inline void
MakeForsythe( DistMatrix<T,U,V>& J, T alpha, T lambda )
{
DEBUG_ONLY(CallStackEntry cse("MakeForsythe"))
MakeJordan( J, lambda );
const Int m = J.Height();
const Int n = J.Width();
if( m > 0 && n > 0 )
J.Set( m-1, 0, alpha );
}
int
main( int argc, char* argv[] )
{
Initialize( argc, argv );
try
{
const Int n = Input("--size","size of matrix",10);
const bool display = Input("--display","display matrix?",true);
const bool print = Input("--print","print matrix?",false);
ProcessInput();
PrintInputReport();
auto J = Legendre<double>( DefaultGrid(), n );
if( display )
{
Display( J, "Jacobi matrix for Legendre polynomials" );
#ifdef HAVE_QT5
Spy( J, "Spy plot for Jacobi matrix" );
#endif
}
if( print )
Print( J, "Jacobi matrix for Legendre polynomials" );
#ifdef HAVE_PMRRR
// We will compute Gaussian quadrature points and weights over [-1,+1]
// using the eigenvalue decomposition of the Jacobi matrix for the
// Legendre polynomials.
DistMatrix<double,VR, STAR> points;
DistMatrix<double,STAR,VR > X;
HermitianTridiagEig
( J.GetDiagonal(), J.GetDiagonal(-1), points, X, ASCENDING );
if( display )
Display( points, "Quadrature points" );
if( print )
Print( points, "points" );
auto firstRow = View( X, 0, 0, 1, n );
DistMatrix<double,STAR,STAR> weights = firstRow;
for( Int j=0; j<n; ++j )
{
const double gamma = weights.Get( 0, j );
weights.Set( 0, j, 2*gamma*gamma );
}
if( display )
Display( weights, "Quadrature weights" );
if( print )
Print( weights, "weights" );
#endif
}
catch( std::exception& e ) { ReportException(e); }
Finalize();
return 0;
}
inline void
MakeForsythe( DistMatrix<T,U,V>& J, T alpha, T lambda )
{
#ifndef RELEASE
CallStackEntry entry("MakeForsythe");
#endif
MakeJordan( J, lambda );
const Int m = J.Height();
const Int n = J.Width();
if( m > 0 && n > 0 )
J.Set( m-1, 0, alpha );
}
HessenbergSchurInfo
MultiBulge
( DistMatrix<F,MC,MR,BLOCK>& H,
DistMatrix<Complex<Base<F>>,STAR,STAR>& w,
DistMatrix<F,MC,MR,BLOCK>& Z,
const HessenbergSchurCtrl& ctrl )
{
DEBUG_CSE
typedef Base<F> Real;
const Real zero(0);
const Grid& grid = H.Grid();
const Int n = H.Height();
Int winBeg = ( ctrl.winBeg==END ? n : ctrl.winBeg );
Int winEnd = ( ctrl.winEnd==END ? n : ctrl.winEnd );
const Int winSize = winEnd - winBeg;
const Int blockSize = H.BlockHeight();
// TODO(poulson): Implement a more reasonable/configurable means of deciding
// when to call the sequential implementation
Int minMultiBulgeSize = Max( ctrl.minMultiBulgeSize, 2*blockSize );
// This maximum is meant to account for parallel overheads and needs to be
// more principled (and perhaps based upon the number of workers and the
// cluster characteristics)
// TODO(poulson): Re-enable this
//minMultiBulgeSize = Max( minMultiBulgeSize, 500 );
HessenbergSchurInfo info;
w.Resize( n, 1 );
if( winSize < minMultiBulgeSize )
{
return multibulge::RedundantlyHandleWindow( H, w, Z, ctrl );
}
auto ctrlShifts( ctrl );
ctrlShifts.winBeg = 0;
ctrlShifts.winEnd = END;
ctrlShifts.fullTriangle = false;
Int numIterSinceDeflation = 0;
const Int numStaleIterBeforeExceptional = 5;
// Cf. LAPACK's DLAQR0 for this choice
const Int maxIter =
Max(30,2*numStaleIterBeforeExceptional) * Max(10,winSize);
Int iterBegLast=-1, winEndLast=-1;
DistMatrix<F,STAR,STAR> hMainWin(grid), hSuperWin(grid);
DistMatrix<Real,STAR,STAR> hSubWin(grid);
while( winBeg < winEnd )
{
if( info.numIterations >= maxIter )
{
if( ctrl.demandConverged )
RuntimeError("MultiBulge QR iteration did not converge");
else
break;
}
auto winInd = IR(winBeg,winEnd);
// Detect an irreducible Hessenberg window, [iterBeg,winEnd)
// ---------------------------------------------------------
// TODO(poulson): Have the interblock chase from the previous sweep
// collect the main and sub diagonal of H along the diagonal workers
// and then broadcast across the "cross" communicator.
