void TestGemv
( El::Int height,
El::Int width,
El::Orientation orientation,
const El::Grid& grid,
bool print )
{
El::DistMatrix<Field> A(grid);
El::Uniform( A, height, width );
// Draw the entries of the original x and y from uniform distributions
// over the complex unit ball
El::DistMatrix<Field,El::VC,El::STAR> x(grid), y(grid);
if( orientation == El::NORMAL )
{
El::Uniform( x, width, 1 );
El::Uniform( y, height, 1 );
}
else
{
El::Uniform( x, height, 1 );
El::Uniform( y, width, 1 );
}
if( print )
{
El::Print( A, "A" );
El::Print( x, "x" );
El::Print( y, "y" );
}
// Run the matrix-vector product
if( grid.Rank() == 0 )
El::Output("Starting Gemv (with Field=",El::TypeName<Field>(),")");
El::Timer gemvElem;
gemvElem.Start();
// Form y := 3 A x + 4 y
El::Gemv( orientation, Field(3), A, x, Field(4), y );
gemvElem.Stop();
if( grid.Rank() == 0 )
El::Output(" Time: ",gemvElem.Total());
if( print )
{
if( orientation == El::NORMAL )
El::Print( y, "y := 3 A x + 4 y" );
else
El::Print( y, "y := 3 A^H x + 4 y" );
}
}
int
main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
El::mpi::Comm comm = El::mpi::COMM_WORLD;
try
{
typedef double Real;
typedef El::Complex<Real> Scalar;
const El::Int m = El::Input("--height","height of matrix",100);
const El::Int n = El::Input("--width","width of matrix",100);
const bool print = El::Input("--print","print matrices?",false);
El::ProcessInput();
El::PrintInputReport();
El::DistMatrix<Scalar> A;
El::Uniform( A, m, n );
El::Timer timer;
// Compute the pseudoinverseof A (but do not overwrite A)
El::DistMatrix<Scalar> pinvA( A );
if( El::mpi::Rank(comm) == 0 )
timer.Start();
El::Pseudoinverse( pinvA );
if( El::mpi::Rank(comm) == 0 )
timer.Stop();
if( print )
{
El::Print( A, "A" );
El::Print( pinvA, "pinv(A)" );
}
const Real frobA = El::FrobeniusNorm( A );
const Real frobPinvA = El::FrobeniusNorm( pinvA );
if( El::mpi::Rank(comm) == 0 )
{
El::Output("PseudoInverse time: ",timer.Total()," secs");
El::Output
("|| A ||_F = ",frobA,"\n",
"|| pinv(A) ||_F = ",frobPinvA,"\n");
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
El::mpi::Comm comm = El::mpi::COMM_WORLD;
const int commRank = El::mpi::Rank( comm );
const int commSize = El::mpi::Size( comm );
try
{
typedef double Real;
typedef El::Complex<Real> Scalar;
const El::Int n = El::Input("--size","size of matrix",100);
const El::Int numRhs = El::Input("--numRhs","# of right-hand sides",1);
const El::Int blocksize =
El::Input("--blocksize","algorithmic blocksize",64);
const El::Int numTests = El::Input("--numTests","number of tests",3);
const bool error = El::Input("--error","test Elemental error?",true);
El::Int gridHeight = El::Input("--gridHeight","grid height",0);
const bool details = El::Input("--details","print norm details?",false);
const bool print = El::Input("--print","print matrices?",false);
El::ProcessInput();
El::PrintInputReport();
El::SetBlocksize( blocksize );
// If the grid height wasn't specified, then we should attempt to build
// a nearly-square process grid
if( gridHeight == 0 )
gridHeight = El::Grid::FindFactor( commSize );
El::Grid grid( comm, gridHeight );
if( commRank == 0 )
El::Output("Grid is: ",grid.Height()," x ",grid.Width());
// Set up random A and B, then make the copies X := B
El::Timer timer;
El::DistMatrix<Scalar> A(grid), B(grid), X(grid);
for( El::Int test=0; test<numTests; ++test )
{
El::Uniform( A, n, n );
El::Uniform( B, n, numRhs );
X = B;
if( print )
{
El::Print( A, "A" );
El::Print( B, "B" );
}
// Perform the LU factorization and simultaneous solve
if( commRank == 0 )
El::Output("Starting Elemental linear solve");
El::mpi::Barrier( comm );
if( commRank == 0 )
timer.Start();
El::LinearSolve( A, X );
El::mpi::Barrier( comm );
if( commRank == 0 )
El::Output(timer.Stop()," seconds");
if( error )
{
// Form R := A X - B
auto R( B );
El::Gemm
( El::NORMAL, El::NORMAL, Scalar(1), A, X, Scalar(-1), R );
// Compute infinity norms and a relative residual
const Real eps = El::limits::Epsilon<Real>();
const Real AInfNorm = El::InfinityNorm( A );
const Real BInfNorm = El::InfinityNorm( B );
const Real XInfNorm = El::InfinityNorm( X );
const Real RInfNorm = El::InfinityNorm( R );
const Real infResidual = RInfNorm / (AInfNorm*XInfNorm*eps*n);
if( commRank == 0 )
{
if( details )
{
El::Output("");
El::Output("||A||_oo = ",AInfNorm);
El::Output("||B||_oo = ",BInfNorm);
El::Output("||X||_oo = ",XInfNorm);
El::Output("||A X - B||_oo = ",RInfNorm);
}
El::Output
("||A X - B||_oo / (||A||_oo ||X||_oo eps n) = ",
infResidual);
}
// Compute one norms and a relative residual
const Real AOneNorm = El::OneNorm( A );
const Real BOneNorm = El::OneNorm( B );
