void test_svd(const std::string & fn, ScalarType EPS)
{
std::size_t sz1, sz2;
//read matrix
// sz1 = 2048, sz2 = 2048;
// std::vector<ScalarType> in(sz1 * sz2);
// random_fill(in);
// read file
std::fstream f(fn.c_str(), std::fstream::in);
//read size of input matrix
read_matrix_size(f, sz1, sz2);
std::size_t to = std::min(sz1, sz2);
viennacl::matrix<ScalarType> Ai(sz1, sz2), Aref(sz1, sz2), QL(sz1, sz1), QR(sz2, sz2);
read_matrix_body(f, Ai);
std::vector<ScalarType> sigma_ref(to);
read_vector_body(f, sigma_ref);
f.close();
// viennacl::fast_copy(&in[0], &in[0] + in.size(), Ai);
Aref = Ai;
viennacl::tools::timer timer;
timer.start();
viennacl::linalg::svd(Ai, QL, QR);
viennacl::backend::finish();
double time_spend = timer.get();
viennacl::matrix<ScalarType> result1(sz1, sz2), result2(sz1, sz2);
result1 = viennacl::linalg::prod(QL, Ai);
result2 = viennacl::linalg::prod(result1, trans(QR));
ScalarType sigma_diff = sigmas_compare(Ai, sigma_ref);
ScalarType prods_diff = matrix_compare(result2, Aref);
bool sigma_ok = (fabs(sigma_diff) < EPS)
&& (fabs(prods_diff) < std::sqrt(EPS)); //note: computing the product is not accurate down to 10^{-16}, so we allow for a higher tolerance here
printf("%6s [%dx%d] %40s sigma_diff = %.6f; prod_diff = %.6f; time = %.6f\n", sigma_ok?"[[OK]]":"[FAIL]", (int)Aref.size1(), (int)Aref.size2(), fn.c_str(), sigma_diff, prods_diff, time_spend);
if (!sigma_ok)
exit(EXIT_FAILURE);
}
int main(int argc, char *argv[]) {
if(argc != 6 && argc != 7) {
printf("usage: rrg N n s D seed (id_string)\n");
return 1;
}
time_t t1,t2,tI,tF;
ITensor U,Dg,P,S;
Index ei;
// RRG structure parameters
const int N = atoi(argv[1]); // should be n*(power of 2)
const int n = atoi(argv[2]); // initial blocking size
int w = n; // block size (scales with m)
int ll = 0; // lambda block index
int m = 0; // RG scale factor
// AGSP and subspace parameters
const double t = 0.3; // Trotter temperature
const int M = 100; // num Trotter steps
const int k = 1; // power of Trotter op (just use 1)
const int s = atoi(argv[3]); // formal s param
const int D = atoi(argv[4]); // formal D param
// computational settings
const bool doI = true; // diag restricted Hamiltonian iteratively?
// setup random sampling
std::random_device r;
const int seed = atoi(argv[5]);
fprintf(stderr,"seed is %d\n",seed);
std::mt19937 gen(seed);
std::uniform_real_distribution<double> udist(0.0,1.0);
FILE *sxfl,*syfl,*szfl,*gsfl;
char id[128],sxnm[256],synm[256],sznm[256],gsnm[256];
if(argc == 6) sprintf(id,"rrg-L%d-s%d-D%d",N,s,D);
else sprintf(id,"%s",argv[6]);
strcat(sxnm,id); strcat(sxnm,"-sx.dat");
strcat(synm,id); strcat(synm,"-sy.dat");
strcat(sznm,id); strcat(sznm,"-sz.dat");
strcat(gsnm,id); strcat(gsnm,"-gs.dat");
sxfl = fopen(sxnm,"a");
syfl = fopen(synm,"a");
szfl = fopen(sznm,"a");
gsfl = fopen(gsnm,"a");
// initialize Hilbert subspaces for each level m = 0,...,log(N/n)
vector<SpinHalf> hsps;
for(int x = n ; x <= N ; x *= 2) hsps.push_back(SpinHalf(x));
SpinHalf hs = hsps.back();
// generate product basis over m=0 Hilbert space
auto p = int(pow(2,n));
vector<MPS> V1;
for(int i = 0 ; i < p ; ++i) {