util::GatherTridiagonal( H, winInd, hMainWin, hSubWin, hSuperWin );
Output("winBeg=",winBeg,", winEnd=",winEnd);
Print( H, "H" );
Print( hMainWin, "hMainWin" );
Print( hSubWin, "hSubWin" );
Print( hSuperWin, "hSuperWin" );
const Int iterOffset =
DetectSmallSubdiagonal
( hMainWin.Matrix(), hSubWin.Matrix(), hSuperWin.Matrix() );
const Int iterBeg = winBeg + iterOffset;
const Int iterWinSize = winEnd-iterBeg;
if( iterOffset > 0 )
{
H.Set( iterBeg, iterBeg-1, zero );
hSubWin.Set( iterOffset-1, 0, zero );
}
if( iterWinSize == 1 )
{
if( ctrl.progress )
Output("One-by-one window at ",iterBeg);
w.Set( iterBeg, 0, hMainWin.GetLocal(iterOffset,0) );
winEnd = iterBeg;
numIterSinceDeflation = 0;
continue;
}
else if( iterWinSize == 2 )
{
if( ctrl.progress )
Output("Two-by-two window at ",iterBeg);
const F eta00 = hMainWin.GetLocal(iterOffset,0);
const F eta01 = hSuperWin.GetLocal(iterOffset,0);
const Real eta10 = hSubWin.GetLocal(iterOffset,0);
const F eta11 = hMainWin.GetLocal(iterOffset+1,0);
multibulge::TwoByTwo
( H, eta00, eta01, eta10, eta11, w, Z, iterBeg, ctrl );
winEnd = iterBeg;
numIterSinceDeflation = 0;
continue;
}
else if( iterWinSize < minMultiBulgeSize )
{
// The window is small enough to switch to the simple scheme
if( ctrl.progress )
Output("Redundantly handling window [",iterBeg,",",winEnd,"]");
auto ctrlIter( ctrl );
ctrlIter.winBeg = iterBeg;
ctrlIter.winEnd = winEnd;
auto iterInfo =
multibulge::RedundantlyHandleWindow( H, w, Z, ctrlIter );
info.numIterations += iterInfo.numIterations;
winEnd = iterBeg;
numIterSinceDeflation = 0;
continue;
}
const Int numShiftsRec = ctrl.numShifts( n, iterWinSize );
if( ctrl.progress )
{
Output("Iter. ",info.numIterations,": ");
Output(" window is [",iterBeg,",",winEnd,")");
Output(" recommending ",numShiftsRec," shifts");
}
// NOTE(poulson): In the case where exceptional shifts are used, the
// main and subdiagonals of H in the window are currently redundantly
// gathered. It could be worthwhile to pass in hMainWin and hSubWin.
const Int shiftBeg = multibulge::ComputeShifts
( H, w, iterBeg, winBeg, winEnd, numShiftsRec, numIterSinceDeflation,
numStaleIterBeforeExceptional, ctrlShifts );
auto shiftInd = IR(shiftBeg,winEnd);
auto wShifts = w(shiftInd,ALL);
// Perform a small-bulge sweep
auto ctrlSweep( ctrl );
ctrlSweep.winBeg = iterBeg;
ctrlSweep.winEnd = winEnd;
multibulge::Sweep( H, wShifts, Z, ctrlSweep );
++info.numIterations;
if( iterBeg == iterBegLast && winEnd == winEndLast )
++numIterSinceDeflation;
iterBegLast = iterBeg;
winEndLast = winEnd;
}
info.numUnconverged = winEnd-winBeg;
return info;
}
inline void UnblockedBidiagU
( DistMatrix<Complex<R> >& A,
DistMatrix<Complex<R>,MD,STAR>& tP,
DistMatrix<Complex<R>,MD,STAR>& tQ )
{
#ifndef RELEASE
PushCallStack("BidiagU");
#endif
const int tPHeight = std::max(A.Width()-1,0);
const int tQHeight = A.Width();
#ifndef RELEASE
if( A.Grid() != tP.Grid() || tP.Grid() != tQ.Grid() )
throw std::logic_error("Process grids do not match");
if( A.Height() < A.Width() )
throw std::logic_error("A must be at least as tall as it is wide");
if( tP.Viewing() && (tP.Height() != tPHeight || tP.Width() != 1) )
throw std::logic_error("tP is the wrong height");
if( tQ.Viewing() && (tQ.Height() != tQHeight || tQ.Width() != 1) )
throw std::logic_error("tQ is the wrong height");
#endif
typedef Complex<R> C;
const Grid& g = A.Grid();
if( !tP.Viewing() )
tP.ResizeTo( tPHeight, 1 );
if( !tQ.Viewing() )
tQ.ResizeTo( tQHeight, 1 );
// Matrix views
DistMatrix<C>
ATL(g), ATR(g), A00(g), a01(g), A02(g), alpha12L(g), a12R(g),