const Real XOneNorm = El::OneNorm( X );
const Real ROneNorm = El::OneNorm( R );
const Real oneResidual = ROneNorm / (AOneNorm*XOneNorm*eps*n);
if( commRank == 0 )
{
if( details )
{
El::Output("");
El::Output("||A||_1 = ",AOneNorm);
El::Output("||B||_1 = ",BOneNorm);
El::Output("||X||_1 = ",XOneNorm);
El::Output("||A X - B||_1 = ",ROneNorm);
}
El::Output
("||A X - B||_1 / (||A||_1 ||X||_1 eps n) = ",
oneResidual,"\n");
}
}
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
try
{
typedef double Real;
typedef El::Complex<Real> Scalar;
const El::Int m = El::Input("--height","height of matrix",20);
const El::Int n = El::Input("--width","width of matrix",100);
const El::Int r = El::Input("--rank","rank of matrix",5);
const El::Int maxSteps =
El::Input("--maxSteps","max # of steps of QR",10);
const Real tol = El::Input("--tol","tolerance for ID",Real(-1));
const bool print = El::Input("--print","print matrices?",false);
const bool smallestFirst =
El::Input("--smallestFirst","smallest norm first?",false);
El::ProcessInput();
El::PrintInputReport();
El::mpi::Comm comm = El::mpi::COMM_WORLD;
const El::Grid& grid = El::Grid::Default();
El::DistMatrix<Scalar> U(grid), V(grid), A(grid);
El::Uniform( U, m, r );
El::Uniform( V, n, r );
El::Gemm( El::NORMAL, El::ADJOINT, Scalar(1), U, V, A );
const Real frobA = El::FrobeniusNorm( A );
if( print )
El::Print( A, "A" );
El::DistPermutation Omega(grid);
El::DistMatrix<Scalar,El::STAR,El::VR> Z(grid);
El::QRCtrl<Real> ctrl;
ctrl.boundRank = true;
ctrl.maxRank = maxSteps;
if( tol != -1. )
{
ctrl.adaptive = true;
ctrl.tol = tol;
}
ctrl.smallestFirst = smallestFirst;
El::Timer timer;
if( El::mpi::Rank(comm) == 0 )
timer.Start();
El::ID( A, Omega, Z, ctrl );
if( El::mpi::Rank(comm) == 0 )
timer.Stop();
const El::Int rank = Z.Height();
if( print )
{
El::DistMatrix<El::Int> OmegaFull(grid);
Omega.ExplicitMatrix( OmegaFull );
El::Print( OmegaFull, "Omega" );
El::Print( Z, "Z" );
}
// Pivot A and form the matrix of its (hopefully) dominant columns
El::Timer permTimer("permTimer");
permTimer.Start();
Omega.PermuteCols( A );
permTimer.Stop();
auto hatA( A );
hatA.Resize( m, rank );
if( print )
{
El::Print( A, "A Omega^T" );
El::Print( hatA, "\\hat{A}" );
}
// Check || A Omega^T - \hat{A} [I, Z] ||_F / || A ||_F
El::DistMatrix<Scalar> AL(grid), AR(grid);
El::PartitionRight( A, AL, AR, rank );
El::Zero( AL );
{
El::DistMatrix<Scalar,El::MC,El::STAR> hatA_MC_STAR(grid);
El::DistMatrix<Scalar,El::STAR,El::MR> Z_STAR_MR(grid);
hatA_MC_STAR.AlignWith( AR );
Z_STAR_MR.AlignWith( AR );
hatA_MC_STAR = hatA;
Z_STAR_MR = Z;
El::LocalGemm
( El::NORMAL, El::NORMAL,
Scalar(-1), hatA_MC_STAR, Z_STAR_MR, Scalar(1), AR );
}
const Real frobError = El::FrobeniusNorm( A );
if( print )
El::Print( A, "A Omega^T - \\hat{A} [I, Z]" );
if( El::mpi::Rank(comm) == 0 )
{
El::Output(" ID time: ",timer.Total()," secs");
El::Output
("|| A ||_F = ",frobA,"\n",
"|| A Omega^T - \\hat{A} [I, Z] ||_F / || A ||_F = ",
frobError/frobA);
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
try
{
const El::Int n = El::Input("--n","problem size",200);
const El::Int maxIter =
El::Input("--maxIter","maximum # of iter's",500);
const Real lb = El::Input("--lb","lower bound for x",0.5);
const Real ub = El::Input("--ub","upper bound for x",1.0);
const Real lbEig = El::Input("--lbEig","spectral lower bound",1.);
const Real ubEig = El::Input("--ubEig","spectral upper bound",2.);
const Real rho = El::Input("--rho","augmented Lagrangian param.",1.);
const Real alpha = El::Input("--alpha","over-relaxation",1.2);
const Real absTol = El::Input("--absTol","absolute tolerance",1e-6);
const Real relTol = El::Input("--relTol","relative tolerance",1e-4);
const bool inv = El::Input("--inv","form inv(LU) to avoid trsv?",true);
const bool progress = El::Input("--progress","print progress?",true);
const bool display = El::Input("--display","display matrices?",false);
const bool print = El::Input("--print","print matrices",false);
El::ProcessInput();
El::PrintInputReport();
El::ADMMCtrl<Real> ctrl;
ctrl.rho = rho;
ctrl.alpha = alpha;
ctrl.maxIter = maxIter;
ctrl.absTol = absTol;
ctrl.relTol = relTol;
ctrl.inv = inv;
ctrl.print = progress;
El::DistMatrix<Real> Q, c, xTrue;
El::HermitianUniformSpectrum( Q, n, lbEig, ubEig );
// Alternate the entries of xTrue between ub and lb
El::Zeros( xTrue, n, 1 );
if( xTrue.LocalWidth() == 1 )
for( El::Int iLoc=0; iLoc<xTrue.LocalHeight(); ++iLoc )
xTrue.SetLocal( iLoc, 0,
( xTrue.GlobalRow(iLoc)%2==0 ? lb : ub ) );
// Set c := - Q xTrue - du + dl
El::Zeros( c, n, 1 );
El::Hemv( El::LOWER, Real(-1), Q, xTrue, Real(0), c );
if( c.LocalWidth() == 1 )
for( El::Int iLoc=0; iLoc<c.LocalHeight(); ++iLoc )