InitState istate(hsps[0],"Dn");
for(int j = 1 ; j <= n ; ++j)
if(i/(int)pow(2,j-1)%2) istate.set(j,"Up");
V1.push_back(MPS(istate));
}
MPS bSpaceL(hsps[0]);
MPS bSpaceR(hsps[0]);
makeVS(V1,bSpaceL,LEFT);
makeVS(V1,bSpaceR,RIGHT);
// Hamiltonian parameters
const double Gamma = 2.0;
vector<double> J(2*(N-1));
fprintf(stdout,"# Hamiltonian terms Jx1,Jy1,Jx2,... (seed=%d)\n",seed);
for(int i = 0 ; i < N-1 ; ++i) {
J[2*i+0] = pow(udist(gen),Gamma);
J[2*i+1] = pow(udist(gen),Gamma);
fprintf(stdout,"%16.14f,%16.14f",J[2*i],J[2*i+1]);
if(i != N-2) fprintf(stdout,",");
}
fprintf(stdout,"\n");
fflush(stdout);
// initialize H for full system and extract block Hamiltonians
AutoMPO autoH(hs);
std::stringstream sts;
auto out = std::cout.rdbuf(sts.rdbuf());
vector<vector<MPO> > Hs(hsps.size());
for(int i = 1 ; i < N ; ++i) {
autoH += (J[2*(i-1)]-J[2*(i-1)+1]),"S+",i,"S+",i+1;
autoH += (J[2*(i-1)]-J[2*(i-1)+1]),"S-",i,"S-",i+1;
autoH += (J[2*(i-1)]+J[2*(i-1)+1]),"S+",i,"S-",i+1;
autoH += (J[2*(i-1)]+J[2*(i-1)+1]),"S-",i,"S+",i+1;
}
auto H = toMPO<ITensor>(autoH,{"Exact",true});
std::cout.rdbuf(out);
for(auto i : args(hsps)) extractBlocks(autoH,Hs[i],hsps[i]);
vector<MPO> prodSz,prodSx,projSzUp,projSzDn,projSxUp,projSxDn;
for(auto& it : hsps) {
auto curSz = sysOp(it,"Sz",2.0).toMPO(); prodSz.push_back(curSz);
auto curSx = sysOp(it,"Sx",2.0).toMPO(); prodSx.push_back(curSx);
auto curSzUp = sysOp(it,"Id").toMPO(); curSzUp.plusEq(curSz); curSzUp /= 2.0;
auto curSzDn = sysOp(it,"Id").toMPO(); curSzDn.plusEq(-1.0*curSz); curSzDn /= 2.0;
auto curSxUp = sysOp(it,"Id").toMPO(); curSxUp.plusEq(curSx); curSxUp /= 2.0;
auto curSxDn = sysOp(it,"Id").toMPO(); curSxDn.plusEq(-1.0*curSx); curSxDn /= 2.0;
projSzUp.push_back(curSzUp); projSzDn.push_back(curSzDn);
projSxUp.push_back(curSxUp); projSxDn.push_back(curSxDn);
}
// approximate the thermal operator exp(-H/t)^k using Trotter
// and MPO multiplication; temperature of K is k/t
time(&tI);
MPO eH(hs);
twoLocalTrotter(eH,t,M,autoH);
auto K = eH;
for(int i = 1 ; i < k ; ++i) {
nmultMPO(eH,K,K,{"Cutoff",eps,"Maxm",MAXBD});
K.Aref(1) *= 1.0/norm(K.A(1));
}
// INITIALIZATION: reduce dimension by sampling from initial basis, either
// bSpaceL or bSpaceR depending on how the merge will work
vector<MPS> Spre;
for(ll = 0 ; ll < N/n ; ll++) {
auto xs = ll % 2 ? 1 : n; // location of dangling Select index
auto cur = ll % 2 ? bSpaceR : bSpaceL;
Index si("ext",s,Select);
// return orthonormal basis of evecs
auto eigs = diagHermitian(-overlapT(cur,Hs[0][ll],cur),P,S,{"Maxm",s});
cur.Aref(xs) *= P*delta(commonIndex(P,S),si);
regauge(cur,xs,{"Truncate",false});
Spre.push_back(cur);
}
time(&t2);
fprintf(stderr,"initialization: %.f s\n",difftime(t2,tI));
// ITERATION: proceed through RRG hierarchy, increasing the scale m
vector<MPS> Spost;
for(m = 0 ; (int)Spre.size() > 1 ; ++m,w*=2) {
fprintf(stderr,"Level %d (w = %d)\n",m,w);
auto hs = hsps[m];
auto DD = D;//max(4,D/(int(log2(N/n)-m)));
auto thr = 1e-8;
Spost.clear();
// EXPAND STEP: for each block, expand dimension of subspace with AGSP operators
for(ll = 0 ; ll < N/w ; ++ll) {
