void HermitianPseudoinverse
( UpperOrLower uplo, ElementalMatrix<F>& APre, Base<F> tolerance )
{
DEBUG_CSE
typedef Base<F> Real;
DistMatrixReadWriteProxy<F,F,MC,MR> AProx( APre );
auto& A = AProx.Get();
const Grid& g = A.Grid();
// Get the EVD of A
// TODO: Use a relative eigenvalue lower-bound
DistMatrix<Real,VR,STAR> w(g);
DistMatrix<F> Z(g);
HermitianEig( uplo, A, w, Z );
if( tolerance == Real(0) )
{
// Set the tolerance equal to n ||A||_2 eps
const Int n = Z.Height();
const Real eps = limits::Epsilon<Real>();
const Real twoNorm = MaxNorm( w );
tolerance = n*twoNorm*eps;
}
// Invert above the tolerance
auto omegaMap =
[=]( Real omega ) { return ( omega < tolerance ? Real(0) : 1/omega ); };
EntrywiseMap( w, function<Real(Real)>(omegaMap) );
// Form the pseudoinverse
HermitianFromEVD( uplo, A, w, Z );
}
void HermitianSign
( UpperOrLower uplo, Matrix<Field>& A, const HermitianEigCtrl<Field>& ctrl )
{
EL_DEBUG_CSE
typedef Base<Field> Real;
// Get the EVD of A
Matrix<Real> w;
Matrix<Field> Q;
auto ctrlMod( ctrl );
ctrlMod.tridiagEigCtrl.sort = UNSORTED;
HermitianEig( uplo, A, w, Q, ctrlMod );
const Int n = A.Height();
for( Int i=0; i<n; ++i )
{
const Real omega = w(i);
if( omega >= 0 )
w(i) = Real(1);
else
w(i) = Real(-1);
}
// Reform the Hermitian matrix with the modified eigenvalues
HermitianFromEVD( uplo, A, w, Q );
}
void HermitianPseudoinverse
( UpperOrLower uplo, Matrix<F>& A, Base<F> tolerance )
{
DEBUG_CSE
typedef Base<F> Real;
// Get the EVD of A
// TODO: Use a relative eigenvalue lower bound
Matrix<Real> w;
Matrix<F> Z;
HermitianEig( uplo, A, w, Z );
if( tolerance == Real(0) )
{
// Set the tolerance equal to n ||A||_2 eps
const Int n = Z.Height();
const Real eps = limits::Epsilon<Real>();
const Real twoNorm = MaxNorm( w );
tolerance = n*twoNorm*eps;
}
// Invert above the tolerance
auto omegaMap =
[=]( Real omega ) { return ( omega < tolerance ? Real(0) : 1/omega ); };
EntrywiseMap( w, function<Real(Real)>(omegaMap) );
// Form the pseudoinverse
HermitianFromEVD( uplo, A, w, Z );
}
inline void
HermitianSign( UpperOrLower uplo, Matrix<F>& A )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSign");
#endif
typedef BASE(F) R;
// Get the EVD of A
Matrix<R> w;
Matrix<F> Z;
HermitianEig( uplo, A, w, Z );
// Compute the two-norm of A as the maximum absolute value of its eigvals
const R twoNorm = MaxNorm( w );
// Set the tolerance equal to n ||A||_2 eps, and invert values above it
const int n = A.Height();
const R eps = lapack::MachineEpsilon<R>();
const R tolerance = n*twoNorm*eps;
for( int i=0; i<n; ++i )
{
const R omega = w.Get(i,0);
if( Abs(omega) < tolerance )
w.Set(i,0,0);
else if( omega > 0 )
w.Set(i,0,R(1));
else
w.Set(i,0,R(-1));
}
// Reform the Hermitian matrix with the modified eigenvalues
hermitian_function::ReformHermitianMatrix( uplo, A, w, Z );
}
void HermitianSign
( UpperOrLower uplo,
AbstractDistMatrix<Field>& APre,
const HermitianEigCtrl<Field>& ctrl )
{
EL_DEBUG_CSE
DistMatrixReadWriteProxy<Field,Field,MC,MR> AProx( APre );
auto& A = AProx.Get();
// Get the EVD of A
typedef Base<Field> Real;
const Grid& g = A.Grid();
DistMatrix<Real,VR,STAR> w(g);
DistMatrix<Field> Q(g);
auto ctrlMod( ctrl );
ctrlMod.tridiagEigCtrl.sort = UNSORTED;
HermitianEig( uplo, A, w, Q, ctrlMod );