c.UpdateLocal( iLoc, 0,
( c.GlobalRow(iLoc)%2==0 ? 0.5 : -0.5 ) );
if( print )
{
El::Print( Q, "Q" );
El::Print( c, "c" );
El::Print( xTrue, "xTrue" );
}
if( display )
El::Display( Q, "Q" );
El::Timer timer;
El::DistMatrix<Real> z;
if( El::mpi::Rank() == 0 )
timer.Start();
El::qp::box::ADMM( Q, c, lb, ub, z, ctrl );
if( El::mpi::Rank() == 0 )
timer.Stop();
if( print )
El::Print( z, "z" );
if( El::mpi::Rank() == 0 )
El::Output("QPBox time: ",timer.Total(),"secs");
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int
main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
El::mpi::Comm comm = El::mpi::COMM_WORLD;
const int commRank = El::mpi::Rank( comm );
try
{
typedef double Real;
typedef El::Complex<Real> Scalar;
El::Timer timer;
const El::Int n1 = El::Input("--n1","first grid dimension",30);
const El::Int n2 = El::Input("--n2","second grid dimension",30);
const El::Int n3 = El::Input("--n3","third grid dimension",30);
const Real omega = El::Input("--omega","angular frequency",Real(18));
const Real damping = El::Input("--damping","damping parameter",Real(7));
const bool selInv = El::Input("--selInv","selectively invert?",false);
const bool intraPiv = El::Input("--intraPiv","frontal pivoting?",false);
const bool natural = El::Input("--natural","analytic partitions?",true);
const bool sequential = El::Input
("--sequential","sequential partitions?",true);
const int numDistSeps = El::Input
("--numDistSeps",
"number of separators to try per distributed partition",1);
const int numSeqSeps = El::Input
("--numSeqSeps",
"number of separators to try per sequential partition",1);
const int cutoff =
El::Input("--cutoff","cutoff for nested dissection",128);
const bool print = El::Input("--print","print matrix?",false);
const bool display = El::Input("--display","display matrix?",false);
El::ProcessInput();
const El::Grid grid(comm);
El::DistSparseMatrix<Scalar> A(grid);
Scalar dampedOmega( omega, damping );
El::Helmholtz( A, n1, n2, n3, dampedOmega*dampedOmega );
const El::Int N = A.Height();
if( display )
El::Display( A, "A" );
if( print )
El::Print( A, "A" );
if( commRank == 0 )
El::Output("Generating random vector x and forming y := A x");
timer.Start();
El::DistMultiVec<Scalar> x( N, 1, grid ), y( N, 1, grid );
El::MakeUniform( x );
El::Zero( y );
El::Multiply( El::NORMAL, Scalar(1), A, x, Scalar(0), y );
const Real yOrigNorm = El::FrobeniusNorm( y );
El::mpi::Barrier(comm);
if( commRank == 0 )
El::Output(timer.Stop()," seconds");
El::BisectCtrl ctrl;
ctrl.sequential = sequential;
ctrl.numSeqSeps = numSeqSeps;
ctrl.numDistSeps = numDistSeps;
ctrl.cutoff = cutoff;
El::DistSparseLDLFactorization<Scalar> sparseLDLFact;
const bool hermitian = false;
if( commRank == 0 )
El::Output("Running analysis...");
timer.Start();
if( natural )
{
sparseLDLFact.Initialize3DGridGraph
( n1, n2, n3, A, hermitian, ctrl );
}
else
{
sparseLDLFact.Initialize( A, hermitian, ctrl );
}
El::mpi::Barrier(comm);
if( commRank == 0 )
El::Output(timer.Stop()," seconds");
auto& front = sparseLDLFact.Front();
auto& info = sparseLDLFact.NodeInfo();
const El::Int rootSepSize = info.size;
if( commRank == 0 )
El::Output(rootSepSize," vertices in root separator\n");
if( commRank == 0 )
El::Output("Running LDL factorization...");
El::mpi::Barrier(comm);
timer.Start();
El::LDLFrontType type;
if( intraPiv )
type = selInv ? El::LDL_INTRAPIV_SELINV_2D : El::LDL_INTRAPIV_2D;
else
type = selInv ? El::LDL_SELINV_2D : El::LDL_2D;
sparseLDLFact.Factor( type );
El::mpi::Barrier(comm);
if( commRank == 0 )
El::Output(timer.Stop()," seconds");
if( info.child != nullptr && info.child->onLeft )
{
if( commRank == 0 )
El::Output
("Computing SVD of connectivity of second separator to "
"the root separator...");
timer.Start();
const auto& FL = front.child->L2D;
const El::Grid& grid = FL.Grid();
const El::Int height = FL.Height();
const El::Int width = FL.Width();
auto B = FL( El::IR(width,height), El::IR(0,width) );
El::DistMatrix<Real,El::VR,El::STAR> singVals_VR_STAR( grid );
El::SVD( B, singVals_VR_STAR );
El::DistMatrix<Real,El::CIRC,El::CIRC> singVals( singVals_VR_STAR );
El::mpi::Barrier( grid.Comm() );
const Real twoNorm = El::MaxNorm( singVals_VR_STAR );
const El::Int minDim = singVals_VR_STAR.Height();
if( grid.Rank() == singVals.Root() )
{
El::Output
(" two-norm is ",twoNorm," (took ",timer.Stop()," seconds)");
for( Real tol=1e-1; tol>=Real(1e-10); tol/=10 )
{
El::Int numRank = minDim;
for( El::Int j=0; j<minDim; ++j )
{
if( singVals.GetLocal(j,0) <= twoNorm*tol )
{
numRank = j;
break;
}
}
El::Output(" rank (",tol,")=",numRank,"/",minDim);
}
}
}
if( commRank == 0 )
El::Output