MPO A(hs) , Hc = Hs[m][ll];
MPS pre = Spre[ll] , ret(hs);
int xs = ll % 2 ? 1 : w;
// STEP 1: extract filtering operators A from AGSP K
time(&t1);
restrictMPO(K,A,w*ll+1,DD,ll%2);
time(&t2);
fprintf(stderr,"trunc AGSP: %.f s\n",difftime(t2,t1));
// STEP 2: expand subspace using the mapping A:pre->ret
time(&t1);
ret = applyMPO(A,pre,ll%2,{"Cutoff",eps,"Maxm",MAXBD});
time(&t2);
fprintf(stderr,"apply AGSP: %.f s\n",difftime(t2,t1));
// rotate into principal components of subspace, poxsibly reducing dimension
// and stabilizing numerics, then store subspace in eigenbasis of block H
time(&t1);
diagHermitian(overlapT(ret,ret),U,Dg,{"Cutoff",thr});
time(&t2);
ei = Index("ext",int(commonIndex(Dg,U)),Select);
Dg.apply(invsqrt);
ret.Aref(xs) *= dag(U)*Dg*delta(prime(commonIndex(Dg,U)),ei);
fprintf(stderr,"rotate MPS: %.f s\n",difftime(t2,t1));
auto eigs = diagHermitian(-overlapT(ret,Hs[m][ll],ret),P,S);
ret.Aref(xs) *= P*delta(commonIndex(P,S),ei);
ret.Aref(xs) *= 1.0/sqrt(overlapT(ret,ret).real(ei(1),prime(ei)(1)));
regauge(ret,xs,{"Cutoff",eps});
fprintf(stderr,"max m: %d\n",maxM(ret));
Spost.push_back(ret);
}
// MERGE/REDUCE STEP: construct tensor subspace, sample to reduce dimension
Spre.clear();
for(ll = 0 ; ll < N/w ; ll+=2) {
auto spL = Spost[ll]; // L subspace
auto spR = Spost[ll+1]; // R subspace
// STEP 1: find s lowest eigenpairs of restricted H
time(&t1);
auto tpH = tensorProdContract(spL,spR,Hs[m+1][ll/2]);
tensorProdH<ITensor> resH(tpH);
resH.diag(s,doI);
P = resH.eigenvectors();
time(&t2);
fprintf(stderr,"diag restricted H: %.f s\n",difftime(t2,t1));
// STEP 2: tensor viable sets on each side and reduce dimension
MPS ret(hsps[m+1]);
time(&t1);
tensorProduct(spL,spR,ret,P,(ll/2)%2);
time(&t2);
fprintf(stderr,"tensor product (ll=%d): %.f s\n",ll,difftime(t2,t1));
Spre.push_back(ret);
}
}
// EXIT: extract two lowest energy candidate states to determine gap
auto res = Spre[0];
auto fi = Index("ext",s/2,Select);
vector<MPS> resSz = {res,res};
// project to Sz sectors of the eigenspace
diagHermitian(overlapT(res,prodSz[m],res),U,Dg);
resSz[0].Aref(N) *= U*delta(commonIndex(U,Dg),fi);
diagHermitian(overlapT(res,-1.0*prodSz[m],res),U,Dg);
resSz[1].Aref(N) *= U*delta(commonIndex(U,Dg),fi);
vector<MPS> evecs(2);
for(int i : range(2)) {
auto fc = resSz[i];
// diagonalize H within the Sz sectors
auto eigs = diagHermitian(-overlapT(fc,H,fc),P,S);
fc.Aref(N) *= (P*setElt(commonIndex(P,S)(1)));
fc.orthogonalize({"Cutoff",epx,"Maxm",MAXBD});
fc.normalize();
if(i == 0)
fprintf(stderr,"RRG gs energy: %17.14f\n",overlap(fc,H,fc));
evecs[i] = fc;
}
time(&t2);
Real vz,vx;
vz = overlap(evecs[0],prodSz[m],evecs[0]); vx = overlap(evecs[0],prodSx[m],evecs[0]);
fprintf(stderr,"Vz,vx of 0 is: %17.14f,%17.14f\n",vz,vx);
vz = overlap(evecs[1],prodSz[m],evecs[1]); vx = overlap(evecs[1],prodSx[m],evecs[1]);
fprintf(stderr,"Vz,vx of 1 is: %17.14f,%17.14f\n",vz,vx);
int x1_up = (vx > 0.0 ? 1 : 0);
evecs[0] = exactApplyMPO(evecs[0],projSzUp[m],{"Cutoff",epx});
evecs[0] = exactApplyMPO(evecs[0],projSxUp[m],{"Cutoff",epx});
evecs[1] = exactApplyMPO(evecs[1],projSzDn[m],{"Cutoff",epx});
