TYPE nrm2( //
const ArrayListV<TYPE>& ax //[in]
) {
int n = ax.size();
const TYPE* x = ax.getPointer();
return nrm2(n, x, 1);
}
VALUE nrmp( //
int n, //size of the array, sx.size==sy.size
const VALUE* x, //
VALUE2 p, //
int incx //
) {
if (p == 1.0) {
return asum(n, x, incx);
}
if (p == 2.0) {
return nrm2(n, x, incx);
}
VALUE zero = 0.0e+0;
VALUE norm = zero;
if (n <= 0 || incx <= 0) {
return norm;
} else if (n == 1) {
norm = Abs(x[0]);
} else {
for (int i = 0; i < (n - 1) * incx; i += incx) {
if (x[i] != zero) {
norm += pow(Abs(x[i]), p);
}
}
norm = pow(norm, 1.0 / double(p));
}
return norm;
} // << ------------------------------------------
ColumnBundleTransform::ColumnBundleTransform(const vector3<>& kC, const Basis& basisC, const vector3<>& kD,
const ColumnBundleTransform::BasisWrapper& basisDwrapper, int nSpinor, const matrix3<int>& sym, int invert, const matrix3<int>& super)
: kC(kC), basisC(basisC), kD(kD), basisD(basisDwrapper.basis), nSpinor(nSpinor), invert(invert)
{
//Check k-point transformation and determine offset
const matrix3<>& metricC = basisC.gInfo->RTR;
myassert(nrm2(metricC - (~sym)*metricC*sym) < symmThreshold * nrm2(metricC)); //check symmetry
myassert(abs(invert) == 1); //check inversion
myassert(nrm2(basisC.gInfo->R * super - basisD.gInfo->R) < symmThreshold * nrm2(basisD.gInfo->R)); //check supercell
matrix3<int> affine = sym * invert * super; //net affine transformation
double offsetErr;
vector3<int> offset = round(kC * affine - kD, &offsetErr);
myassert(offsetErr < symmThreshold);
//Initialize index map:
index.resize(basisC.nbasis);
for(size_t n=0; n<basisC.nbasis; n++)
{ const vector3<int>& iG_C = basisC.iGarr[n]; //C recip lattice coords
vector3<int> iG_D = iG_C * affine + offset; //corresponding D recip lattice coords
index[n] = basisDwrapper.table[dot(basisDwrapper.pitch, iG_D + basisDwrapper.iGbox)]; //use lookup table to get D index
}
myassert(*std::min_element(index.begin(), index.end()) >= 0); //make sure all entries were found
#ifdef GPU_ENABLED
cudaMalloc(&indexGpu, sizeof(int)*index.size()); gpuErrorCheck();
cudaMemcpy(indexGpu, index.data(), sizeof(int)*index.size(), cudaMemcpyHostToDevice); gpuErrorCheck();
indexPref = indexGpu;
#else
indexPref = index.data();
#endif
//Initialize spinor transformation:
switch(nSpinor)
{ case 1: spinorRot = eye(1); break;
case 2:
{ spinorRot = Symmetries::getSpinorRotation(~(basisC.gInfo->R * sym * inv(basisC.gInfo->R)));
if(invert<0)
{ matrix sInvert = zeroes(2,2);
sInvert.set(0,1, 1.);
sInvert.set(1,0, -1.);
spinorRot = conj(sInvert * spinorRot);
}
break;
}
default: myassert(!"Invalid value for nSpinor");
}
}
/*! */
inline dquater vt2q(const dcovec3& v, const double& theta)
{VERBOSE_REPORT;
return vr2q( v/(nrm2(v)+DBL_MIN)*std::sin(0.5*theta), std::cos(0.5*theta) );
}
/*! inverse */
inline dquater inv(const dquater& q)
{VERBOSE_REPORT;
return conj(q)/pow(nrm2(q),2);
}
/***************************************
* Conjugate Gradient *
* This function will do the CG *
* algorithm without preconditioning. *
* For optimiziation you must not *
* change the algorithm. *
***************************************
r(0) = b - Ax(0)
p(0) = r(0)
rho(0) = <r(0),r(0)>
***************************************
for k=0,1,2,...,n-1
q(k) = A * p(k)
dot_pq = <p(k),q(k)>
alpha = rho(k) / dot_pq
x(k+1) = x(k) + alpha*p(k)
r(k+1) = r(k) - alpha*q(k)
check convergence ||r(k+1)||_2 < eps
rho(k+1) = <r(k+1), r(k+1)>
beta = rho(k+1) / rho(k)
p(k+1) = r(k+1) + beta*p(k)
***************************************/
void cg(const int n, const int nnz, const int maxNNZ, const floatType* data, const int* indices, const int* length, const floatType* b, floatType* x, struct SolverConfig* sc){
floatType* r, *p, *q;
floatType alpha, beta, rho, rho_old, dot_pq, bnrm2;
int iter;
double timeMatvec_s;
double timeMatvec=0;
int i;
floatType temp;
/* allocate memory */
r = (floatType*)malloc(n * sizeof(floatType));
p = (floatType*)malloc(n * sizeof(floatType));
q = (floatType*)malloc(n * sizeof(floatType));
#pragma acc data copyin(data[0:n*maxNNZ], indices[0:n*maxNNZ], length[0:n], n, nnz, maxNNZ, b[0:n]) copy(x[0:n]) create(alpha, beta, r[0:n], p[0:n], q[0:n], i, temp) //eigentlich auch copy(x[0:n]) aber error: not found on device???