("Computing SVD of the largest off-diagonal block of "
"numerical Green's function on root separator...");
{
timer.Start();
const auto& FL = front.L2D;
const El::Grid& grid = FL.Grid();
const El::Int lHalf = rootSepSize/2;
const El::Int uHalf = rootSepSize - lHalf;
if( commRank == 0 )
El::Output("lower half=",lHalf,", upper half=",uHalf);
auto offDiagBlock =
FL( El::IR(lHalf,rootSepSize), El::IR(0,lHalf) );
El::DistMatrix<Real,El::VR,El::STAR> singVals_VR_STAR( grid );
El::SVD( offDiagBlock, singVals_VR_STAR );
El::DistMatrix<Real,El::CIRC,El::CIRC> singVals( singVals_VR_STAR );
El::mpi::Barrier( grid.Comm() );
const Real twoNorm = El::MaxNorm( singVals_VR_STAR );
if( grid.Rank() == singVals.Root() )
{
El::Output(timer.Stop()," seconds");
for( Real tol=1e-1; tol>=Real(1e-10); tol/=10 )
{
El::Int numRank = lHalf;
for( El::Int j=0; j<lHalf; ++j )
{
if( singVals.GetLocal(j,0) <= twoNorm*tol )
{
numRank = j;
break;
}
}
El::Output(" rank (",tol,")=",numRank,"/",lHalf);
}
}
}
if( commRank == 0 )
El::Output("Solving against y...");
timer.Start();
sparseLDLFact.Solve( y );
El::mpi::Barrier(comm);
if( commRank == 0 )
El::Output(timer.Stop()," seconds");
if( commRank == 0 )
El::Output("Checking error in computed solution...");
const Real xNorm = El::FrobeniusNorm( x );
const Real yNorm = El::FrobeniusNorm( y );
y -= x;
const Real errorNorm = El::FrobeniusNorm( y );
if( commRank == 0 )
El::Output
("|| x ||_2 = ",xNorm,"\n",
"|| xComp ||_2 = ",yNorm,"\n",
"|| A x ||_2 = ",yOrigNorm,"\n",
"|| error ||_2 / || x ||_2 = ",errorNorm/xNorm,"\n",
"|| error ||_2 / || A x ||_2 = ",errorNorm/yOrigNorm);
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int
main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
try
{
const El::Int m = El::Input("--m","height of matrix",100);
const El::Int n = El::Input("--n","width of matrix",200);
// TODO(poulson): Add options for controlling IPM
const El::Int maxIter =
El::Input("--maxIter","maximum # of iter's",500);
const Real lambda = El::Input("--lambda","DS parameter",0.5);
const bool display = El::Input("--display","display matrices?",false);
const bool print = El::Input("--print","print matrices",false);
El::ProcessInput();
El::PrintInputReport();
El::SparseMatrix<Real> A;
El::Matrix<Real> b, xTrue;
El::Identity( A, m, n );
El::Uniform( b, m, 1 );
if( print )
{
El::Print( A, "A" );
El::Print( b, "b" );
}
if( display )
El::Display( A, "A" );
El::lp::affine::Ctrl<Real> affineCtrl;
affineCtrl.mehrotraCtrl.print = true;
El::Matrix<Real> x;
El::Timer timer;
if( El::mpi::Rank() == 0 )
timer.Start();
El::DS( A, b, lambda, x, affineCtrl );
if( El::mpi::Rank() == 0 )
timer.Stop();
if( print )
El::Print( x, "x" );
const El::Int xZeroNorm = El::ZeroNorm( x );
if( El::mpi::Rank() == 0 )
{
El::Output("Dantzig Selector time: ",timer.Total()," secs");
El::Output("|| x ||_0 = ",xZeroNorm);
}
SoftThreshold( x, El::Sqrt(El::limits::Epsilon<Real>()) );
if( print )
El::Print( x, "xThresh" );
const El::Int xZeroNormThresh = El::ZeroNorm( x );
if( El::mpi::Rank() == 0 )
{
El::Output("|| xThresh ||_0 = ",xZeroNormThresh);
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int
main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
try
{
const El::Int n = El::Input("--size","width of matrix",100);
const bool display = El::Input("--display","display matrices?",false);
const bool print = El::Input("--print","print matrices?",false);
const bool smallestFirst =
El::Input("--smallestFirst","smallest norm first?",false);
El::ProcessInput();
El::PrintInputReport();
const El::Int m = n;
El::QRCtrl<double> ctrl;
ctrl.smallestFirst = smallestFirst;
El::DistMatrix<El::Complex<double>> A;
El::GKS( A, n );
const double frobA = El::FrobeniusNorm( A );
if( display )
El::Display( A, "A" );
if( print )
El::Print( A, "A" );
// Compute the pivoted QR decomposition of A, but do not overwrite A
auto QRPiv( A );
El::DistMatrix<El::Complex<double>> householderScalarsPiv;
El::DistMatrix<double> signaturePiv;
El::DistPermutation Omega;
El::Timer PQRtimer;
if( El::mpi::Rank() == 0 )
PQRtimer.Start();
El::QR( QRPiv, householderScalarsPiv, signaturePiv, Omega );
if( El::mpi::Rank() == 0 )
PQRtimer.Stop();
if( display )
{
El::Display( QRPiv, "QRPiv" );
El::Display( householderScalarsPiv, "householderScalarsPiv" );
El::Display( signaturePiv, "signaturePiv" );
El::DistMatrix<El::Int> OmegaFull;
Omega.ExplicitMatrix( OmegaFull );
El::Display( OmegaFull, "Omega" );
}
if( print )
{
El::Print( QRPiv, "QRPiv" );
El::Print( householderScalarsPiv, "householderScalarsPiv" );
El::Print( signaturePiv, "signaturePiv" );
El::DistMatrix<El::Int> OmegaFull;
Omega.ExplicitMatrix( OmegaFull );
El::Print( OmegaFull, "Omega" );
}
