pair<Base<F>,Base<F>>
HermitianExtremalSingValEst( const SparseMatrix<F>& A, Int basisSize )
{
EL_DEBUG_CSE
typedef Base<F> Real;
Matrix<Real> T;
Lanczos( A, T, basisSize );
const Int k = T.Height();
if( k == 0 )
return pair<Real,Real>(0,0);
Matrix<Real> d, dSub;
d = GetDiagonal( T );
dSub = GetDiagonal( T, -1 );
Matrix<Real> w;
HermitianTridiagEig( d, dSub, w );
pair<Real,Real> extremal;
extremal.second = MaxNorm(w);
extremal.first = extremal.second;
for( Int i=0; i<k; ++i )
extremal.first = Min(extremal.first,Abs(w(i)));
return extremal;
}
pair<Base<F>,Base<F>>
HermitianHelper( const DistSparseMatrix<F>& A, Int basisSize )
{
typedef Base<F> Real;
Grid grid( A.Comm() );
DistMatrix<Real,STAR,STAR> T(grid);
Lanczos( A, T, basisSize );
const Int k = T.Height();
if( k == 0 )
return pair<Real,Real>(0,0);
auto d = GetDiagonal( T.Matrix() );
auto dSub = GetDiagonal( T.Matrix(), -1 );
Matrix<Real> w;
HermitianTridiagEig( d, dSub, w );
pair<Real,Real> extremal;
extremal.second = MaxNorm(w);
extremal.first = extremal.second;
for( Int i=0; i<k; ++i )
extremal.first = Min(extremal.first,Abs(w.Get(i,0)));
return extremal;
}
SafeProduct<Base<F>> AfterCholesky
( UpperOrLower uplo, const ElementalMatrix<F>& APre )
{
DEBUG_CSE
DistMatrixReadProxy<F,F,MC,MR> AProx( APre );
auto& A = AProx.GetLocked();
typedef Base<F> Real;
const Int n = A.Height();
const Grid& g = A.Grid();
DistMatrix<F,MD,STAR> d(g);
GetDiagonal( A, d );
Real localKappa = 0;
if( d.Participating() )
{
const Real scale = Real(n)/Real(2);
const Int nLocalDiag = d.LocalHeight();
for( Int iLoc=0; iLoc<nLocalDiag; ++iLoc )
{
const Real delta = RealPart(d.GetLocal(iLoc,0));
localKappa += Log(delta)/scale;
}
}
SafeProduct<Real> det( n );
det.kappa = mpi::AllReduce( localKappa, g.VCComm() );
det.rho = Real(1);
return det;
}
DistMatrix<T,DiagCol<U,V>(),DiagRow<U,V>()>
GetDiagonal( const DistMatrix<T,U,V>& A, Int offset )
{
DistMatrix<T,DiagCol<U,V>(),DiagRow<U,V>()> d(A.Grid());
GetDiagonal( A, d, offset );
return d;
}
inline LDLPivot
BunchParlett( const DistMatrix<F>& A, Base<F> gamma )
{
DEBUG_CSE
typedef Base<F> Real;
if( gamma == Real(0) )
gamma = LDLPivotConstant<Real>( BUNCH_PARLETT );
const ValueInt<Real> diagMax = VectorMaxAbsLoc( GetDiagonal(A) );
const Entry<Real> offDiagMax = SymmetricMaxAbsLoc( LOWER, A );
LDLPivot pivot;
if( diagMax.value >= gamma*offDiagMax.value )
{
pivot.nb = 1;
pivot.from[0] = diagMax.index;
return pivot;
}
else
{
pivot.nb = 2;
pivot.from[0] = offDiagMax.i;
pivot.from[1] = offDiagMax.j;
return pivot;
}
}
bool TriangIsNormal( const Matrix<F>& U, Base<F> tol )
{
const Base<F> diagFrob = FrobeniusNorm(GetDiagonal(U));
const Base<F> upperFrob = FrobeniusNorm( U );
const Base<F> offDiagFrob = Sqrt(upperFrob*upperFrob-diagFrob*diagFrob);
return offDiagFrob <= tol*diagFrob;
}
void CheckRealSchur( const AbstractDistMatrix<Real>& UPre, bool standardForm )
{
EL_DEBUG_CSE
DistMatrixReadProxy<Real,Real,MC,MR> UProx( UPre );
auto& U = UProx.GetLocked();
auto uMain = GetDiagonal(U);
auto uSub = GetDiagonal(U,-1);
DistMatrix<Real,STAR,STAR> uMain_STAR_STAR( uMain ),
uSub_STAR_STAR( uSub );
auto& uMainLoc = uMain_STAR_STAR.Matrix();
auto& uSubLoc = uSub_STAR_STAR.Matrix();
const Int n = U.Height();
if( standardForm )
{
auto uSup = GetDiagonal(U,+1);