const Int numLocalEigs = w.LocalHeight();
for( Int iLoc=0; iLoc<numLocalEigs; ++iLoc )
{
const Real omega = w.GetLocal(iLoc,0);
if( omega >= 0 )
w.SetLocal(iLoc,0,Real(1));
else
w.SetLocal(iLoc,0,Real(-1));
}
// Reform the Hermitian matrix with the modified eigenvalues
HermitianFromEVD( uplo, A, w, Q );
}
inline void
HermitianPseudoinverse
( UpperOrLower uplo, DistMatrix<Complex<R>,MC,MR>& A )
{
#ifndef RELEASE
PushCallStack("HermitianPseudoinverse");
#endif
// Get the EVD of A
const Grid& g = A.Grid();
DistMatrix<R,VR,STAR> w(g);
DistMatrix<Complex<R>,MC,MR> Z(g);
HermitianEig( uplo, A, w, Z );
// Compute the two-norm of A as the maximum absolute value of its
// eigenvalues
R maxLocalAbsEig = 0;
const int localHeight = w.LocalHeight();
for( int iLocal=0; iLocal<localHeight; ++iLocal )
maxLocalAbsEig =
std::max(maxLocalAbsEig,Abs(w.GetLocalEntry(iLocal,0)));
R twoNorm;
mpi::AllReduce( &maxLocalAbsEig, &twoNorm, 1, mpi::MAX, g.VCComm() );
// Set the tolerance equal to n ||A||_2 eps
const int n = A.Height();
const R eps = lapack::MachineEpsilon<R>();
const R tolerance = n*twoNorm*eps;
// Form the pseudoinverse
hermitian_pseudoinverse::Functor<R> f( tolerance );
hermitian_function::ReformHermitianMatrix( uplo, A, w, Z, f );
#ifndef RELEASE
PopCallStack();
#endif
}
void SkewHermitianEig
( UpperOrLower uplo, const Matrix<F>& G, Matrix<Base<F>>& wImag, SortType sort,
const HermitianEigSubset<Base<F>>& subset,
const HermitianEigCtrl<Complex<Base<F>>>& ctrl )
{
DEBUG_ONLY(CSE cse("SkewHermitianEig"))
Matrix<Complex<Base<F>>> A;
Copy( G, A );
ScaleTrapezoid( Complex<Base<F>>(0,-1), uplo, A );
HermitianEig( uplo, A, wImag, sort, subset, ctrl );
}
void SkewHermitianEig
( UpperOrLower uplo, const AbstractDistMatrix<F>& G,
AbstractDistMatrix<Base<F>>& wImag, SortType sort,
const HermitianEigSubset<Base<F>>& subset,
const HermitianEigCtrl<Complex<Base<F>>>& ctrl )
{
DEBUG_ONLY(CallStackEntry cse("SkewHermitianEig"))
DistMatrix<Complex<Base<F>>> A(G.Grid());
Copy( G, A );
ScaleTrapezoid( Complex<Base<F>>(0,-1), uplo, A );
HermitianEig( uplo, A, wImag, sort, subset, ctrl );
}
inline void HermitianSVD
( UpperOrLower uplo, DistMatrix<F>& A,
DistMatrix<BASE(F),VR,STAR>& s, DistMatrix<F>& U, DistMatrix<F>& V )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#ifdef HAVE_PMRRR
typedef BASE(F) R;
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Redistribute the singular values into an [MR,* ] distribution
const Grid& grid = A.Grid();
DistMatrix<R,MR,STAR> s_MR_STAR( grid );
s_MR_STAR.AlignWith( V.DistData() );
s_MR_STAR = s;
// Set the singular values to the absolute value of the eigenvalues
const Int numLocalVals = s.LocalHeight();
for( Int iLoc=0; iLoc<numLocalVals; ++iLoc )
{
const R sigma = s.GetLocal(iLoc,0);
s.SetLocal(iLoc,0,Abs(sigma));
}
// Copy V into U (flipping the sign as necessary)
U.AlignWith( V );
U.ResizeTo( V.Height(), V.Width() );
const Int localHeight = V.LocalHeight();
const Int localWidth = V.LocalWidth();
for( Int jLoc=0; jLoc<localWidth; ++jLoc )
{
const R sigma = s_MR_STAR.GetLocal( jLoc, 0 );
F* UCol = U.Buffer( 0, jLoc );
const F* VCol = V.LockedBuffer( 0, jLoc );
if( sigma >= 0 )
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
UCol[iLoc] = VCol[iLoc];
else
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