// Compute the standard QR decomposition of A
auto QRNoPiv( A );
El::DistMatrix<El::Complex<double>> householderScalarsNoPiv;
El::DistMatrix<double> signatureNoPiv;
El::Timer QRtimer;
if( El::mpi::Rank() == 0 )
QRtimer.Start();
El::QR( QRNoPiv, householderScalarsNoPiv, signatureNoPiv );
if( El::mpi::Rank() == 0 )
QRtimer.Start();
if( display )
{
El::Display( QRNoPiv, "QRNoPiv" );
El::Display( householderScalarsNoPiv, "householderScalarsNoPiv" );
El::Display( signatureNoPiv, "signatureNoPiv" );
}
if( print )
{
El::Print( QRNoPiv, "QRNoPiv" );
El::Print( householderScalarsNoPiv, "householderScalarsNoPiv" );
El::Print( signatureNoPiv, "signatureNoPiv" );
}
// Check the error in the pivoted QR factorization,
// || A Omega^T - Q R ||_F / || A ||_F
auto E( QRPiv );
El::MakeTrapezoidal( El::UPPER, E );
El::qr::ApplyQ
( El::LEFT, El::NORMAL, QRPiv, householderScalarsPiv, signaturePiv, E );
Omega.InversePermuteCols( E );
E -= A;
const double frobQRPiv = El::FrobeniusNorm( E );
if( display )
El::Display( E, "A P - Q R" );
if( print )
El::Print( E, "A P - Q R" );
// Check the error in the standard QR factorization,
// || A - Q R ||_F / || A ||_F
E = QRNoPiv;
El::MakeTrapezoidal( El::UPPER, E );
El::qr::ApplyQ
( El::LEFT, El::NORMAL,
QRNoPiv, householderScalarsNoPiv, signatureNoPiv, E );
E -= A;
const double frobQRNoPiv = El::FrobeniusNorm( E );
if( display )
El::Display( E, "A - Q R" );
if( print )
El::Print( E, "A - Q R" );
// Check orthogonality of pivoted Q, || I - Q^H Q ||_F / || A ||_F
El::Identity( E, m, n );
El::qr::ApplyQ
( El::LEFT, El::NORMAL,
QRPiv, householderScalarsPiv, signaturePiv, E );
El::qr::ApplyQ
( El::LEFT, El::ADJOINT,
QRPiv, householderScalarsPiv, signaturePiv, E );
const El::Int k = El::Min(m,n);
auto EUpper = E( El::IR(0,k), El::IR(0,k) );
El::ShiftDiagonal( EUpper, El::Complex<double>(-1) );
const double frobOrthogPiv = El::FrobeniusNorm( EUpper );
if( display )
El::Display( E, "pivoted I - Q^H Q" );
if( print )
El::Print( E, "pivoted I - Q^H Q" );
// Check orthogonality of unpivoted Q, || I - Q^H Q ||_F / || A ||_F
El::Identity( E, m, n );
El::qr::ApplyQ
( El::LEFT, El::NORMAL,
QRPiv, householderScalarsPiv, signaturePiv, E );
El::qr::ApplyQ
( El::LEFT, El::ADJOINT,
QRPiv, householderScalarsPiv, signaturePiv, E );
EUpper = E( El::IR(0,k), El::IR(0,k) );
El::ShiftDiagonal( EUpper, El::Complex<double>(-1) );
const double frobOrthogNoPiv = El::FrobeniusNorm( EUpper );
if( display )
El::Display( E, "unpivoted I - Q^H Q" );
if( print )
El::Print( E, "unpivoted I - Q^H Q" );
if( El::mpi::Rank() == 0 )
{
El::Output("Pivot QR time: ",PQRtimer.Total()," secs\n",
" QR time: ",QRtimer.Total()," secs");
El::Output
("|| A ||_F = ",frobA,"\n\n",
"With pivoting: \n",
" || A P - Q R ||_F / || A ||_F = ",frobQRPiv/frobA,"\n",
" || I - Q^H Q ||_F / || A ||_F = ",frobOrthogPiv/frobA,"\n\n",
"Without pivoting: \n",
" || A - Q R ||_F / || A ||_F = ",frobQRNoPiv/frobA,"\n",
" || I - Q^H Q ||_F / || A ||_F = ",frobOrthogNoPiv/frobA,"\n");
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int
main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
try
{
El::Int gridHeight =
El::Input("--gridHeight","process grid height",0);
const bool colMajor =
El::Input("--colMajor","column-major ordering?",true);
const El::Int matType =
El::Input("--matType","0:uniform,\n"
"1:Haar,\n"
"2:Lotkin,\n"
"3:Grcar,\n"
"4:FoxLi,\n"
"5:HelmholtzPML1D,\n"
"6:HelmholtzPML2D,\n"
"7:TrefethenEmbree,\n"
"8:Bull's head,\n"
"9:Triangle,\n"
"10:Whale,\n"
"11:UniformHelmholtzGreen's,\n"
"12:HatanoNelson,\n"
"13:EhrenfestDecay,\n"
"14:RiffleDecay\n"
"15:Jordan\n",4);
const El::Int normInt = El::Input("--norm","0:two norm,1:one norm",0);
const El::Int n = El::Input("--size","height of matrix",100);
const El::Int nbAlg = El::Input("--nbAlg","algorithmic blocksize",96);
// QR algorithm options
const El::Int nbDist =
El::Input("--nbDist","distribution blocksize",32);
// Spectral Divide and Conquer options
const bool sdc = El::Input("--sdc","use Spectral D&C?",false);
const El::Int cutoff = El::Input("--cutoff","problem size for QR",256);
const El::Int maxInnerIts = El::Input("--maxInnerIts","SDC limit",2);
const El::Int maxOuterIts = El::Input("--maxOuterIts","SDC limit",10);
const bool random = El::Input("--random","Random RRQR in SDC",true);
const double sdcTol = El::Input("--sdcTol","Rel. tol. for SDC",1e-6);
const double spreadFactor =
El::Input("--spreadFactor","median pert.",1e-6);
const double signTol =
El::Input("--signTol","Sign tolerance for SDC",1e-9);
const double realCenter = El::Input("--realCenter","real center",0.);
const double imagCenter = El::Input("--imagCenter","imag center",0.);