{
DBGMAT("Start matrix A = ", n, nnz, maxNNZ, data, indices, length)
DBGVEC("b = ", b, n);
DBGVEC("x = ", x, n);
/* r(0) = b - Ax(0) */
timeMatvec_s = getWTime();
matvec(n, nnz, maxNNZ, data, indices, length, x, r);
//hier inline ausprobieren
/*int i, j, k;
#pragma acc parallel loop present(data, indices, length, x)
for (i = 0; i < n; i++) {
r[i] = 0;
for (j = 0; j < length[i]; j++) {
k = j * n + i;
r[i] += data[k] * x[indices[k]];
}
}*/
timeMatvec += getWTime() - timeMatvec_s;
xpay(b, -1.0, n, r);
DBGVEC("r = b - Ax = ", r, n);
/* Calculate initial residuum */
nrm2(r, n, &bnrm2);
bnrm2 = 1.0 /bnrm2;
/* p(0) = r(0) */
memcpy(p, r, n*sizeof(floatType));
DBGVEC("p = r = ", p, n);
/* rho(0) = <r(0),r(0)> */
vectorDot(r, r, n, &rho);
printf("rho_0=%e\n", rho);
for(iter = 0; iter < sc->maxIter; iter++){
DBGMSG("=============== Iteration %d ======================\n", iter);
/* q(k) = A * p(k) */
timeMatvec_s = getWTime();
matvec(n, nnz, maxNNZ, data, indices, length, p, q);
timeMatvec += getWTime() - timeMatvec_s;
DBGVEC("q = A * p= ", q, n);
/* dot_pq = <p(k),q(k)> */
vectorDot(p, q, n, &dot_pq);
DBGSCA("dot_pq = <p, q> = ", dot_pq);
/* alpha = rho(k) / dot_pq */
alpha = rho / dot_pq;
DBGSCA("alpha = rho / dot_pq = ", alpha);
/* x(k+1) = x(k) + alpha*p(k) */
axpy(alpha, p, n, x);
#pragma acc update host(x[0:n])
DBGVEC("x = x + alpha * p= ", x, n);
/* r(k+1) = r(k) - alpha*q(k) */
axpy(-alpha, q, n, r);
DBGVEC("r = r - alpha * q= ", r, n);
rho_old = rho;
DBGSCA("rho_old = rho = ", rho_old);
/* rho(k+1) = <r(k+1), r(k+1)> */
vectorDot(r, r, n, &rho);
DBGSCA("rho = <r, r> = ", rho);
/* Normalize the residual with initial one */
sc->residual= sqrt(rho) * bnrm2;
/* Check convergence ||r(k+1)||_2 < eps
* If the residual is smaller than the CG
* tolerance specified in the CG_TOLERANCE
* environment variable our solution vector
* is good enough and we can stop the
* algorithm. */
printf("res_%d=%e\n", iter+1, sc->residual);
if(sc->residual <= sc->tolerance)
break;
/* beta = rho(k+1) / rho(k) */
beta = rho / rho_old;
DBGSCA("beta = rho / rho_old= ", beta);
/* p(k+1) = r(k+1) + beta*p(k) */
xpay(r, beta, n, p);
DBGVEC("p = r + beta * p> = ", p, n);
}
/* Store the number of iterations and the
* time for the sparse matrix vector
* product which is the most expensive
* function in the whole CG algorithm. */
sc->iter = iter;
sc->timeMatvec = timeMatvec;
/* Clean up */
free(r);
free(p);
free(q);
}//ende data region
}
void Phonon::setup(bool printDefaults)
{
//Parse input to initialize unit cell:
parse(input, e, printDefaults);
logSuspend();
parse(input, eSupTemplate); //silently create a copy by re-parsing input (Everything is not trivially copyable)
logResume();
//Ensure phonon command specified:
if(!sup.length())
die("phonon supercell must be specified using the phonon command.\n");
if(!e.gInfo.S.length_squared())
die("Manual fftbox setting required for phonon. If supercell grid\n"
"initialization fails, specify slightly larger manual fftbox.\n");
//Check kpoint and supercell compatibility:
if(e.eInfo.qnums.size()>1 || e.eInfo.qnums[0].k.length_squared())
die("phonon requires a Gamma-centered uniform kpoint mesh.\n");
for(int j=0; j<3; j++)
{ if(!sup[j] || e.eInfo.kfold[j] % sup[j])
{ die("kpoint folding %d is not a multiple of supercell count %d for lattice direction %d.\n",
e.eInfo.kfold[j], sup[j], j);
}
eSupTemplate.eInfo.kfold[j] = e.eInfo.kfold[j] / sup[j];
}
logPrintf("########### Unit cell calculation #############\n");
SpeciesInfo::Constraint constraintFull;
constraintFull.moveScale = 0;
constraintFull.type = SpeciesInfo::Constraint::None;
for(size_t sp=0; sp<e.iInfo.species.size(); sp++)
e.iInfo.species[sp]->constraints.assign(e.iInfo.species[sp]->atpos.size(), constraintFull);
e.setup();
if(!e.coulombParams.supercell) e.updateSupercell(true); //force supercell generation
nSpins = e.eInfo.spinType==SpinZ ? 2 : 1;
nSpinor = e.eInfo.spinorLength();
//Initialize state of unit cell:
if(e.cntrl.dumpOnly)
{ //Single energy calculation so that all dependent quantities have been initialized:
logPrintf("\n----------- Energy evaluation at fixed state -------------\n"); logFlush();
e.eVars.elecEnergyAndGrad(e.ener, 0, 0, true);
}
else elecFluidMinimize(e);
logPrintf("# Energy components:\n"); e.ener.print(); logPrintf("\n");
//Determine optimum number of bands for supercell calculation:
nBandsOpt = 0;
for(int q=e.eInfo.qStart; q<e.eInfo.qStop; q++)
{ int nBands_q = std::upper_bound(e.eVars.F[q].begin(), e.eVars.F[q].end(), Fcut, std::greater<double>()) - e.eVars.F[q].begin();
nBandsOpt = std::max(nBandsOpt, nBands_q);
}
mpiUtil->allReduce(nBandsOpt, MPIUtil::ReduceMax);
logPrintf("Fcut=%lg reduced nBands from %d to %d per unit cell.\n", Fcut, e.eInfo.nBands, nBandsOpt);
//Make unit cell state available on all processes
//(since MPI division of qSup and q are different and independent of the map)
for(int q=0; q<e.eInfo.nStates; q++)
{ //Allocate:
if(!e.eInfo.isMine(q))
{ e.eVars.C[q].init(e.eInfo.nBands, e.basis[q].nbasis * e.eInfo.spinorLength(), &e.basis[q], &e.eInfo.qnums[q]);
e.eVars.F[q].resize(e.eInfo.nBands);