UCol[iLoc] = -VCol[iLoc];
}
#else
U = A;
MakeHermitian( uplo, U );
SVD( U, s, V );
#endif // ifdef HAVE_PMRRR
}
inline void HermitianSVD
( UpperOrLower uplo,
DistMatrix<F>& A, DistMatrix<typename Base<F>::type,VR,STAR>& s,
DistMatrix<F>& U, DistMatrix<F>& V )
{
#ifndef RELEASE
PushCallStack("HermitianSVD");
#endif
typedef typename Base<F>::type R;
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Redistribute the singular values into an [MR,* ] distribution
const Grid& grid = A.Grid();
DistMatrix<R,MR,STAR> s_MR_STAR( grid );
s_MR_STAR.AlignWith( V );
s_MR_STAR = s;
// Set the singular values to the absolute value of the eigenvalues
const int numLocalVals = s.LocalHeight();
for( int iLocal=0; iLocal<numLocalVals; ++iLocal )
{
const R sigma = s.GetLocal(iLocal,0);
s.SetLocal(iLocal,0,Abs(sigma));
}
// Copy V into U (flipping the sign as necessary)
U.AlignWith( V );
U.ResizeTo( V.Height(), V.Width() );
const int localHeight = V.LocalHeight();
const int localWidth = V.LocalWidth();
for( int jLocal=0; jLocal<localWidth; ++jLocal )
{
const R sigma = s_MR_STAR.GetLocal( jLocal, 0 );
F* UCol = U.LocalBuffer( 0, jLocal );
const F* VCol = V.LockedLocalBuffer( 0, jLocal );
if( sigma >= 0 )
for( int iLocal=0; iLocal<localHeight; ++iLocal )
UCol[iLocal] = VCol[iLocal];
else
for( int iLocal=0; iLocal<localHeight; ++iLocal )
UCol[iLocal] = -VCol[iLocal];
}
#ifndef RELEASE
PopCallStack();
#endif
}
void HermitianSign
( UpperOrLower uplo,
ElementalMatrix<F>& APre,
ElementalMatrix<F>& NPre,
const HermitianEigCtrl<F>& ctrl )
{
DEBUG_CSE
DistMatrixReadWriteProxy<F,F,MC,MR> AProx( APre );
DistMatrixWriteProxy<F,F,MC,MR> NProx( NPre );
auto& A = AProx.Get();
auto& N = NProx.Get();
// Get the EVD of A
typedef Base<F> Real;
const Grid& g = A.Grid();
DistMatrix<Real,VR,STAR> w(g);
DistMatrix<F> Q(g);
auto ctrlMod( ctrl );
ctrlMod.tridiagEigCtrl.sort = UNSORTED;
HermitianEig( uplo, A, w, Q, ctrlMod );
const Int n = A.Height();
const Int numLocalEigs = w.LocalHeight();
DistMatrix<Real,VR,STAR> wSgn(g), wAbs(g);
wSgn.AlignWith( w );
wAbs.AlignWith( w );
wSgn.Resize( n, 1 );
wAbs.Resize( n, 1 );
for( Int iLoc=0; iLoc<numLocalEigs; ++iLoc )
{
const Real omega = w.GetLocal(iLoc,0);
if( omega >= 0 )
{
wSgn.SetLocal(iLoc,0,Real(1));
wAbs.SetLocal(iLoc,0,omega);
}
else
{
wSgn.SetLocal(iLoc,0,Real(-1));
wAbs.SetLocal(iLoc,0,-omega);
}
}
// Form the Hermitian matrix with the modified eigenvalues
HermitianFromEVD( uplo, A, wSgn, Q );
HermitianFromEVD( uplo, N, wAbs, Q );
}
inline void
HermitianPseudoinverse
( UpperOrLower uplo, DistMatrix<F>& A )
{
#ifndef RELEASE
PushCallStack("HermitianPseudoinverse");
#endif
typedef typename Base<F>::type R;
// Get the EVD of A
const Grid& g = A.Grid();
DistMatrix<R,VR,STAR> w(g);
DistMatrix<F> Z(g);
HermitianEig( uplo, A, w, Z );
// Compute the two-norm of A as the maximum absolute value of its
// eigenvalues
R maxLocalAbsEig = 0;
const int numLocalEigs = w.LocalHeight();
for( int iLocal=0; iLocal<numLocalEigs; ++iLocal )
{
const R omega = w.GetLocal(iLocal,0);
maxLocalAbsEig = std::max(maxLocalAbsEig,Abs(omega));
}
R twoNorm;
mpi::AllReduce( &maxLocalAbsEig, &twoNorm, 1, mpi::MAX, g.VCComm() );
// Set the tolerance equal to n ||A||_2 eps, and invert values above it
const int n = A.Height();