double realWidth = El::Input("--realWidth","x width of image",0.);
double imagWidth = El::Input("--imagWidth","y width of image",0.);
const El::Int numReal = El::Input("--numReal","num real chunks",2);
const El::Int numImag = El::Input("--numImag","num imag chunks",2);
const El::Int realSize =
El::Input("--realSize","number of x samples",100);
const El::Int imagSize =
El::Input("--imagSize","number of y samples",100);
const bool arnoldi = El::Input("--arnoldi","use Arnoldi?",true);
const El::Int basisSize =
El::Input("--basisSize","num basis vectors",10);
const El::Int maxIts =
El::Input("--maxIts","maximum pseudospec iter's",200);
const double psTol =
El::Input("--psTol","tolerance for pseudospectra",1e-6);
// Uniform options
const double uniformRealCenter =
El::Input("--uniformRealCenter","real center of uniform dist",0.);
const double uniformImagCenter =
El::Input("--uniformImagCenter","imag center of uniform dist",0.);
const double uniformRadius =
El::Input("--uniformRadius","radius of uniform dist",1.);
// Grcar options
const El::Int numBands =
El::Input("--numBands","num bands for Grcar",3);
// Fox-Li options
const double omega =
El::Input("--omega","frequency for Fox-Li/Helm",16*M_PI);
// Helmholtz-PML options [also uses omega from Fox-Li]
const El::Int mx =
El::Input("--mx","number of x points for HelmholtzPML",30);
const El::Int my =
El::Input("--my","number of y points for HelmholtzPML",30);
const El::Int numPmlPoints =
El::Input("--numPml","num PML points for Helm",5);
const double sigma = El::Input("--sigma","PML amplitude",1.5);
const double pmlExp =
El::Input("--pmlExp","PML takeoff exponent",3.);
// Uniform Helmholtz Green's options
const double lambda =
El::Input("--lambda","wavelength of U.H.Green's",0.1);
// Hatano-Nelson options
const double gHatano =
El::Input("--gHatano","g in Hatano-Nelson",0.5);
const bool periodic =
El::Input("--periodic","periodic HatanoNelson?",true);
// Input/Output options
const bool progress = El::Input("--progress","print progress?",true);
const bool deflate = El::Input("--deflate","deflate?",true);
const bool display = El::Input("--display","display matrices?",false);
const bool write = El::Input("--write","write matrices?",false);
const bool saveSchur =
El::Input("--saveSchur","save Schur factor?",true);
const El::Int numSaveFreq =
El::Input("--numSaveFreq","numerical save frequency",-1);
const El::Int imgSaveFreq =
El::Input("--imgSaveFreq","image save frequency",-1);
const El::Int imgDispFreq =
El::Input("--imgDispFreq","image display frequency",-1);
const std::string numBase =
El::Input("--numBase","numerical save basename",std::string("num"));
const std::string imgBase =
El::Input("--imgBase","image save basename",std::string("img"));
const El::Int numFormatInt =
El::Input("--numFormat","numerical format",2);
const El::Int imgFormatInt =
El::Input("--imgFormat","image format",8);
const El::Int colorMapInt =
El::Input("--colorMap","color map",0);
const bool itCounts =
El::Input("--itCounts","display iter. counts?",true);
El::ProcessInput();
El::PrintInputReport();
El::mpi::Comm comm = El::mpi::COMM_WORLD;
if( gridHeight == 0 )
gridHeight = El::Grid::DefaultHeight( El::mpi::Size(comm) );
const El::GridOrder order = colMajor ? El::COLUMN_MAJOR : El::ROW_MAJOR;
const El::Grid grid( comm, gridHeight, order );
El::SetBlocksize( nbAlg );
if( normInt < 0 || normInt > 1 )
El::LogicError("Invalid pseudospec norm type");
if( numFormatInt < 1 || numFormatInt >= El::FileFormat_MAX )
El::LogicError
("Invalid numerical format integer, should be in [1,",
El::FileFormat_MAX,")");
if( imgFormatInt < 1 || imgFormatInt >= El::FileFormat_MAX )
El::LogicError
("Invalid image format integer, should be in [1,",
El::FileFormat_MAX,")");
const auto psNorm = static_cast<El::PseudospecNorm>(normInt);
const auto numFormat = static_cast<El::FileFormat>(numFormatInt);
const auto imgFormat = static_cast<El::FileFormat>(imgFormatInt);
const auto colorMap = static_cast<El::ColorMap>(colorMapInt);
El::SetColorMap( colorMap );
const El::Complex<double>
center(realCenter,imagCenter),
uniformCenter(uniformRealCenter,uniformImagCenter);
bool isReal = true;
std::string matName;
El::DistMatrix<double> AReal(grid);
El::DistMatrix<El::Complex<double>> ACpx(grid);
switch( matType )
{
case 0: matName="uniform";
El::Uniform( ACpx, n, n, uniformCenter, uniformRadius );
isReal = false;
break;
case 1: matName="Haar";
El::Haar( ACpx, n );
isReal = false;
break;
case 2: matName="Lotkin";
El::Lotkin( AReal, n );
isReal = true;
break;
case 3: matName="Grcar";
El::Grcar( AReal, n, numBands );
isReal = true;
break;
case 4: matName="FoxLi";
El::FoxLi( ACpx, n, omega );