DistMatrix<Real,STAR,STAR> uSup_STAR_STAR( uSup );
auto& uSupLoc = uSup_STAR_STAR.Matrix();
for( Int j=0; j<n-1; ++j )
{
const Real thisDiag = uMainLoc(j);
const Real nextDiag = uMainLoc(j+1);
const Real thisSub = uSubLoc(j);
const Real thisSup = uSupLoc(j);
if( thisSub != Real(0) && thisDiag != nextDiag )
LogicError
("Diagonal of 2x2 block was not constant: ",thisDiag," and ",
nextDiag);
if( thisSub*thisSup >= 0 )
LogicError("b*c >= 0: b=",thisSup," and c=",thisSub);
}
}
if( n < 3 )
return;
for( Int j=0; j<n-2; ++j )
{
const Real thisSub = uSubLoc(j);
const Real nextSub = uSubLoc(j+1);
if( thisSub != Real(0) && nextSub != Real(0) )
LogicError
("Quasi-triangular assumption broken at j=",j,
": subdiagonals were ",thisSub," and ",nextSub);
}
}
bool TriangIsNormal( const ElementalMatrix<F>& UPre, Base<F> tol )
{
DistMatrixReadProxy<F,F,MC,MR> UProx( UPre );
auto& U = UProx.GetLocked();
const Base<F> diagFrob = FrobeniusNorm(GetDiagonal(U));
const Base<F> upperFrob = FrobeniusNorm( U );
const Base<F> offDiagFrob = Sqrt(upperFrob*upperFrob-diagFrob*diagFrob);
return offDiagFrob <= tol*diagFrob;
}
inline bool
TriangIsNormal( const ElementalMatrix<F>& UPre, Base<F> tol )
{
auto UPtr = ReadProxy<F,MC,MR>( &UPre );
auto& U = *UPtr;
const Base<F> diagFrob = FrobeniusNorm(GetDiagonal(U));
const Base<F> upperFrob = FrobeniusNorm( U );
const Base<F> offDiagFrob = Sqrt(upperFrob*upperFrob-diagFrob*diagFrob);
return offDiagFrob <= tol*diagFrob;
}
pair<Base<F>,Base<F>>
Helper( const DistSparseMatrix<F>& A, Int basisSize )
{
typedef Base<F> Real;
Grid grid( A.Comm() );
DistMatrix<Real,STAR,STAR> T(grid);
ProductLanczos( A, T, basisSize );
const Int k = T.Height();
if( k == 0 )
return pair<Real,Real>(0,0);
auto d = GetDiagonal( T.Matrix() );
auto dSub = GetDiagonal( T.Matrix(), -1 );
Matrix<Real> w;
HermitianTridiagEig( d, dSub, w, ASCENDING );
pair<Real,Real> extremal;
extremal.first = Sqrt( Max(w.Get(0,0),Real(0)) );
extremal.second = Sqrt( Max(w.Get(k-1,0),Real(0)) );
return extremal;
}
pair<Base<F>,Base<F>>
Helper( const SparseMatrix<F>& A, Int basisSize )
{
typedef Base<F> Real;
Matrix<Real> T;
ProductLanczos( A, T, basisSize );
const Int k = T.Height();
if( k == 0 )
return pair<Real,Real>(0,0);
Matrix<Real> d, dSub;
d = GetDiagonal( T );
dSub = GetDiagonal( T, -1 );
Matrix<Real> w;
HermitianTridiagEig( d, dSub, w, ASCENDING );
pair<Real,Real> extremal;
extremal.first = Sqrt( Max(w.Get(0,0),Real(0)) );
extremal.second = Sqrt( Max(w.Get(k-1,0),Real(0)) );
return extremal;
}
pair<Base<F>,Base<F>>
ExtremalSingValEst( const SparseMatrix<F>& A, Int basisSize )
{
EL_DEBUG_CSE
typedef Base<F> Real;
Matrix<Real> T;
ProductLanczos( A, T, basisSize );
const Int k = T.Height();
if( k == 0 )
return pair<Real,Real>(0,0);
Matrix<Real> d, dSub;
d = GetDiagonal( T );
dSub = GetDiagonal( T, -1 );
Matrix<Real> w;
HermitianTridiagEig( d, dSub, w );
pair<Real,Real> extremal;
extremal.first = Sqrt( Max(w(0),Real(0)) );
extremal.second = Sqrt( Max(w(k-1),Real(0)) );
return extremal;
}
void CheckRealSchur( const Matrix<Real>& U, bool standardForm )
{
EL_DEBUG_CSE
const Int n = U.Height();
auto uMain = GetDiagonal(U);
auto uSub = GetDiagonal(U,-1);
if( standardForm )
{
auto uSup = GetDiagonal(U,+1);
for( Int j=0; j<n-1; ++j )
{
const Real thisDiag = uMain(j);
