void HermitianUniformSpectrum
( DistMatrix<F,U,V>& A, Int n, Base<F> lower, Base<F> upper )
{
DEBUG_ONLY(CallStackEntry cse("HermitianUniformSpectrum"))
A.Resize( n, n );
const Grid& grid = A.Grid();
typedef Base<F> Real;
const bool isComplex = IsComplex<F>::val;
const bool standardDist = ( U == MC && V == MR );
// Form d and D
std::vector<F> d( n );
if( grid.Rank() == 0 )
for( Int j=0; j<n; ++j )
d[j] = SampleUniform<Real>( lower, upper );
mpi::Broadcast( d.data(), n, 0, grid.Comm() );
DistMatrix<F> ABackup( grid );
ABackup.AlignWith( A );
Diagonal( ABackup, d );
// Apply a Haar matrix from both sides
DistMatrix<F> Q(grid);
DistMatrix<F,MD,STAR> t(grid);
DistMatrix<Real,MD,STAR> s(grid);
ImplicitHaar( Q, t, s, n );
// Copy the result into the correct distribution
qr::ApplyQ( LEFT, NORMAL, Q, t, s, ABackup );
qr::ApplyQ( RIGHT, ADJOINT, Q, t, s, ABackup );
A = ABackup;
// Force the diagonal to be real-valued
if( isComplex )
{
const Int localHeight = A.LocalHeight();
const Int localWidth = A.LocalWidth();
for( Int jLoc=0; jLoc<localWidth; ++jLoc )
{
const Int j = A.GlobalCol(jLoc);
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
{
const Int i = A.GlobalRow(iLoc);
if( i == j )
A.SetLocalImagPart( iLoc, jLoc, Real(0) );
}
}
}
}
inline void
MakeNormalUniformSpectrum
( DistMatrix<Complex<R>,U,V>& A, Complex<R> center=0, R radius=1 )
{
#ifndef RELEASE
CallStackEntry entry("MakeNormalUniformSpectrum");
#endif
typedef Complex<R> C;
if( A.Height() != A.Width() )
LogicError("Cannot make a non-square matrix normal");
const Grid& grid = A.Grid();
const bool standardDist = ( U == MC && V == MR );
// Sample the diagonal matrix D from the ball B_radius(center)
// and then rotate it with a random Householder similarity transformation:
//
// (I-2uu^H) D (I-2uu^H)^H = D - 2(u (Conj(D) u)^H + (D u) u^H) +
// (4 u^H D u) u u^H
//
// Form d and D
const Int n = A.Height();
std::vector<C> d( n );
if( grid.Rank() == 0 )
for( Int j=0; j<n; ++j )
d[j] = SampleBall<C>( center, radius );
mpi::Broadcast( &d[0], n, 0, grid.Comm() );
DistMatrix<C> ABackup( grid );
if( standardDist )
Diagonal( A, d );
else
{
ABackup.AlignWith( A );
Diagonal( ABackup, d );
}
// Form u
DistMatrix<C> u( grid );
if( standardDist )
u.AlignWith( A );
else
u.AlignWith( ABackup );
Uniform( u, n, 1 );
const R origNorm = Nrm2( u );
Scale( 1/origNorm, u );
// Form v := D u
DistMatrix<C> v( grid );
if( standardDist )
v.AlignWith( A );
else
v.AlignWith( ABackup );
v.ResizeTo( n, 1 );
if( v.LocalWidth() == 1 )
{
const Int colShift = v.ColShift();
const Int colStride = v.ColStride();
const Int localHeight = v.LocalHeight();
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
{
const Int i = colShift + iLoc*colStride;
v.SetLocal( iLoc, 0, d[i]*u.GetLocal(iLoc,0) );
}
}
// Form w := Conj(D) u
DistMatrix<C> w( grid );
if( standardDist )
w.AlignWith( A );
else
w.AlignWith( ABackup );
w.ResizeTo( n, 1 );
if( w.LocalWidth() == 1 )
{
const Int colShift = w.ColShift();
const Int colStride = w.ColStride();
const Int localHeight = w.LocalHeight();
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
{
const Int i = colShift + iLoc*colStride;
w.SetLocal( iLoc, 0, Conj(d[i])*u.GetLocal(iLoc,0) );
}
}
// Update A := A - 2(u w^H + v u^H)
if( standardDist )
{
Ger( C(-2), u, w, A );
Ger( C(-2), v, u, A );
}
else
{
Ger( C(-2), u, w, ABackup );
Ger( C(-2), v, u, ABackup );
}
// Form \gamma := 4 u^H (D u) = 4 (u,Du)
const C gamma = 4*Dot(u,v);
// Update A := A + gamma u u^H
if( standardDist )
Ger( gamma, u, u, A );
else
Ger( gamma, u, u, ABackup );
// Copy the result into the correct distribution
if( !standardDist )
A = ABackup;
}