e.eVars.Hsub_eigs[q].resize(e.eInfo.nBands);
if(e.eInfo.fillingsUpdate==ElecInfo::FermiFillingsAux)
e.eVars.B[q].init(e.eInfo.nBands, e.eInfo.nBands);
}
//Broadcast from owner:
int qSrc = e.eInfo.whose(q);
e.eVars.C[q].bcast(qSrc);
e.eVars.F[q].bcast(qSrc);
e.eVars.Hsub_eigs[q].bcast(qSrc);
if(e.eInfo.fillingsUpdate==ElecInfo::FermiFillingsAux)
e.eVars.B[q].bcast(qSrc);
}
logPrintf("\n------- Configuring supercell and perturbation modes -------\n");
//Grid:
eSupTemplate.gInfo.S = Diag(sup) * e.gInfo.S; //ensure exact supercell
eSupTemplate.gInfo.R = e.gInfo.R * Diag(sup);
prodSup = sup[0] * sup[1] * sup[2];
//Replicate atoms (and related properties):
for(size_t sp=0; sp<e.iInfo.species.size(); sp++)
{ const SpeciesInfo& spIn = *(e.iInfo.species[sp]);
SpeciesInfo& spOut = *(eSupTemplate.iInfo.species[sp]);
spOut.atpos.clear();
spOut.initialMagneticMoments.clear();
matrix3<> invSup = inv(Diag(vector3<>(sup)));
vector3<int> iR;
for(iR[0]=0; iR[0]<sup[0]; iR[0]++)
for(iR[1]=0; iR[1]<sup[1]; iR[1]++)
for(iR[2]=0; iR[2]<sup[2]; iR[2]++)
{ for(vector3<> pos: spIn.atpos)
spOut.atpos.push_back(invSup * (pos + iR));
for(vector3<> M: spIn.initialMagneticMoments)
spOut.initialMagneticMoments.push_back(M); //needed only to determine supercell symmetries
}
spOut.constraints.assign(spOut.atpos.size(), constraintFull);
}
//Supercell symmetries:
eSupTemplate.symm.setup(eSupTemplate);
const std::vector< matrix3<int> >& symSup = eSupTemplate.symm.getMatrices();
symSupCart.clear();
eSupTemplate.gInfo.invR = inv(eSupTemplate.gInfo.R);
for(const matrix3<int>& m: symSup)
symSupCart.push_back(eSupTemplate.gInfo.R * m * eSupTemplate.gInfo.invR);
//Pick maximally symmetric orthogonal basis:
logPrintf("\nFinding maximally-symmetric orthogonal basis for displacements:\n");
std::vector< vector3<> > dirBasis;
{ std::multimap<int, vector3<> > dirList; //directions indexed by their stabilizer group cardinality
vector3<int> iR;
for(iR[0]=0; iR[0]<=+1; iR[0]++)
for(iR[1]=-1; iR[1]<=+1; iR[1]++)
for(iR[2]=-1; iR[2]<=+1; iR[2]++)
if(iR.length_squared())
{ //Try low-order lattice vector linear combination:
vector3<> n = eSupTemplate.gInfo.R * iR; n *= (1./n.length());
dirList.insert(std::make_pair(nStabilizer(n, symSupCart), n));
//Try low-order reciprocal lattice vector linear combination:
n = iR * eSupTemplate.gInfo.invR; n *= (1./n.length());
dirList.insert(std::make_pair(nStabilizer(n, symSupCart), n));
}
dirBasis.push_back(dirList.rbegin()->second);
//Pick second driection orthogonal to first:
std::multimap<int, vector3<> > dirList2;
for(auto entry: dirList)
{ vector3<> n = entry.second;
n -= dot(n, dirBasis[0]) * dirBasis[0];
if(n.length_squared() < symmThresholdSq) continue;
n *= (1./n.length());
dirList2.insert(std::make_pair(nStabilizer(n, symSupCart), n));
}
dirBasis.push_back(dirList2.rbegin()->second);
dirBasis.push_back(cross(dirBasis[0], dirBasis[1])); //third direction constrained by orthogonality
}
for(const vector3<>& n: dirBasis)
logPrintf(" [ %+lf %+lf %+lf ] |Stabilizer|: %d\n", n[0], n[1], n[2], nStabilizer(n,symSupCart));
//List all modes:
modes.clear();
for(size_t sp=0; sp<e.iInfo.species.size(); sp++)
for(size_t at=0; at<e.iInfo.species[sp]->atpos.size(); at++) //only need to move atoms in first unit cell
for(int iDir=0; iDir<3; iDir++)
{ Mode mode;
mode.sp = sp;
mode.at = at;
mode.dir[iDir] = 1.;
modes.push_back(mode);
}
//Find irreducible modes:
perturbations.clear();
for(unsigned sp=0; sp<e.iInfo.species.size(); sp++)
{ int nAtoms = e.iInfo.species[sp]->atpos.size();
int nPert = nAtoms * dirBasis.size();
//generate all perturbations first:
std::vector<Perturbation> pertSp(nPert); //perturbations of this species
std::vector<matrix> proj(nPert); //projection operator into subspace spanned by star of current perturbation
matrix projTot;
const auto& atomMap = eSupTemplate.symm.getAtomMap()[sp];
for(int iPert=0; iPert<nPert; iPert++)
{ pertSp[iPert].sp = sp;
pertSp[iPert].at = iPert / dirBasis.size();
pertSp[iPert].dir = dirBasis[iPert % dirBasis.size()];
pertSp[iPert].weight = 1./symSupCart.size();
for(unsigned iSym=0; iSym<symSupCart.size(); iSym++)
{ int at = atomMap[pertSp[iPert].at][iSym] % nAtoms; //map back to first cell
vector3<> dir = symSupCart[iSym] * pertSp[iPert].dir;
matrix nHat = zeroes(nPert,1);
for(int iDir=0; iDir<3; iDir++)
nHat.set(at*3+iDir,0, dir[iDir]);
proj[iPert] += pertSp[iPert].weight * nHat * dagger(nHat);
}
projTot += proj[iPert];
}
myassert(nrm2(projTot - eye(nPert)) < symmThreshold);
//only select perturbations with distinct subspace projections:
std::vector<bool> irred(nPert, true); //whether each perturbation is in irreducible set
for(int iPert=0; iPert<nPert; iPert++)
{ for(int jPert=0; jPert<iPert; jPert++)
if(irred[jPert] && nrm2(proj[iPert]-proj[jPert])<symmThreshold)
{ pertSp[jPert].weight += pertSp[iPert].weight; //send weight of current mode to its image in irreducible set
irred[iPert] = false; //this mode will be accounted for upon symmetrization
break;
}
}
for(int iPert=0; iPert<nPert; iPert++)
if(irred[iPert])
perturbations.push_back(pertSp[iPert]);
}
