omega(&Grid);
LatticeColourMatrix ShiftedOmega(&Grid);
LatticeGaugeField U_prime(&Grid);
U_prime = zero;
LatticeColourMatrix U_prime_mu(&Grid);
U_prime_mu = zero;
SU<Nc>::LieRandomize(pRNG2, Omega, 1.0);
for (int mu = 0; mu < Nd; mu++)
{
U[mu] = peekLorentz(Umu, mu);
ShiftedOmega = Cshift(Omega, mu, 1);
U_prime_mu = Omega * U[mu] * adj(ShiftedOmega);
pokeLorentz(U_prime, U_prime_mu, mu);
}
/////////////////
WilsonCloverFermionR Dwc_prime(U_prime, Grid, RBGrid, mass, csw_r, csw_t, anis, params);
Dwc_prime.ImportGauge(U_prime);
tmp = Omega * src;
pickCheckerboard(Even, src_e, tmp);
pickCheckerboard(Odd, src_o, tmp);
Dwc_prime.Mooee(src_e, phi_e);
Dwc_prime.Mooee(src_o, phi_o);
setCheckerboard(phi, phi_e);
setCheckerboard(phi, phi_o);
err = chi - adj(Omega) * phi;
std::cout << GridLogMessage << "norm diff " << norm2(err) << std::endl;
std::cout << GridLogMessage << "=================================================================" << std::endl;
std::cout << GridLogMessage << "= Testing gauge covariance Clover term w/o EO preconditioning " << std::endl;
std::cout << GridLogMessage << "================================================================" << std::endl;
chi = zero;
phi = zero;
WilsonFermionR Dw(Umu, Grid, RBGrid, mass, params);
Dw.ImportGauge(Umu);
Dw.M(src, result);
Dwc.M(src, chi);
Dwc_prime.M(Omega * src, phi);
WilsonFermionR Dw_prime(U_prime, Grid, RBGrid, mass, params);
Dw_prime.ImportGauge(U_prime);
Dw_prime.M(Omega * src, result2);
err = chi - adj(Omega) * phi;
err2 = result - adj(Omega) * result2;
std::cout << GridLogMessage << "norm diff Wilson " << norm2(err) << std::endl;
std::cout << GridLogMessage << "norm diff WilsonClover " << norm2(err2) << std::endl;
std::cout << GridLogMessage << "==========================================================" << std::endl;
std::cout << GridLogMessage << "= Testing Mooee(csw=0) Clover to reproduce Mooee Wilson " << std::endl;
std::cout << GridLogMessage << "==========================================================" << std::endl;
chi = zero;
phi = zero;
err = zero;
WilsonCloverFermionR Dwc_csw0(Umu, Grid, RBGrid, mass, 0.0, 0.0, anis, params); // <-- Notice: csw=0
Dwc_csw0.ImportGauge(Umu);
pickCheckerboard(Even, phi_e, phi);
pickCheckerboard(Odd, phi_o, phi);
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
Dw.Mooee(src_e, chi_e);
Dw.Mooee(src_o, chi_o);
Dwc_csw0.Mooee(src_e, phi_e);
Dwc_csw0.Mooee(src_o, phi_o);
setCheckerboard(chi, chi_e);
setCheckerboard(chi, chi_o);
setCheckerboard(phi, phi_e);
setCheckerboard(phi, phi_o);
setCheckerboard(src, src_e);
setCheckerboard(src, src_o);
err = chi - phi;
std::cout << GridLogMessage << "norm diff " << norm2(err) << std::endl;
std::cout << GridLogMessage << "==========================================================" << std::endl;
std::cout << GridLogMessage << "= Testing EO operator is equal to the unprec " << std::endl;
std::cout << GridLogMessage << "==========================================================" << std::endl;
chi = zero;
phi = zero;
err = zero;
pickCheckerboard(Even, phi_e, phi);
pickCheckerboard(Odd, phi_o, phi);
pickCheckerboard(Even, chi_e, chi);
pickCheckerboard(Odd, chi_o, chi);
// M phi = (Mooee src_e + Meooe src_o , Meooe src_e + Mooee src_o)
Dwc.M(src, ref); // Reference result from the unpreconditioned operator
// EO matrix
Dwc.Mooee(src_e, chi_e);
Dwc.Mooee(src_o, chi_o);
Dwc.Meooe(src_o, phi_e);
Dwc.Meooe(src_e, phi_o);
phi_o += chi_o;
phi_e += chi_e;
setCheckerboard(phi, phi_e);
setCheckerboard(phi, phi_o);
err = ref - phi;
std::cout << GridLogMessage << "ref (unpreconditioned operator) diff :" << norm2(ref) << std::endl;
std::cout << GridLogMessage << "phi (EO decomposition) diff :" << norm2(phi) << std::endl;
std::cout << GridLogMessage << "norm diff :" << norm2(err) << std::endl;
Grid_finalize();
}