QDWHInfo QDWH( Matrix<F>& A, const QDWHCtrl& ctrl )
{
EL_DEBUG_CSE
typedef Base<F> Real;
const Real twoEst = TwoNormEstimate( A );
A *= 1/twoEst;
// The one-norm of the inverse can be replaced with an estimate which is
// a few times cheaper, e.g., via Higham and Tisseur's block algorithm
// from "A Block Algorithm for Matrix 1-Norm Estimation, with an Application
// to 1-Norm Pseudospectra".
Real sMinUpper;
Matrix<F> Y( A );
if( A.Height() > A.Width() )
{
qr::ExplicitTriang( Y );
try
{
TriangularInverse( UPPER, NON_UNIT, Y );
sMinUpper = Real(1) / OneNorm( Y );
} catch( SingularMatrixException& e ) { sMinUpper = 0; }
}
else
{
try
{
Inverse( Y );
sMinUpper = Real(1) / OneNorm( Y );
} catch( SingularMatrixException& e ) { sMinUpper = 0; }
}
return QDWHInner( A, sMinUpper, ctrl );
}
inline void
CholeskyUVar2( Matrix<F>& A )
{
#ifndef RELEASE
PushCallStack("hpd_inverse::CholeskyUVar2");
if( A.Height() != A.Width() )
throw std::logic_error("Nonsquare matrices cannot be triangular");
#endif
// Matrix views
Matrix<F>
ATL, ATR, A00, A01, A02,
ABL, ABR, A10, A11, A12,
A20, A21, A22;
// Start the algorithm
PartitionDownDiagonal
( A, ATL, ATR,
ABL, ABR, 0 );
while( ATL.Height() < A.Height() )
{
RepartitionDownDiagonal
( ATL, /**/ ATR, A00, /**/ A01, A02,
/*************/ /******************/
/**/ A10, /**/ A11, A12,
ABL, /**/ ABR, A20, /**/ A21, A22 );
//--------------------------------------------------------------------//
Cholesky( UPPER, A11 );
Trsm( RIGHT, UPPER, NORMAL, NON_UNIT, F(1), A11, A01 );
Trsm( LEFT, UPPER, ADJOINT, NON_UNIT, F(1), A11, A12 );
Herk( UPPER, NORMAL, F(1), A01, F(1), A00 );
Gemm( NORMAL, NORMAL, F(-1), A01, A12, F(1), A02 );
Herk( UPPER, ADJOINT, F(-1), A12, F(1), A22 );
Trsm( RIGHT, UPPER, ADJOINT, NON_UNIT, F(1), A11, A01 );
Trsm( LEFT, UPPER, NORMAL, NON_UNIT, F(-1), A11, A12 );
TriangularInverse( UPPER, NON_UNIT, A11 );
Trtrmm( ADJOINT, UPPER, A11 );
//--------------------------------------------------------------------//
SlidePartitionDownDiagonal
( ATL, /**/ ATR, A00, A01, /**/ A02,
/**/ A10, A11, /**/ A12,
/*************/ /******************/
ABL, /**/ ABR, A20, A21, /**/ A22 );
}
#ifndef RELEASE
PopCallStack();
#endif
}
Int ADMM
( const Matrix<Real>& A,
const Matrix<Real>& b,
const Matrix<Real>& c,
Matrix<Real>& z,
const ADMMCtrl<Real>& ctrl )
{
EL_DEBUG_CSE
// Cache a custom partially-pivoted LU factorization of
// | rho*I A^H | = | B11 B12 |
// | A 0 | | B21 B22 |
// by (justifiably) avoiding pivoting in the first n steps of
// the factorization, so that
// [I,rho*I] = lu(rho*I).
// The factorization would then proceed with
// B21 := B21 U11^{-1} = A (rho*I)^{-1} = A/rho
// B12 := L11^{-1} B12 = I A^H = A^H.
// The Schur complement would then be
// B22 := B22 - B21 B12 = 0 - (A*A^H)/rho.
// We then factor said matrix with LU with partial pivoting and
// swap the necessary rows of B21 in order to implicitly commute
// the row pivots with the Gauss transforms in the manner standard
// for GEPP. Unless A A' is singular, pivoting should not be needed,
// as Cholesky factorization of the negative matrix should be valid.
//
// The result is the factorization
// | I 0 | | rho*I A^H | = | I 0 | | rho*I U12 |,
// | 0 P22 | | A 0 | | L21 L22 | | 0 U22 |
// where [L22,U22] are stored within B22.
Matrix<Real> U12, L21, B22, bPiv;
Adjoint( A, U12 );
L21 = A;
L21 *= 1/ctrl.rho;
Herk( LOWER, NORMAL, -1/ctrl.rho, A, B22 );
MakeHermitian( LOWER, B22 );
// TODO: Replace with sparse-direct Cholesky version?