logPrintf("\n%d perturbations of the unit cell reduced to %d under symmetries:\n", int(modes.size()), int(perturbations.size()));
for(const Perturbation& pert: perturbations)
logPrintf("%s %d [ %+lf %+lf %+lf ] %lf\n", e.iInfo.species[pert.sp]->name.c_str(),
pert.at, pert.dir[0], pert.dir[1], pert.dir[2], pert.weight*symSupCart.size());
//Determine wavefunction unitary rotations:
logPrintf("\nCalculating unitary rotations of unit cell states under symmetries:\n");
stateRot.resize(nSpins);
double unitarityErr = 0.;
for(int iSpin=0; iSpin<nSpins; iSpin++)
{ //Find states involved in the supercell Gamma-point:
struct Kpoint : public Supercell::KmeshTransform
{ vector3<> k; //also store k-point for convenience (KmeshTransform doesn't have it)
};
std::vector<Kpoint> kpoints; kpoints.reserve(prodSup);
const Supercell& supercell = *(e.coulombParams.supercell);
for(unsigned ik=0; ik<supercell.kmesh.size(); ik++)
{ double kSupErr; round(matrix3<>(Diag(sup)) * supercell.kmesh[ik], &kSupErr);
if(kSupErr < symmThreshold) //maps to Gamma point
{ Kpoint kpoint;
(Supercell::KmeshTransform&)kpoint = supercell.kmeshTransform[ik]; //copy base class
kpoint.k = supercell.kmesh[ik];
kpoint.iReduced += iSpin*(e.eInfo.nStates/nSpins); //point to source k-point with appropriate spin
kpoints.push_back(kpoint);
}
}
myassert(int(kpoints.size()) == prodSup);
//Initialize basis and qnum for these states:
std::vector<QuantumNumber> qnums(prodSup);
std::vector<Basis> basis(prodSup);
logSuspend();
for(int ik=0; ik<prodSup; ik++)
{ qnums[ik].k = kpoints[ik].k;
qnums[ik].spin = (nSpins==1 ? 0 : (iSpin ? +1 : -1));
qnums[ik].weight = 1./prodSup;
basis[ik].setup(e.gInfo, e.iInfo, e.cntrl.Ecut, kpoints[ik].k);
}
logResume();
//Get wavefunctions for all these k-points:
#define whose_ik(ik) (((ik) * mpiUtil->nProcesses())/prodSup) //local MPI division
std::vector<ColumnBundle> C(prodSup);
std::vector<std::shared_ptr<ColumnBundleTransform::BasisWrapper> > basisWrapper(prodSup);
auto sym = e.symm.getMatrices(); //unit cell symmetries
for(int ik=0; ik<prodSup; ik++)
{ C[ik].init(e.eInfo.nBands, basis[ik].nbasis*nSpinor, &basis[ik], &qnums[ik], isGpuEnabled());
if(whose_ik(ik) == mpiUtil->iProcess())
{ int q = kpoints[ik].iReduced;
C[ik].zero();
basisWrapper[ik] = std::make_shared<ColumnBundleTransform::BasisWrapper>(basis[ik]);
ColumnBundleTransform(e.eInfo.qnums[q].k, e.basis[q], qnums[ik].k, *(basisWrapper[ik]),
nSpinor, sym[kpoints[ik].iSym], kpoints[ik].invert).scatterAxpy(1., e.eVars.C[q], C[ik],0,1);
}
}
for(int ik=0; ik<prodSup; ik++) C[ik].bcast(whose_ik(ik)); //make available on all processes
//Determine max eigenvalue:
int nBands = e.eInfo.nBands;
double Emax = -INFINITY;
for(int q=e.eInfo.qStart; q<e.eInfo.qStop; q++)
Emax = std::max(Emax, e.eVars.Hsub_eigs[q].back());
mpiUtil->allReduce(Emax, MPIUtil::MPIUtil::ReduceMax);
double EmaxValid = +INFINITY;
//Loop over supercell symmetry operations:
PeriodicLookup<QuantumNumber> plook(qnums, e.gInfo.GGT);
stateRot[iSpin].resize(symSupCart.size());
for(size_t iSym=0; iSym<symSupCart.size(); iSym++)
{ matrix3<> symUnitTmp = e.gInfo.invR * symSupCart[iSym] * e.gInfo.R; //in unit cell lattice coordinates
#define SymmErrMsg \
"Supercell symmetries do not map unit cell k-point mesh onto itself.\n" \
"This implies that the supercell is more symmetric than the unit cell!\n" \
"Please check to make sure that you have used the minimal unit cell.\n\n"
matrix3<int> symUnit;
for(int j1=0; j1<3; j1++)
for(int j2=0; j2<3; j2++)
{ symUnit(j1,j2) = round(symUnitTmp(j1,j2));
if(fabs(symUnit(j1,j2) - symUnitTmp(j1,j2)) > symmThreshold)
die(SymmErrMsg)
}
//Find image kpoints under rotation: (do this for all k-points so that all processes exit together if necessary)
std::vector<int> ikRot(prodSup);
for(int ik=0; ik<prodSup; ik++)
{ size_t ikRotCur = plook.find(qnums[ik].k * symUnit);
if(ikRotCur==string::npos) die(SymmErrMsg)
ikRot[ik] = ikRotCur;
}
#undef SymmErrMsg
//Calculate unitary transformation matrix:
stateRot[iSpin][iSym].init(prodSup, nBands);
for(int ik=0; ik<prodSup; ik++)
if(whose_ik(ikRot[ik]) == mpiUtil->iProcess()) //MPI division by target k-point
{ ColumnBundle Crot = C[ikRot[ik]].similar();
Crot.zero();
ColumnBundleTransform(qnums[ik].k, basis[ik], qnums[ikRot[ik]].k, *(basisWrapper[ikRot[ik]]),
nSpinor, symUnit, +1).scatterAxpy(1., C[ik], Crot,0,1);
matrix Urot = Crot ^ O(C[ikRot[ik]]); //will be unitary if Crot is a strict unitary rotation of C[ikRot[ik]]
//Check maximal subspace that is unitary: (remiander must be incomplete degenerate subspace)
int nBandsValid = nBands;
while(nBandsValid && !isUnitary(Urot(0,nBandsValid, 0,nBandsValid)))
nBandsValid--;
if(nBandsValid<nBands)
{ //Update energy range of validity:
EmaxValid = std::min(EmaxValid, e.eVars.Hsub_eigs[kpoints[ik].iReduced][nBandsValid]);
//Make valid subspace exactly unitary:
matrix UrotSub = Urot(0,nBandsValid, 0,nBandsValid);
matrix UrotOverlap = dagger(UrotSub) * UrotSub;
UrotSub = UrotSub * invsqrt(UrotOverlap); //make exactly unitary
unitarityErr += std::pow(nrm2(UrotOverlap - eye(nBandsValid)), 2);
//Zero out invalid subspace:
Urot.zero();