const R eps = lapack::MachineEpsilon<R>();
const R tolerance = n*twoNorm*eps;
for( int iLocal=0; iLocal<numLocalEigs; ++iLocal )
{
const R omega = w.GetLocal(iLocal,0);
if( Abs(omega) < tolerance )
w.SetLocal(iLocal,0,0);
else
w.SetLocal(iLocal,0,1/omega);
}
// Form the pseudoinverse
hermitian_function::ReformHermitianMatrix( uplo, A, w, Z );
#ifndef RELEASE
PopCallStack();
#endif
}
inline void
HPSDSquareRoot( UpperOrLower uplo, DistMatrix<R,MC,MR>& A )
{
#ifndef RELEASE
PushCallStack("HPSDSquareRoot");
#endif
// Get the EVD of A
const Grid& g = A.Grid();
DistMatrix<R,VR,STAR> w(g);
DistMatrix<R,MC,MR> Z(g);
HermitianEig( uplo, A, w, Z );
// Compute the two-norm of A as the maximum absolute value
// of its eigenvalues
R maxLocalAbsEig = 0;
const int localHeight = w.LocalHeight();
for( int iLocal=0; iLocal<localHeight; ++iLocal )
maxLocalAbsEig =
std::max(maxLocalAbsEig,Abs(w.GetLocalEntry(iLocal,0)));
R twoNorm;
mpi::AllReduce( &maxLocalAbsEig, &twoNorm, 1, mpi::MAX, g.VCComm() );
// Compute the smallest eigenvalue of A
R minLocalEig = twoNorm;
for( int iLocal=0; iLocal<localHeight; ++iLocal )
minLocalEig = std::min(minLocalEig,w.GetLocalEntry(iLocal,0));
R minEig;
mpi::AllReduce( &minLocalEig, &minEig, 1, mpi::MIN, g.VCComm() );
// Set the tolerance equal to n ||A||_2 eps
const int n = A.Height();
const R eps = lapack::MachineEpsilon<R>();
const R tolerance = n*twoNorm*eps;
// Ensure that the minimum eigenvalue is not less than - n ||A||_2 eps
if( minEig < -tolerance )
throw NonHPSDMatrixException();
// Form the pseudoinverse
square_root::Functor<R> f( tolerance );
hermitian_function::ReformHermitianMatrix( uplo, A, w, Z, f );
#ifndef RELEASE
PopCallStack();
#endif
}
inline void HermitianSVD
( UpperOrLower uplo, Matrix<F>& A, Matrix<BASE(F)>& s )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#if 1
// Grab the eigenvalues of A
HermitianEig( uplo, A, s );
// Set the singular values to the absolute value of the eigenvalues
for( Int i=0; i<s.Height(); ++i )
s.Set(i,0,Abs(s.Get(i,0)));
#else
MakeHermitian( uplo, A );
SVD( A, s );
#endif
}
inline void
RealHermitianFunction
( UpperOrLower uplo, DistMatrix<R,MC,MR>& A, const RealFunctor& f )
{
#ifndef RELEASE
PushCallStack("RealHermitianFunction");
#endif
if( A.Height() != A.Width() )
throw std::logic_error("Hermitian matrices must be square");
// Get the EVD of A
const Grid& g = A.Grid();
DistMatrix<R,VR,STAR> w(g);
DistMatrix<R,MC,MR> Z(g);
HermitianEig( uplo, A, w, Z );
// Form the custom outer product, Z f(Omega) Z^T
hermitian_function::ReformHermitianMatrix( uplo, A, w, Z, f );
#ifndef RELEASE
PopCallStack();
#endif
}
inline void HermitianSVD
( UpperOrLower uplo,
Matrix<F>& A, Matrix<BASE(F)>& s, Matrix<F>& U, Matrix<F>& V )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#if 1
typedef BASE(F) R;
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Set the singular values to the absolute value of the eigenvalues
for( Int i=0; i<s.Height(); ++i )
s.Set(i,0,Abs(s.Get(i,0)));
// Copy V into U (flipping the sign as necessary)
const Int n = A.Height();
U.ResizeTo( n, n );
for( Int j=0; j<n; ++j )
{
const R sigma = s.Get( j, 0 );
F* UCol = U.Buffer( 0, j );
const F* VCol = V.LockedBuffer( 0, j );
if( sigma >= 0 )
for( Int i=0; i<n; ++i )
UCol[i] = VCol[i];
else