isReal = false;
break;
case 5: matName="HelmholtzPML";
El::HelmholtzPML
( ACpx, n, El::Complex<double>(omega), numPmlPoints, sigma,
pmlExp );
isReal = false;
break;
case 6: matName="HelmholtzPML2D";
El::HelmholtzPML
( ACpx, mx, my, El::Complex<double>(omega), numPmlPoints, sigma,
pmlExp );
isReal = false;
break;
case 7: matName="TrefethenEmbree";
El::TrefethenEmbree( ACpx, n );
isReal = false;
break;
case 8: matName="BullsHead";
El::BullsHead( ACpx, n );
isReal = false;
break;
case 9: matName="Triangle";
El::Triangle( AReal, n );
isReal = true;
break;
case 10: matName="Whale";
El::Whale( ACpx, n );
isReal = false;
break;
case 11: matName="UniformHelmholtzGreens";
El::UniformHelmholtzGreens( ACpx, n, lambda );
isReal = false;
break;
case 12: matName="HatanoNelson";
El::HatanoNelson
( AReal, n, realCenter, uniformRadius, gHatano, periodic );
isReal = true;
break;
case 13: matName="EhrenfestDecay";
// Force the complex matrix to allow for one-norm pseudospectra
El::EhrenfestDecay( ACpx, n );
isReal = false;
break;
case 14: matName="RiffleDecay";
// Force the complex matrix to allow for one-norm pseudospectra
El::RiffleDecay( ACpx, n );
isReal = false;
break;
case 15: matName="Jordan";
El::Jordan( AReal, n, 0. );
isReal = true;
break;
default: El::LogicError("Invalid matrix type");
}
if( display )
{
if( isReal )
El::Display( AReal, "A" );
else
El::Display( ACpx, "A" );
}
if( write )
{
if( isReal )
{
El::Write( AReal, "A", numFormat );
El::Write( AReal, "A", imgFormat );
}
else
{
El::Write( ACpx, "A", numFormat );
El::Write( ACpx, "A", imgFormat );
}
}
// Begin by computing the Schur decomposition
El::Timer timer;
El::DistMatrix<El::Complex<double>> w(grid);
El::mpi::Barrier( comm );
const bool formATR = true;
El::DistMatrix<double> QReal(grid);
El::DistMatrix<El::Complex<double>> QCpx(grid);
El::SchurCtrl<double> ctrl;
ctrl.hessSchurCtrl.fullTriangle = formATR;
ctrl.hessSchurCtrl.blockHeight = nbDist;
ctrl.hessSchurCtrl.scalapack = false;
ctrl.useSDC = sdc;
// Spectral D&C options (only relevant if 'sdc' is true)
ctrl.sdcCtrl.cutoff = cutoff;
ctrl.sdcCtrl.maxInnerIts = maxInnerIts;
ctrl.sdcCtrl.maxOuterIts = maxOuterIts;
ctrl.sdcCtrl.tol = sdcTol;
ctrl.sdcCtrl.spreadFactor = spreadFactor;
ctrl.sdcCtrl.random = random;
ctrl.sdcCtrl.progress = progress;
ctrl.sdcCtrl.signCtrl.tol = signTol;
ctrl.sdcCtrl.signCtrl.progress = progress;
timer.Start();
if( isReal )
{
if( psNorm == El::PS_TWO_NORM )
El::Schur( AReal, w, ctrl );
else
El::Schur( AReal, w, QReal, ctrl );
}
else
{
if( psNorm == El::PS_TWO_NORM )
El::Schur( ACpx, w, ctrl );
else
El::Schur( ACpx, w, QCpx, ctrl );
}
El::mpi::Barrier( comm );
const double schurTime = timer.Stop();
if( El::mpi::Rank(comm) == 0 )
El::Output("Schur decomposition took ",schurTime," seconds");
if( saveSchur )
{
if( El::mpi::Rank(comm) == 0 )
El::Output("Writing Schur decomposition to file...");
timer.Start();
if( isReal )
{
auto schurTitle =
El::BuildString
(matName,"_",AReal.ColStride(),"x",AReal.RowStride(),
"_",AReal.DistRank());
El::Write( AReal.LockedMatrix(), schurTitle, El::BINARY );
if( psNorm == El::PS_ONE_NORM )
{
auto QTitle =
El::BuildString
(matName,"_Q_",QReal.ColStride(),"x",QReal.RowStride(),
"_",QReal.DistRank());
El::Write( QReal.LockedMatrix(), QTitle, El::BINARY );
}
}
else
{
auto schurTitle =
El::BuildString
(matName,"_",ACpx.ColStride(),"x",ACpx.RowStride(),
"_",ACpx.DistRank());
El::Write( ACpx.LockedMatrix(), schurTitle, El::BINARY );
if( psNorm == El::PS_ONE_NORM )
{
auto QTitle =
El::BuildString
(matName,"_Q_",QCpx.ColStride(),"x",QCpx.RowStride(),
"_",QCpx.DistRank());
El::Write( QCpx.LockedMatrix(), QTitle, El::BINARY );
}
}
El::mpi::Barrier( comm );
const double saveSchurTime = timer.Stop();
if( El::mpi::Rank(comm) == 0 )
El::Output("Saving took ",saveSchurTime," seconds");
}
// Find a window if none is specified
if( realWidth == 0. || imagWidth == 0. )
{
const double radius = El::MaxNorm( w );
const double oneNorm =
isReal ? El::OneNorm(AReal) : El::OneNorm(ACpx);
double width;
if( oneNorm == 0. && radius == 0. )
{
width = 1;
if( El::mpi::Rank(comm) == 0 )
El::Output("Setting width to 1 to handle zero matrix");
}
else if( radius >= 0.2*oneNorm )
{
width = 2.5*radius;
if( El::mpi::Rank(comm) == 0 )
El::Output
("Setting width to ",width,
" based on the spectral radius, ",radius);
}
else
{
width = 0.8*oneNorm;
if( El::mpi::Rank(comm) == 0 )
El::Output
("Setting width to ",width," based on the one norm, ",
oneNorm);
}
realWidth = width;
imagWidth = width;
}
El::PseudospecCtrl<double> psCtrl;
psCtrl.norm = psNorm;
psCtrl.schur = true;
psCtrl.maxIts = maxIts;
psCtrl.tol = psTol;
psCtrl.deflate = deflate;