Urot.set(0,nBandsValid, 0,nBandsValid, UrotSub);
}
stateRot[iSpin][iSym].set(ik, ikRot[ik], Urot);
}
stateRot[iSpin][iSym].allReduce();
}
#undef whose_ik
mpiUtil->allReduce(EmaxValid, MPIUtil::ReduceMin);
if(nSpins>1) logPrintf("\tSpin %+d: ", iSpin ? +1 : -1); else logPrintf("\t");
logPrintf("Matrix elements valid for ");
if(std::isfinite(EmaxValid)) logPrintf("E < %+.6lf (Emax = %+.6lf) due to incomplete degenerate subspaces.\n", EmaxValid, Emax);
else logPrintf("all available states (all degenerate subspaces are complete).\n");
}
mpiUtil->allReduce(unitarityErr, MPIUtil::ReduceSum);
unitarityErr = sqrt(unitarityErr / (nSpins * prodSup * symSupCart.size()));
logPrintf("\tRMS unitarity error in valid subspaces: %le\n", unitarityErr);
}
inline bool isUnitary(const matrix& U) { return nrm2(U*dagger(U) - eye(U.nCols())) < symmThreshold; }
/*! solve */
int32_t gmres
(
const CPPL::dgsmatrix& A,
const CPPL::dcovector& b,
CPPL::dcovector& x,
const double& eps
)
{
///////////////////////////////////////////////
//////////////// preconditioner ///////////////
///////////////////////////////////////////////
CPPL::dgbmatrix Minv(x.l, x.l, 0, 0);
//////// no precondition ////////
Minv.identity();
///////////////////////////////////////////////
///////////////// mid values //////////////////
///////////////////////////////////////////////
long m(10);//restart number
CPPL::dcovector r(b-A*x);
CPPL::dcovector s(m+1), co(m+1), si(m+1), w;
std::vector<CPPL::dcovector> v(m+1);
CPPL::dgematrix H(m+1,m);
//H.zero();
//co.zero();
//si.zero();
//s.zero();
//////// norm ////////
double norm_r, norm_r_min(DBL_MAX);
const double norm_r_ini(fabs(damax(r)));
std::cerr << "[NOTE]@gmres: norm_r_ini=" << norm_r_ini << ", eps=" << eps<< std::endl;
if( norm_r_ini<DBL_MIN ){
std::cerr << "[NOTE]@gmres: already converged. v(^^)" << std::endl;
return 0;
}
///////////////////////////////////////////////
//////////////////// loop /////////////////////
///////////////////////////////////////////////
int itc(1);
//int itmax(int(2.1*x.l));
int itmax(int(1.1*x.l));
//int itmax(int(0.6*x.l));
do{
std::cerr << "** itc=" << itc << " ********************************************" << std::endl;
//////// 0 ////////
v[0] =r/nrm2(r);
s.zero();
s(0) =nrm2(r);
for(long i=0; i<m; i++){
//std::cerr << "++++ i=" << i << " ++++" << std::endl;
w =A*v[i];
w =Minv*w;
for(long k=0; k<i+1; k++){
H(k,i) =w%v[k];
w -=H(k,i)*v[k];
}
H(i+1,i) =nrm2(w);
v[i+1] =w/H(i+1,i);
//// J,s ////
for(long k=0; k<i; k++){
rotate(H(k,i), H(k+1,i), co(k), si(k));
}
make_rotator( H(i,i), H(i+1,i), co(i), si(i) );
//std::cerr << "co = " << t(co) << std::endl; std::cerr << "si = " << t(si) << std::endl;
rotate( H(i,i), H(i+1,i), co(i), si(i) );//necessary
//std::cerr << "H =\n" << H << std::endl;
rotate( s(i), s(i+1), co(i), si(i) );
//std::cerr << "s = " << t(s) << std::endl;
}
//for(long i=0; i<m+1; i++){ for(long j=i+1; j<m+1; j++){ std::cerr << "vv = " << v[i]%v[j] << std::endl; } }// v check
//std::cerr << "H =\n" << H << std::endl;
//std::cerr << "s =" << t(s) << std::endl;
//for(long i=0; i<m+1; i++){ std::cerr << "v["<<i<<"] =" << t(v[i]) << std::flush; }
//////// y ////////
CPPL::dcovector y(s);
for(long i=m-1; i>=0; i--){
y(i) /= H(i,i);
for(long j=i-1; j>=0; j--){
y(j) -= H(j,i) * y(i);
}
}
//std::cerr << "H*y = " << t(H*y) << std::endl;
//std::cerr << "s = " << t(s) << std::endl;
//std::cerr << "y = " << t(s) << std::endl;
//////// update ////////
for(long i=0; i<m; i++){
x += v[i] * y(i);
}
//std::cerr << "x = " << t(x) << std::endl;
//////// residual ////////
r =b-A*x;
r =Minv*r;
//std::cerr << "r = " << t(r) << std::endl;
//////// convergence check ////////
norm_r =fabs(damax(r));
std::cerr << "norm_r = " << norm_r << std::endl;
if( isnan(norm_r) ){ break; }//failed
if( !std::isnormal(norm_r) ){ break; }//failed
if( !std::isfinite(norm_r) ){ break; }//failed
if( norm_r>1e3*norm_r_ini ){ break; }//failed (getting so worse)
if( norm_r<=eps ){//r satistied
std::cerr << "[NOTE]@gmres: converged. v(^^) itc=" << itc << "/" << itmax << ", norm=" << norm_r << std::endl;
return 0;
}
}while(++itc<itmax);
//////// failed ////////
std::cerr << "[NOTE]@gmres: itc=" << itc << ", norm=" << norm_r << ", r_satisfied=" << (norm_r<=eps) << std::endl;
std::cerr << "[NOTE]@gmres: failed to converge. orz" << std::endl;
return 1;
}
void
KPMLinalg::normalize( my::scalar* x )
{
const my::real norm=1./nrm2(x);
scale(norm,x);
};
Quaternion inverse() const {
const double inv_nrm2 = 1.0 / nrm2();
return Quaternion(_real * inv_nrm2, - _imaginary * inv_nrm2);
}
double nrm() const {
return std::sqrt(nrm2());
}
structmass1 += mass;
COMVel1 += Sim.particleList[ID].getVelocity() * mass;
COMPos1 += Sim.particleList[ID].getPosition() * mass;
}
BOOST_FOREACH(const size_t& ID, range2)
{
double mass = Sim.dynamics.getSpecies(Sim.particleList[ID]).getMass(ID);
structmass2 += mass;
COMVel2 += Sim.particleList[ID].getVelocity() * mass;
COMPos2 += Sim.particleList[ID].getPosition() * mass;
}
COMVel1 /= structmass1;
COMVel2 /= structmass2;
COMPos1 /= structmass1;
COMPos2 /= structmass2;
rij = COMPos1 - COMPos2;