Permutation P2;
LU( B22, P2 );
P2.PermuteRows( L21 );
bPiv = b;
P2.PermuteRows( bPiv );
// Possibly form the inverse of L22 U22
Matrix<Real> X22;
if( ctrl.inv )
{
X22 = B22;
MakeTrapezoidal( LOWER, X22 );
FillDiagonal( X22, Real(1) );
TriangularInverse( LOWER, UNIT, X22 );
Trsm( LEFT, UPPER, NORMAL, NON_UNIT, Real(1), B22, X22 );
}
Int numIter=0;
const Int m = A.Height();
const Int n = A.Width();
Matrix<Real> g, xTmp, y, t;
Zeros( g, m+n, 1 );
PartitionDown( g, xTmp, y, n );
Matrix<Real> x, u, zOld, xHat;
Zeros( z, n, 1 );
Zeros( u, n, 1 );
Zeros( t, n, 1 );
while( numIter < ctrl.maxIter )
{
zOld = z;
// Find x from
// | rho*I A^H | | x | = | rho*(z-u)-c |
// | A 0 | | y | | b |
// via our cached custom factorization:
//
// |x| = inv(U) inv(L) P' |rho*(z-u)-c|
// |y| |b |
// = |rho*I U12|^{-1} |I 0 | |I 0 | |rho*(z-u)-c|
// = |0 U22| |L21 L22| |0 P22'| |b |
// = " " |rho*(z-u)-c|
// | P22' b |
xTmp = z;
xTmp -= u;
xTmp *= ctrl.rho;
xTmp -= c;
y = bPiv;
Gemv( NORMAL, Real(-1), L21, xTmp, Real(1), y );
if( ctrl.inv )
{
Gemv( NORMAL, Real(1), X22, y, t );
y = t;
}
else
{
Trsv( LOWER, NORMAL, UNIT, B22, y );
Trsv( UPPER, NORMAL, NON_UNIT, B22, y );
}
Gemv( NORMAL, Real(-1), U12, y, Real(1), xTmp );
xTmp *= 1/ctrl.rho;
// xHat := alpha*x + (1-alpha)*zOld
xHat = xTmp;
xHat *= ctrl.alpha;
Axpy( 1-ctrl.alpha, zOld, xHat );
// z := pos(xHat+u)
z = xHat;
z += u;
LowerClip( z, Real(0) );
// u := u + (xHat-z)
u += xHat;
u -= z;
const Real objective = Dot( c, xTmp );
// rNorm := || x - z ||_2
t = xTmp;
t -= z;
const Real rNorm = FrobeniusNorm( t );
// sNorm := |rho| || z - zOld ||_2
t = z;
t -= zOld;
const Real sNorm = Abs(ctrl.rho)*FrobeniusNorm( t );
const Real epsPri = Sqrt(Real(n))*ctrl.absTol +
ctrl.relTol*Max(FrobeniusNorm(xTmp),FrobeniusNorm(z));
const Real epsDual = Sqrt(Real(n))*ctrl.absTol +
ctrl.relTol*Abs(ctrl.rho)*FrobeniusNorm(u);
if( ctrl.print )
{
t = xTmp;
LowerClip( t, Real(0) );
t -= xTmp;
const Real clipDist = FrobeniusNorm( t );
cout << numIter << ": "
<< "||x-z||_2=" << rNorm << ", "
<< "epsPri=" << epsPri << ", "
<< "|rho| ||z-zOld||_2=" << sNorm << ", "
<< "epsDual=" << epsDual << ", "
<< "||x-Pos(x)||_2=" << clipDist << ", "
<< "c'x=" << objective << endl;
}
if( rNorm < epsPri && sNorm < epsDual )
break;
++numIter;
}
if( ctrl.maxIter == numIter )
cout << "ADMM failed to converge" << endl;
x = xTmp;
return numIter;
}
inline void
Inverse( DistMatrix<F>& A )
{
#ifndef RELEASE
PushCallStack("Inverse");
if( A.Height() != A.Width() )
throw std::logic_error("Cannot invert non-square matrices");
#endif
const Grid& g = A.Grid();
DistMatrix<int,VC,STAR> p( g );
LU( A, p );
TriangularInverse( UPPER, NON_UNIT, A );
// Solve inv(A) L = inv(U) for inv(A)
DistMatrix<F> ATL(g), ATR(g),
ABL(g), ABR(g);
DistMatrix<F> A00(g), A01(g), A02(g),
A10(g), A11(g), A12(g),
A20(g), A21(g), A22(g);
DistMatrix<F> A1(g), A2(g);
DistMatrix<F,VC, STAR> A1_VC_STAR(g);
DistMatrix<F,STAR,STAR> L11_STAR_STAR(g);
DistMatrix<F,VR, STAR> L21_VR_STAR(g);
DistMatrix<F,STAR,MR > L21Trans_STAR_MR(g);