evecs[1] = exactApplyMPO(evecs[1],(x1_up?projSxUp[m]:projSxDn[m]),{"Cutoff",epx});
for(auto& it : evecs) it.normalize();
fprintf(stderr,"gs candidate energy: %17.14f\nRRG BD ",overlap(evecs[0],H,evecs[0]));
for(const auto& it : evecs) fprintf(stderr,"%d ",maxM(it));
fprintf(stderr,"\telapsed: %.f s\n",difftime(t2,tI));
// CLEANUP: use DMRG to improve discovered evecs
vector<Real> evals(2),e_prev(2);
int max_iter = 30 , used_max = 0;
Real flr = 1e-13 , over_conv = 1e-1 , gap = 1.0 , conv = over_conv*gap , max_conv = 1.0;
for(int i = 0 ; i < (int)evecs.size() ; ++i) evals[i] = overlap(evecs[i],H,evecs[i]);
for(int i = 0 ; (i < 2 || conv < max_conv) && i < max_iter ; ++i) {
e_prev = evals;
time(&t1);
evals = dmrgMPO(H,evecs,8,{"Penalty",0.1,"Cutoff",epx});
time(&t2);
gap = evals[1]-evals[0];
max_conv = 0.0;
for(auto& j : range(2))
if(fabs(e_prev[j]-evals[j]) > max_conv) max_conv = e_prev[j]-evals[j];
fprintf(stderr,"DMRG BD ");
for(const auto& it : evecs) fprintf(stderr,"%3d ",maxM(it));
fprintf(stderr,"\tgap: %e\tconv=%9.2e,%9.2e\telapsed: %.f s\n",gap,
e_prev[0]-evals[0],e_prev[1]-evals[1],difftime(t2,t1));
conv = max(over_conv*gap,flr);
if(i == max_iter) used_max = 1;
}
for(int i = 0 ; i < (int)evecs.size() ; ++i) {
vz = overlap(evecs[i],prodSz[m],evecs[i]); vx = overlap(evecs[i],prodSx[m],evecs[i]);
fprintf(stderr,"Vz,vx of %d is: %12.9f,%12.9f\n",i,vz,vx);
}
evecs[0] = exactApplyMPO(evecs[0],projSzUp[m],{"Cutoff",1e-16});
evecs[0] = exactApplyMPO(evecs[0],projSxUp[m],{"Cutoff",1e-16});
evecs[1] = exactApplyMPO(evecs[1],projSzDn[m],{"Cutoff",1e-16});
evecs[1] = exactApplyMPO(evecs[1],(x1_up?projSxUp[m]:projSxDn[m]),{"Cutoff",1e-16});
for(auto& it : evecs) it.normalize();
for(auto i : range(evecs.size())) evals[i] = overlap(evecs[i],H,evecs[i]);
time(&tF);
auto gsR = evecs[0];
auto ee = measEE(gsR,N/2);
gap = evals[1]-evals[0];
fprintf(stderr,"gs: %17.14f gap: %15.9e ee: %10.8f\n",evals[0],gap,ee);
fprintf(gsfl,"# GS data (L=%d s=%d D=%d seed=%d time=%.f)\n",N,s,D,seed,difftime(tF,tI));
if(used_max) fprintf(gsfl,"# WARNING max iterations reached\n");
fprintf(gsfl,"%17.14f\t%15.9e\t%10.8f\n",evals[0],gap,ee);
// Compute two-point correlation functions in ground state via usual MPS method
fprintf(sxfl,"# SxSx corr matrix (L=%d s=%d D=%d seed=%d)\n",N,s,D,seed);
fprintf(syfl,"# SySy corr matrix (L=%d s=%d D=%d seed=%d)\n",N,s,D,seed);
fprintf(szfl,"# SzSz corr matrix (L=%d s=%d D=%d seed=%d)\n",N,s,D,seed);
for(int i = 1 ; i <= N ; ++i) {
gsR.position(i,{"Cutoff",0.0});
auto SxA = hs.op("Sx",i); auto SyA = hs.op("Sy",i); auto SzA = hs.op("Sz",i);
for(int j = 1 ; j <= N ; ++j) {
if(j <= i) {
fprintf(sxfl,"%15.12f\t",0.0);
fprintf(syfl,"%15.12f\t",0.0);
fprintf(szfl,"%15.12f\t",0.0);
} else {
auto SxB = hs.op("Sx",j); auto SyB = hs.op("Sy",j); auto SzB = hs.op("Sz",j);
fprintf(sxfl,"%15.12f\t",measOp(gsR,SxA,i,SxB,j));
fprintf(syfl,"%15.12f\t",measOp(gsR,SyA,i,SyB,j));
fprintf(szfl,"%15.12f\t",measOp(gsR,SzA,i,SzB,j));
}
}
fprintf(sxfl,"\n");
fprintf(syfl,"\n");
fprintf(szfl,"\n");
}
fclose(sxfl);
fclose(syfl);
fclose(szfl);
fclose(gsfl);
return 0;
}