vij = COMVel1 - COMVel2;
Sim.dynamics.BCs().applyBC(rij, vij);
rvdot = (rij | vij);
r2 = rij.nrm2();
v2 = vij.nrm2();
}
/* Computes the norm of (I - A). */
double minusIdNrm2(Mat mA) {
addId(-1, mA); // sort of a hack, but it works very well
double norm = nrm2(mA);
addId(1, mA);
return norm;
}
int main(int argc, char** argv) {
MPI_Init(&argc, &argv);
/*# Init #*/
int rankWorld, sizeWorld;
MPI_Comm_size(MPI_COMM_WORLD, &sizeWorld);
MPI_Comm_rank(MPI_COMM_WORLD, &rankWorld);
const HpddmOption* const opt = HpddmOptionGet();
HpddmOptionParse(opt, argc, argv, rankWorld == 0);
{
char* val[4] = { "Nx=<100>", "Ny=<100>", "overlap=<1>", "generate_random_rhs=<0>" };
char* desc[4] = { "Number of grid points in the x-direction.", "Number of grid points in the y-direction.", "Number of grid points in the overlap.", "Number of generated random right-hand sides." };
HpddmOptionParseInts(opt, argc, argv, 4, val, desc);
val[0] = "symmetric_csr=(0|1)"; desc[0] = "Assemble symmetric matrices.";
val[1] = "nonuniform=(0|1)"; desc[1] = "Use a different number of eigenpairs to compute on each subdomain.";
HpddmOptionParseArgs(opt, argc, argv, 2, val, desc);
}
int sizes[8];
int* connectivity[8];
int o[8];
int neighbors = 0;
HpddmMatrixCSR* Mat, *MatNeumann = NULL;
K* f, *sol;
underlying_type* d;
int ndof;
generate(rankWorld, sizeWorld, &neighbors, o, sizes, connectivity, &ndof, &Mat, &MatNeumann, &d, &f, &sol);
unsigned short mu = HpddmOptionApp(opt, "generate_random_rhs");
int status = 0;
if(sizeWorld > 1) {
HpddmSchwarz* A = HpddmSchwarzCreate(Mat, neighbors, o, sizes, connectivity);
for(int i = 0; i < neighbors; ++i)
free(connectivity[i]);
HpddmSchwarzMultiplicityScaling(A, d);
HpddmSchwarzInitialize(A, d);
if(mu != 0)
HpddmSchwarzScaledExchange(A, f, mu);
else
mu = 1;
if(HpddmOptionSet(opt, "schwarz_coarse_correction")) {
double* addr = HpddmOptionAddr(opt, "geneo_nu");
unsigned short nu = *addr;
if(nu > 0) {
if(HpddmOptionApp(opt, "nonuniform"))
*addr += MAX((int)(-*addr + 1), pow(-1, rankWorld) * rankWorld);
HpddmSchwarzSolveGEVP(A, MatNeumann);
nu = HpddmOptionVal(opt, "geneo_nu");
}
else {
nu = 1;
K** deflation = malloc(sizeof(K*));
*deflation = malloc(sizeof(K) * ndof);
for(int i = 0; i < ndof; ++i)
deflation[0][i] = 1.0;
HpddmSetVectors(HpddmSchwarzPreconditioner(A), deflation);
}
HpddmInitializeCoarseOperator(HpddmSchwarzPreconditioner(A), nu);
HpddmSchwarzBuildCoarseOperator(A, MPI_COMM_WORLD);
/*# FactorizationEnd #*/
}
HpddmSchwarzCallNumfact(A);
if(rankWorld != 0)
HpddmOptionRemove(opt, "verbosity");
const MPI_Comm* comm = HpddmGetCommunicator(HpddmSchwarzPreconditioner(A));
/*# Solution #*/
int it = HpddmSolve(A, f, sol, mu, comm);
/*# SolutionEnd #*/
underlying_type* storage = malloc(sizeof(underlying_type) * 2 * mu);
HpddmSchwarzComputeResidual(A, sol, f, storage, mu);
if(rankWorld == 0)
for(unsigned short nu = 0; nu < mu; ++nu) {
if(nu == 0)
printf(" --- residual = ");
else
printf(" ");
printf("%e / %e", storage[1 + 2 * nu], storage[2 * nu]);
if(mu > 1)
printf(" (rhs #%d)", nu + 1);
printf("\n");
}
if(it > ((int)HpddmOptionVal(opt, "krylov_method") == 6 ? 60 : 45))
status = 1;
else {
for(unsigned short nu = 0; nu < mu; ++nu)
if(storage[1 + 2 * nu] / storage[2 * nu] > 1.0e-2)
status = 1;
}
free(storage);
if(HpddmOptionVal(opt, "geneo_nu") == 0)
HpddmDestroyVectors(HpddmSchwarzPreconditioner(A));
HpddmSchwarzDestroy(A);
}
else {
HpddmSubdomain* S = NULL;
HpddmSubdomainNumfact(&S, Mat);
mu = MAX(1, mu);
HpddmSubdomainSolve(S, f, sol, mu);
int one = 1;
underlying_type* nrmb = malloc(sizeof(underlying_type) * 2 * mu);
for(unsigned short nu = 0; nu < mu; ++nu)
nrmb[nu] = nrm2(&ndof, f + nu * ndof, &one);
K* tmp = malloc(sizeof(K) * mu * ndof);
HpddmCSRMM(Mat, sol, tmp, mu);
K minus = -1;
ndof *= mu;
axpy(&ndof, &minus, f, &one, tmp, &one);
ndof /= mu;
underlying_type* nrmAx = nrmb + mu;
for(unsigned short nu = 0; nu < mu; ++nu) {
nrmAx[nu] = nrm2(&ndof, tmp + nu * ndof, &one);
if(nu == 0)
printf(" --- residual = ");
else
printf(" ");
printf("%e / %e", nrmAx[nu], nrmb[nu]);
if(mu > 1)
printf(" (rhs #%d)", nu + 1);
printf("\n");
if(nrmAx[nu] / nrmb[nu] > (sizeof(underlying_type) == sizeof(double) ? 1.0e-6 : 1.0e-2))
status = 1;
}
free(tmp);
free(nrmb);
HpddmSubdomainDestroy(S);
HpddmMatrixCSRDestroy(Mat);
}
free(d);
if(HpddmOptionSet(opt, "schwarz_coarse_correction") && HpddmOptionVal(opt, "geneo_nu") > 0)
HpddmMatrixCSRDestroy(MatNeumann);
free(sol);
free(f);
MPI_Finalize();
return status;
}
LatticeMinimizer::LatticeMinimizer(Everything& e) : e(e), Rorig(e.gInfo.R)
{
logPrintf("\n--------- Lattice Minimization ---------\n");
//Ensure that lattice-move-scale is commensurate with symmetries:
std::vector<matrix3<int>> sym = e.symm.getMatrices();
for(const matrix3<int>& m: sym)
for(int i=0; i<3; i++)
for(int j=0; j<3; j++)
if(m(i,j) && e.cntrl.lattMoveScale[i] != e.cntrl.lattMoveScale[j])
die("latt-move-scale is not commensurate with symmetries:\n"
"\t(Lattice vectors #%d and #%d are connected by symmetry,\n"
"\tbut have different move scale factors %lg != %lg).\n",
i, j, e.cntrl.lattMoveScale[i], e.cntrl.lattMoveScale[j]);