ABL(g), ABR(g), a10(g), alpha11(g), a12(g), aB1(g), AB2(g),
A20(g), a21(g), A22(g);
// Temporary matrices
DistMatrix<C,STAR,MR > a12_STAR_MR(g);
DistMatrix<C,MC, STAR> aB1_MC_STAR(g);
DistMatrix<C,MR, STAR> x12Adj_MR_STAR(g);
DistMatrix<C,MC, STAR> w21_MC_STAR(g);
PushBlocksizeStack( 1 );
PartitionDownDiagonal
( A, ATL, ATR,
ABL, ABR, 0 );
while( ATL.Width() < A.Width() )
{
RepartitionDownDiagonal
( ATL, /**/ ATR, A00, /**/ a01, A02,
/*************/ /**********************/
/**/ a10, /**/ alpha11, a12,
ABL, /**/ ABR, A20, /**/ a21, A22 );
View2x1
( aB1, alpha11,
a21 );
View2x1
( AB2, a12,
A22 );
aB1_MC_STAR.AlignWith( aB1 );
a12_STAR_MR.AlignWith( a12 );
x12Adj_MR_STAR.AlignWith( AB2 );
w21_MC_STAR.AlignWith( A22 );
Zeros( a12.Width(), 1, x12Adj_MR_STAR );
Zeros( a21.Height(), 1, w21_MC_STAR );
const bool thisIsMyRow = ( g.Row() == alpha11.ColAlignment() );
const bool thisIsMyCol = ( g.Col() == alpha11.RowAlignment() );
const bool nextIsMyCol = ( g.Col() == a12.RowAlignment() );
//--------------------------------------------------------------------//
// Find tauQ, u, and epsilonQ such that
// I - conj(tauQ) | 1 | | 1, u^H | | alpha11 | = | epsilonQ |
// | u | | a21 | | 0 |
const C tauQ = Reflector( alpha11, a21 );
tQ.Set(A00.Height(),0,tauQ );
C epsilonQ=0;
if( thisIsMyCol && thisIsMyRow )
epsilonQ = alpha11.GetLocal(0,0);
// Set aB1 = | 1 | and form x12^H := (aB1^H AB2)^H = AB2^H aB1
// | u |
alpha11.Set(0,0,C(1));
aB1_MC_STAR = aB1;
internal::LocalGemv
( ADJOINT, C(1), AB2, aB1_MC_STAR, C(0), x12Adj_MR_STAR );
x12Adj_MR_STAR.SumOverCol();
// Update AB2 := AB2 - conj(tauQ) aB1 x12
// = AB2 - conj(tauQ) aB1 aB1^H AB2
// = (I - conj(tauQ) aB1 aB1^H) AB2
internal::LocalGer( -Conj(tauQ), aB1_MC_STAR, x12Adj_MR_STAR, AB2 );
// Put epsilonQ back instead of the temporary value, 1
if( thisIsMyCol && thisIsMyRow )
alpha11.SetLocal(0,0,epsilonQ);
if( A22.Width() != 0 )
{
// Due to the deficiencies in the BLAS ?gemv routines, this section
// is easier if we temporarily conjugate a12
Conjugate( a12 );
// Expose the subvector we seek to zero, a12R
PartitionRight( a12, alpha12L, a12R );
// Find tauP, v, and epsilonP such that
// I - conj(tauP) | 1 | | 1, v^H | | alpha12L | = | epsilonP |
// | v | | a12R^T | | 0 |
const C tauP = Reflector( alpha12L, a12R );
tP.Set(A00.Height(),0,tauP);
C epsilonP=0;
if( nextIsMyCol && thisIsMyRow )
epsilonP = alpha12L.GetLocal(0,0);
// Set a12^T = | 1 | and form w21 := A22 a12^T = A22 | 1 |
// | v | | v |
alpha12L.Set(0,0,C(1));
a12_STAR_MR = a12;
internal::LocalGemv
( NORMAL, C(1), A22, a12_STAR_MR, C(0), w21_MC_STAR );
w21_MC_STAR.SumOverRow();
// A22 := A22 - tauP w21 conj(a12)
// = A22 - tauP A22 a12^T conj(a12)
// = A22 (I - tauP a12^T conj(a12))
// = A22 conj(I - conj(tauP) a12^H a12)
// which compensates for the fact that the reflector was generated
// on the conjugated a12.
internal::LocalGer( -tauP, w21_MC_STAR, a12_STAR_MR, A22 );
// Put epsilonP back instead of the temporary value, 1
if( nextIsMyCol && thisIsMyRow )
alpha12L.SetLocal(0,0,epsilonP);
// Undue the temporary conjugation
Conjugate( a12 );
}
//--------------------------------------------------------------------//
aB1_MC_STAR.FreeAlignments();
a12_STAR_MR.FreeAlignments();
x12Adj_MR_STAR.FreeAlignments();
w21_MC_STAR.FreeAlignments();
SlidePartitionDownDiagonal
( ATL, /**/ ATR, A00, a01, /**/ A02,
/**/ a10, alpha11, /**/ a12,
/*************/ /**********************/
ABL, /**/ ABR, A20, a21, /**/ A22 );
}
PopBlocksizeStack();
#ifndef RELEASE
PopCallStack();
#endif
}