DistMatrix<F,MC, STAR> Z1(g);
PartitionUpDiagonal
( A, ATL, ATR,
ABL, ABR, 0 );
while( ABR.Height() < A.Height() )
{
RepartitionUpDiagonal
( ATL, /**/ ATR, A00, A01, /**/ A02,
/**/ A10, A11, /**/ A12,
/*************/ /******************/
ABL, /**/ ABR, A20, A21, /**/ A22 );
View( A1, A, 0, A00.Width(), A.Height(), A01.Width() );
View( A2, A, 0, A00.Width()+A01.Width(), A.Height(), A02.Width() );
L21_VR_STAR.AlignWith( A2 );
L21Trans_STAR_MR.AlignWith( A2 );
Z1.AlignWith( A01 );
Z1.ResizeTo( A.Height(), A01.Width() );
//--------------------------------------------------------------------//
// Copy out L1
L11_STAR_STAR = A11;
L21_VR_STAR = A21;
L21Trans_STAR_MR.TransposeFrom( L21_VR_STAR );
// Zero the strictly lower triangular portion of A1
MakeTrapezoidal( LEFT, UPPER, 0, A11 );
Zero( A21 );
// Perform the lazy update of A1
internal::LocalGemm
( NORMAL, TRANSPOSE,
F(-1), A2, L21Trans_STAR_MR, F(0), Z1 );
A1.SumScatterUpdate( F(1), Z1 );
// Solve against this diagonal block of L11
A1_VC_STAR = A1;
internal::LocalTrsm
( RIGHT, LOWER, NORMAL, UNIT, F(1), L11_STAR_STAR, A1_VC_STAR );
A1 = A1_VC_STAR;
//--------------------------------------------------------------------//
Z1.FreeAlignments();
L21Trans_STAR_MR.FreeAlignments();
L21_VR_STAR.FreeAlignments();
SlidePartitionUpDiagonal
( ATL, /**/ ATR, A00, /**/ A01, A02,
/*************/ /*******************/
/**/ A10, /**/ A11, A12,
ABL, /**/ ABR, A20, /**/ A21, A22 );
}
// inv(A) := inv(A) P
ApplyInverseColumnPivots( A, p );
#ifndef RELEASE
PopCallStack();
#endif
}
inline void
Inverse( Matrix<F>& A )
{
#ifndef RELEASE
PushCallStack("Inverse");
if( A.Height() != A.Width() )
throw std::logic_error("Cannot invert non-square matrices");
#endif
Matrix<int> p;
LU( A, p );
TriangularInverse( UPPER, NON_UNIT, A );
// Solve inv(A) L = inv(U) for inv(A)
Matrix<F> ATL, ATR,
ABL, ABR;
Matrix<F> A00, A01, A02,
A10, A11, A12,
A20, A21, A22;
Matrix<F> A1, A2;
Matrix<F> L11,
L21;
PartitionUpDiagonal
( A, ATL, ATR,
ABL, ABR, 0 );
while( ABR.Height() < A.Height() )
{
RepartitionUpDiagonal
( ATL, /**/ ATR, A00, A01, /**/ A02,
/**/ A10, A11, /**/ A12,
/*************/ /******************/
ABL, /**/ ABR, A20, A21, /**/ A22 );
View( A1, A, 0, A00.Width(), A.Height(), A01.Width() );
View( A2, A, 0, A00.Width()+A01.Width(), A.Height(), A02.Width() );
//--------------------------------------------------------------------//
// Copy out L1
L11 = A11;
L21 = A21;
// Zero the strictly lower triangular portion of A1
MakeTrapezoidal( LEFT, UPPER, 0, A11 );
Zero( A21 );
// Perform the lazy update of A1
Gemm( NORMAL, NORMAL, F(-1), A2, L21, F(1), A1 );
// Solve against this diagonal block of L11
Trsm( RIGHT, LOWER, NORMAL, UNIT, F(1), L11, A1 );
//--------------------------------------------------------------------//
SlidePartitionUpDiagonal
( ATL, /**/ ATR, A00, /**/ A01, A02,
/*************/ /*******************/
/**/ A10, /**/ A11, A12,
ABL, /**/ ABR, A20, /**/ A21, A22 );
}
// inv(A) := inv(A) P
ApplyInverseColumnPivots( A, p );
#ifndef RELEASE
PopCallStack();
#endif
}