for( Int i=0; i<n; ++i )
UCol[i] = -VCol[i];
}
#else
U = A;
MakeHermitian( uplo, U );
SVD( U, s, V );
#endif
}
inline void HermitianSingularValues
( UpperOrLower uplo,
DistMatrix<F>& A, DistMatrix<typename Base<F>::type,VR,STAR>& s )
{
#ifndef RELEASE
PushCallStack("HermitianSingularValues");
#endif
typedef typename Base<F>::type R;
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s );
// Replace the eigenvalues with their absolute values
const int numLocalVals = s.LocalHeight();
for( int iLocal=0; iLocal<numLocalVals; ++iLocal )
{
const R sigma = s.GetLocal(iLocal,0);
s.SetLocal(iLocal,0,Abs(sigma));
}
#ifndef RELEASE
PopCallStack();
#endif
}
void HermitianSVD
( UpperOrLower uplo, AbstractDistMatrix<F>& A, AbstractDistMatrix<Base<F>>& s,
AbstractDistMatrix<F>& U, AbstractDistMatrix<F>& V )
{
DEBUG_ONLY(CallStackEntry cse("HermitianSVD"))
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Copy V into U (flipping the sign as necessary)
Copy( U, V );
typedef Base<F> Real;
DistMatrix<Real,VR,STAR> sSgn( s );
auto sgnLambda = []( Real sigma ) { return Sgn(sigma,false); };
EntrywiseMap( sSgn, function<Real(Real)>(sgnLambda) );
DiagonalScale( RIGHT, NORMAL, sSgn, U );
// Set the singular values to the absolute value of the eigenvalues
auto absLambda = []( Real sigma ) { return Abs(sigma); };
EntrywiseMap( s, function<Real(Real)>(absLambda) );
// TODO: Descending sort of triplets
}
void HermitianSign
( UpperOrLower uplo,
Matrix<Field>& A,
Matrix<Field>& N,
const HermitianEigCtrl<Field>& ctrl )
{
EL_DEBUG_CSE
typedef Base<Field> Real;
// Get the EVD of A
Matrix<Real> w;
Matrix<Field> Q;
auto ctrlMod( ctrl );
ctrlMod.tridiagEigCtrl.sort = UNSORTED;
HermitianEig( uplo, A, w, Q, ctrlMod );
const Int n = A.Height();
Matrix<Real> wSgn( n, 1 ), wAbs( n, 1 );
for( Int i=0; i<n; ++i )
{
const Real omega = w(i);
if( omega >= 0 )
{
wSgn(i) = Real(1);
wAbs(i) = omega;
}
else
{
wSgn(i) = Real(-1);
wAbs(i) = -omega;
}
}
// Form the Hermitian matrices with modified eigenvalues
HermitianFromEVD( uplo, A, wSgn, Q );
HermitianFromEVD( uplo, N, wAbs, Q );
}
void HermitianSVD
( UpperOrLower uplo,
Matrix<F>& A, Matrix<Base<F>>& s, Matrix<F>& U, Matrix<F>& V )
{
DEBUG_ONLY(CallStackEntry cse("HermitianSVD"))
#if 1
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Copy V into U (flipping the sign as necessary)
const Int n = A.Height();
U.Resize( n, n );
for( Int j=0; j<n; ++j )
{
const Base<F> sigma = s.Get( j, 0 );
F* UCol = U.Buffer( 0, j );
const F* VCol = V.LockedBuffer( 0, j );
if( sigma >= 0 )
{
for( Int i=0; i<n; ++i )
UCol[i] = VCol[i];
}
else
{
for( Int i=0; i<n; ++i )
UCol[i] = -VCol[i];
s.Set( j, 0, -sigma );
}
}
// TODO: Descending sort of triplets
#else
U = A;
MakeHermitian( uplo, U );
SVD( U, s, V );
#endif
}
inline void HermitianSVD
( UpperOrLower uplo, DistMatrix<F>& A, DistMatrix<BASE(F),VR,STAR>& s )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#ifdef HAVE_PMRRR
typedef BASE(F) R;
// Grab the eigenvalues of A
HermitianEig( uplo, A, s );
// Replace the eigenvalues with their absolute values
const Int numLocalVals = s.LocalHeight();
for( Int iLoc=0; iLoc<numLocalVals; ++iLoc )
{
const R sigma = s.GetLocal(iLoc,0);
s.SetLocal(iLoc,0,Abs(sigma));
}
#else
MakeHermitian( uplo, A );
SVD( A, s );
#endif // ifdef HAVE_PMRRR
}