//Check which lattice vectors can be altered:
vector3<bool> isFixed, isTruncated = e.coulombParams.isTruncated();
for(int k=0; k<3; k++)
isFixed[k] = (e.cntrl.lattMoveScale[k]==0.) || isTruncated[k];
//Create a orthonormal basis for strain commensurate with symmetries:
for(int k=0; k<6; k++)
{ //Initialize a basis element for arbitrary symmetric matrices:
matrix3<int> s; //all zero:
if(k<3) //diagonal strain
{ s(k,k) = 1;
if(isFixed[k]) continue; //strain alters fixed direction
}
else //off-diagonal strain
{ int i=(k+1)%3;
int j=(k+2)%3;
s(i,j) = s(j,i) = 1;
if(isFixed[i] || isFixed[j]) continue; //strain alters fixed direction
}
//Symmetrize:
matrix3<int> sSym;
for(const matrix3<int>& m: sym)
{ matrix3<int> mInv = det(m) * adjugate(m); //since |det(m)| = 1
sSym += mInv * s * m;
}
//Orthonormalize w.r.t previous basis elements:
matrix3<> strain(sSym); //convert from integer to double matrix
for(const matrix3<>& sPrev: strainBasis)
strain -= sPrev * dot(sPrev, strain);
double strainNorm = nrm2(strain);
if(strainNorm < symmThresholdSq) continue; //linearly dependent
strainBasis.push_back((1./strainNorm) * strain);
}
if(!strainBasis.size())
die("All lattice-vectors are constrained by coulomb truncation and/or\n"
"latt-move-scale: please disable lattice minimization.\n");
//Print initialization status:
e.latticeMinParams.nDim = strainBasis.size();
logPrintf("Minimization of dimension %lu over strains spanned by:\n", strainBasis.size());
for(const matrix3<>& s: strainBasis)
{ s.print(globalLog, " %lg ");
logPrintf("\n");
}
h = 1e-5;
}
Results* join_clusters2_restart
(double *x,//array/matrix of data
SymNoDiag *W,//lower triangle of weight matrix
unsigned int Px,//problem size
double lambda,//starting point in regularization path
double join_thresh, //tolerance for equality of points
double opt_thresh, //tolerance for optimality
double lambda_factor,//increase of lambda after optimality
double smooth,//smoothing parameter
int maxit,
int linesearch_freq,//how often to do a linesearch? if 0, never. if
//n>0, do n-1 linesearch steps for every
//decreasing step size step. set this to 2 if
//unsure.
int linesearch_points,//how many points to check along the gradient
//direction. set to 10 if unsure.
int check_splits,
int target_cluster,
int verbose
){
unsigned int N = W->N;
//W->print();
double old_lambda=0;
std::vector<int> rows,rowsj;
std::vector<int>::iterator rowit,ri,rj;
std::list< std::vector<int> > clusters,tocheck;
std::list< std::vector<int> >::iterator it,cj;
unsigned int i,k,j;
int tried_restart;
for(i=0;i<N;i++){
rows.assign(1,i);
clusters.push_back(rows);
}
double *old_alpha = new double[N*Px];
double *alpha = new double[N*Px];
double *xbar = new double[N*Px];
double *dir = new double[N*Px];
for(i=0;i<N*Px;i++){
alpha[i]=xbar[i]=x[i];
}
Matrix amat(alpha,N,Px),xmat(x,N,Px);
SymNoDiag diffs(N);
diffs.calc_diffs(clusters,amat,nrm2);
//store initial trivial solution
Results *results = new Results(N,Px,opt_thresh);
if(target_cluster==0)results->add(alpha,0,0);
double weight,diff,step;
while(clusters.size()>1){
double grad=opt_thresh;
int iteration=1;
tried_restart=0;
//if we use the general (slower) algorithm for any weights, then
//split the clusters to individual points
if(check_splits){
clusters.clear();
//reassign original clusters
for(i=0;i<N;i++){
rows.assign(1,i);
clusters.push_back(rows);
}
//recopy original xbar
for(i=0;i<N*Px;i++){
xbar[i]=x[i];
}
}
while(grad>=opt_thresh){
//first calc gradients
grad = 0;
for(it=clusters.begin();it!=clusters.end();it++){
rows = *it;
i = rows[0];
for(k=0;k<Px;k++){
dir[i+k*N] = xbar[i+k*N] - alpha[i+k*N];
}
for(cj=clusters.begin();cj!=clusters.end();cj++){
if(it!=cj){
rowsj = *cj;
j=rowsj[0];
weight=0;
diff = *diffs(i,j);
if(diff!=0){
if(smooth!=0){
diff *= diff; //now squared l2 norm
diff += smooth; //add smoothing parameter under sqrt
diff = sqrt(diff);//put sqrt back
}
for(ri=rows.begin();ri!=rows.end();ri++){
for(rj=rowsj.begin();rj!=rowsj.end();rj++){
weight += W->getval(*ri,*rj);
}
}
//weight *= lambda / diff / ((double)(N-1)) / ((double)rows.size());
weight *= lambda / diff / ((double)rows.size());
for(k=0;k<Px;k++){
dir[i+k*N] += weight * (alpha[j+k*N]-alpha[i+k*N]);
}
}
}
}
grad += nrm2(Array(dir+i,N,Px));
}
//store this iteration
//results->add(alpha,lambda,grad);
//then take a step
if(linesearch_freq==0 || (iteration % linesearch_freq)==0 ){
//Decreasing step size
//TDH and pierre 18 jan 2011 try sqrt dec step size
step=1/((double)iteration);
//step=1/sqrt((double)iteration);
if(verbose>=2)printf("grad %f step %f it %d\n",grad,step,iteration);
take_step(clusters,alpha,dir,N,Px,step);
}else{
double cost_here,cost_step;
std::map<double,double> cost_steps;
std::map<double,double>::iterator step1,step2;
for(i=0;i<N*Px;i++)old_alpha[i]=alpha[i];//copy alpha
//compare current cost to cost after stepping in gradient direction
cost_here=cost_step=calc_cost(clusters,amat,xmat,W,diffs,lambda);
step = 0;
cost_steps.insert(std::pair<double,double>(cost_here,0));
while(cost_step<=cost_here){
take_step(clusters,alpha,dir,N,Px,1);
step += 1;