const Real nextDiag = uMain(j+1);
const Real thisSub = uSub(j);
const Real thisSup = uSup(j);
if( uSub(j) != Real(0) && thisDiag != nextDiag )
LogicError
("Diagonal of 2x2 block was not constant: ",thisDiag," and ",
nextDiag);
if( thisSub*thisSup >= 0 )
LogicError("b*c >= 0: b=",thisSup," and c=",thisSub);
}
}
if( n < 3 )
return;
for( Int j=0; j<n-2; ++j )
{
const Real thisSub = uSub(j);
const Real nextSub = uSub(j+1);
if( thisSub != Real(0) && nextSub != Real(0) )
LogicError
("Quasi-triangular assumption broken at j=",j,
": subdiagonals were ",thisSub," and ",nextSub);
}
}
SafeProduct<Base<F>> AfterCholesky
( UpperOrLower uplo, const Matrix<F>& A )
{
DEBUG_CSE
typedef Base<F> Real;
const Int n = A.Height();
Matrix<F> d;
GetDiagonal( A, d );
SafeProduct<Real> det( n );
det.rho = Real(1);
const Real scale = Real(n)/Real(2);
for( Int i=0; i<n; ++i )
{
const Real delta = RealPart(d(i));
det.kappa += Log(delta)/scale;
}
return det;
}
( Matrix<F>& U,
const Matrix<F>& shifts,
Matrix<F>& X,
Matrix<F>& scales )
{
typedef Base<F> Real;
DEBUG_ONLY(
CSE cse("safemstrsm::LUNBlock");
if( U.Height() != U.Width() )
LogicError("Triangular matrix must be square");
if( U.Width() != X.Height() )
LogicError("Matrix dimensions do not match");
if( shifts.Height() != X.Width() )
LogicError("Incompatible number of shifts");
)
auto diag = GetDiagonal(U);
const Int n = U.Height();
const Int numShifts = shifts.Height();
auto overflowPair = OverflowParameters<Real>();
const Real smallNum = overflowPair.first;
const Real bigNum = overflowPair.second;
const Real oneHalf = Real(1)/Real(2);
const Real oneQuarter = Real(1)/Real(4);
// Default scale is 1
scales.Resize( numShifts, 1 );
Fill( scales, F(1) );
// Compute infinity norms of columns of U (excluding diagonal)
http://opensource.org/licenses/BSD-2-Clause
*/
#pragma once
#ifndef EL_SCHUR_CHECkREAL_HPP
#define EL_SCHUR_CHECkREAL_HPP
namespace El {
namespace schur {
template<typename Real>
void CheckRealSchur( const Matrix<Real>& U, bool standardForm )
{
DEBUG_ONLY(CSE cse("CheckRealSchur"))
const Int n = U.Height();
auto uMain = GetDiagonal(U);
auto uSub = GetDiagonal(U,-1);
if( standardForm )
{
auto uSup = GetDiagonal(U,+1);
for( Int j=0; j<n-1; ++j )
{
const Real thisDiag = uMain.Get(j, 0);
const Real nextDiag = uMain.Get(j+1,0);
const Real thisSub = uSub.Get(j,0);
const Real thisSup = uSup.Get(j,0);
if( uSub.Get(j,0) != Real(0) && thisDiag != nextDiag )
LogicError
("Diagonal of 2x2 block was not constant: ",thisDiag," and ",
nextDiag);
if( thisSub*thisSup >= 0 )
Matrix<T> GetDiagonal( const Matrix<T>& A, Int offset )
{
Matrix<T> d;
GetDiagonal( A, d, offset );
return d;
}
http://opensource.org/licenses/BSD-2-Clause
*/
#ifndef EL_CHOLESKY_LVAR3PIVOTED_HPP
#define EL_CHOLESKY_LVAR3PIVOTED_HPP
namespace El {
namespace cholesky {
namespace pivot {
template<typename F>
inline LDLPivot
Full( const Matrix<F>& A )
{
DEBUG_ONLY(CSE cse("cholesky::pivot::Full"))
const auto diagMax = VectorMaxAbsLoc( GetDiagonal(A) );
LDLPivot pivot;
pivot.nb = 1;
pivot.from[0] = diagMax.index;
return pivot;
}
template<typename F>
inline LDLPivot
Full( const DistMatrix<F>& A )
{
DEBUG_ONLY(CSE cse("cholesky::pivot::Full"))
const auto diagMax = VectorMaxAbsLoc( GetDiagonal(A) );
LDLPivot pivot;
pivot.nb = 1;
pivot.from[0] = diagMax.index;