psCtrl.arnoldi = arnoldi;
psCtrl.basisSize = basisSize;
psCtrl.progress = progress;
psCtrl.snapCtrl.imgSaveFreq = imgSaveFreq;
psCtrl.snapCtrl.numSaveFreq = numSaveFreq;
psCtrl.snapCtrl.imgDispFreq = imgDispFreq;
psCtrl.snapCtrl.imgFormat = imgFormat;
psCtrl.snapCtrl.numFormat = numFormat;
psCtrl.snapCtrl.itCounts = itCounts;
// Visualize/write the pseudospectra within each window
El::DistMatrix<double> invNormMap(grid);
El::DistMatrix<El::Int> itCountMap(grid);
const El::Int xBlock = realSize / numReal;
const El::Int yBlock = imagSize / numImag;
const El::Int xLeftover = realSize - (numReal-1)*xBlock;
const El::Int yLeftover = imagSize - (numImag-1)*yBlock;
const double realStep = realWidth/realSize;
const double imagStep = imagWidth/imagSize;
const El::Complex<double> corner =
center - El::Complex<double>(realWidth/2,imagWidth/2);
for( El::Int realChunk=0; realChunk<numReal; ++realChunk )
{
const El::Int realChunkSize =
( realChunk==numReal-1 ? xLeftover : xBlock );
const double realChunkWidth = realStep*realChunkSize;
for( El::Int imagChunk=0; imagChunk<numImag; ++imagChunk )
{
auto chunkTag = El::BuildString("_",realChunk,"_",imagChunk);
const El::Int imagChunkSize =
( imagChunk==numImag-1 ? yLeftover : yBlock );
const double imagChunkWidth = imagStep*imagChunkSize;
const El::Complex<double> chunkCorner = corner +
El::Complex<double>
(realStep*realChunk*xBlock,imagStep*imagChunk*yBlock);
const El::Complex<double> chunkCenter = chunkCorner +
El::Complex<double>
(realStep*realChunkSize,imagStep*imagChunkSize)/2.;
if( El::mpi::Rank(comm) == 0 )
El::Output
("Starting computation for chunk centered at ",chunkCenter);
El::mpi::Barrier( comm );
timer.Start();
psCtrl.snapCtrl.numBase = matName+"_"+numBase+chunkTag;
psCtrl.snapCtrl.imgBase = matName+"_"+imgBase+chunkTag;
if( isReal )
{
itCountMap =
El::QuasiTriangularSpectralWindow
( AReal, QReal, invNormMap, chunkCenter,
realChunkWidth, imagChunkWidth,
realChunkSize, imagChunkSize, psCtrl );
}
else
{
itCountMap =
El::TriangularSpectralWindow
( ACpx, QCpx, invNormMap, chunkCenter,
realChunkWidth, imagChunkWidth,
realChunkSize, imagChunkSize, psCtrl );
}
El::mpi::Barrier( comm );
const double pseudoTime = timer.Stop();
const El::Int numIts = MaxNorm( itCountMap );
if( El::mpi::Rank() == 0 )
El::Output
("num seconds=",pseudoTime,"\n",
"num iterations=",numIts);
}
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}
int
main( int argc, char* argv[] )
{
El::Environment env( argc, argv );
try
{
const El::Int m = El::Input("--m","height of matrix",100);
const El::Int n = El::Input("--n","width of matrix",200);
const bool useIPM = El::Input("--useIPM","use Interior Point?",true);
// TODO(poulson): Add options for controlling IPM
const El::Int maxIter =
El::Input("--maxIter","maximum # of iter's",500);
const Real rho = El::Input("--rho","augmented Lagrangian param.",1.);
const Real alpha = El::Input("--alpha","over-relaxation",1.2);
const Real absTol = El::Input("--absTol","absolute tolerance",1e-6);
const Real relTol = El::Input("--relTol","relative tolerance",1e-4);
const Real pinvTol = El::Input("--pinvTol","pseudoinv tolerance",0.);
const bool usePinv =
El::Input("--usePinv","Directly compute pinv(A)",true);
const bool progress = El::Input("--progress","print progress?",true);
const bool display = El::Input("--display","display matrices?",false);
const bool print = El::Input("--print","print matrices",false);
El::ProcessInput();
El::PrintInputReport();
El::DistMatrix<Real> A, b, xTrue;
El::Uniform( A, m, n );
El::Uniform( b, m, 1 );
if( print )
{
El::Print( A, "A" );
El::Print( b, "b" );
}
if( display )
El::Display( A, "A" );
const bool sparse = false;
El::BPCtrl<Real> ctrl(sparse);
ctrl.useIPM = useIPM;
ctrl.admmCtrl.rho = rho;
ctrl.admmCtrl.alpha = alpha;
ctrl.admmCtrl.maxIter = maxIter;
ctrl.admmCtrl.absTol = absTol;
ctrl.admmCtrl.relTol = relTol;
ctrl.admmCtrl.usePinv = usePinv;
ctrl.admmCtrl.pinvTol = pinvTol;
ctrl.admmCtrl.progress = progress;
ctrl.lpIPMCtrl.mehrotraCtrl.print = true;
El::DistMatrix<Real> x;
El::Timer timer;
if( El::mpi::Rank() == 0 )
timer.Start();
El::BP( A, b, x, ctrl );
if( El::mpi::Rank() == 0 )
timer.Stop();
if( print )
El::Print( x, "x" );
const El::Int xZeroNorm = El::ZeroNorm( x );
if( El::mpi::Rank() == 0 )
{
El::Output("Basis Pursuit time: ",timer.Total()," secs");
El::Output("|| x ||_0 = ",xZeroNorm);
}
SoftThreshold( x, El::Sqrt(El::limits::Epsilon<Real>()) );
if( print )
El::Print( x, "xThresh" );
const El::Int xZeroNormThresh = El::ZeroNorm( x );
if( El::mpi::Rank() == 0 )
{
El::Output("|| xThresh ||_0 = ",xZeroNormThresh);
}
}
catch( std::exception& e ) { El::ReportException(e); }
return 0;
}