diffs.calc_diffs(clusters,amat,nrm2);
cost_step=calc_cost(clusters,amat,xmat,W,diffs,lambda);
if(verbose>=2)
printf("cost %.10f step %f cost_here %f\n",cost_step,step,cost_here);
cost_steps.insert(std::pair<double,double>(cost_step,step));
}
for(int cuts=0;cuts<linesearch_points;cuts++){
step1=step2=cost_steps.begin();
step2++;
step = (step1->second + step2->second)/2;
for(i=0;i<N*Px;i++){
alpha[i]=old_alpha[i];
}
take_step(clusters,alpha,dir,N,Px,step);
diffs.calc_diffs(clusters,amat,nrm2);
cost_step=calc_cost(clusters,amat,xmat,W,diffs,lambda);
if(verbose>=2)printf("cost %.10f step %f %d\n",cost_step,step,cuts);
cost_steps.insert(std::pair<double,double>(cost_step,step));
}
cost_steps.clear();
}
if(iteration++ > maxit){
if(tried_restart){
printf("max iteration %d exit\n",maxit);
delete old_alpha;
delete alpha;
delete xbar;
delete dir;
return results;
}else{
if(verbose>=1)printf("max iterations, trying restart from x\n");
tried_restart=1;
iteration=1;
for(i=0;i<N*Px;i++)alpha[i]=x[i];
}
}
//calculate differences
diffs.calc_diffs(clusters,amat,nrm2);
//check for joins
JoinPair tojoin;
while(dojoin(tojoin=check_clusters_thresh(&clusters,diffs,join_thresh))){
//if(verbose>=1)
// printf("join: %d %d\n",tojoin.first->front(),tojoin.second->front());
int ni=tojoin.first->size();
int nj=tojoin.second->size();
i=tojoin.first->front();
j=tojoin.second->front();
tojoin.first->insert(tojoin.first->end(),
tojoin.second->begin(),
tojoin.second->end());
for(k=0;k<Px;k++){
alpha[i+k*N] = (alpha[i+k*N]*ni + alpha[j+k*N]*nj)/(ni+nj);
xbar[i+k*N] = (xbar[i+k*N]*ni + xbar[j+k*N]*nj)/(ni+nj);
}
clusters.erase(tojoin.second);
iteration=1;
if(clusters.size()>1){
diffs.calc_diffs(clusters,amat,nrm2);//inefficient
}else{
grad=0;//so we can escape from the last optimization loop
}
}
}//while(grad>=opt_thresh)
if(verbose>=1)
printf("solution iteration %d lambda %f nclusters %d\n",
iteration,lambda,(int)clusters.size());
if(target_cluster == 0){
//for each cluster, there may be several points. we store the
//alpha value just in the row of the first point. thus here we
//copy this value to the other rows before copying the optimal
//alpha to results.
for(it=clusters.begin();it!=clusters.end();it++){
rows = *it;
if(rows.size()>1){
for(i=1;i<rows.size();i++){
for(k=0;k<Px;k++){
alpha[rows[i]+k*N] = alpha[rows[0]+k*N];
}
}
}
}
results->add(alpha,lambda,grad);
}
//haven't yet reached the target number of clusters, multiply
//lambda by lambda_factor and continue along the path
if((int)clusters.size()>target_cluster){
old_lambda=lambda;
lambda *= lambda_factor;
}
//if we have passed the target cluster number then decrease
//lambda and go look for it!
if((int)clusters.size()<target_cluster){
if(verbose>=1){
printf("missed target %d, going back for it\n",target_cluster);
}
lambda = (lambda+old_lambda)/2;
clusters.clear();
//reassign original clusters
for(i=0;i<N;i++){
rows.assign(1,i);
clusters.push_back(rows);
}
//recopy original xbar
for(i=0;i<N*Px;i++){
xbar[i]=x[i];
}
}
//this is the number of clusters that we were looking for,
//save and quit!
if((int)clusters.size()==target_cluster){
for(it=clusters.begin();it!=clusters.end();it++){
rows = *it;
if(rows.size()>1){
for(i=1;i<rows.size();i++){
for(k=0;k<Px;k++){
alpha[rows[i]+k*N] = alpha[rows[0]+k*N];
}
}
}
}
results->add(alpha,lambda,grad);
if(verbose>=1)printf("got target cluster %d exit\n",target_cluster);
delete old_alpha;
delete alpha;
delete xbar;
delete dir;
return results;
}
}
//TODO: consolidate cleanup... just use data structures that
//automatically clean themselves up when the function exits.
delete old_alpha;
delete alpha;
delete xbar;
delete dir;
return results;
}
//=============================================================================
bool minres
(
const CPPL::dsymatrix& A,
CPPL::dcovector& x,
const double& eps
)
{
const CPPL::dcovector y(x);
CPPL::dcovector r(y);
double beta2(nrm2(r)), beta3;
double rho0(1.0), rho1(beta2), rho2;
double rhop(0.0);
double c0(0.0), c1(-1.0), c2;
double s0(0.0), s1(0.0), s2;
double f(beta2);
CPPL::dcovector p1(x.l), p2(r/beta2), p3;
CPPL::dcovector q0(x.l), q1(x.l), q2;
x.zero();
p1.zero();
q0.zero();
q1.zero();
int itc(0);
const int itmax(2*x.l);
while( (fabs(f)>eps || fabs(damax(y-A*x))>eps) && itc<itmax){
std::cout << itc << " " << fabs(damax(y-A*x)) << std::endl;
//std::cerr << "itc=" << itc << ", fabs(f)=" << fabs(f) << std::endl;
CPPL::dcovector Ap2(A*p2), z;
z =Ap2-beta2*p1;
double alpha;
alpha =Ap2%p2;
p3 =z-alpha*p2;
beta3 =nrm2(p3);
p3 /=beta3;
double d, h;
d =(alpha-rhop*c0)*s1;
h =beta2*s0;
rhop =-beta2*c0*s1 -alpha*c1;
rho2 =sqrt(pow(rhop,2)+pow(beta3,2));
c2 =rhop/rho2;
s2 =beta3/rho2;
CPPL::dcovector zp;
zp =p2 -(h/rho0)*q0;
q2 =zp -(d/rho1)*q1;
double t;
t =f*c2;
f *=s2;
x +=(t/rho2)*q2;
beta2=beta3;
rho0=rho1; rho1=rho2;
c0=c1; c1=c2;
s0=s1; s1=s2;
swap(p1,p2); swap(p2,p3);
swap(q0,q1); swap(q1,q2);
itc++;
}
std::cerr << "itc=" << itc << " fabs(damax(y-A*x))=" << fabs(damax(y-A*x)) << std::endl;
//std::cerr << "itc=" << itc << " fabs(f)=" << fabs(f) << std::endl;
if(itc<itmax){ return 0; }
else{ return 1; }
}