void
LocalSparseTriangularSolver<MatrixType>::
localApply (const MV& X,
MV& Y,
const Teuchos::ETransp mode,
const scalar_type& alpha,
const scalar_type& beta) const
{
using Teuchos::RCP;
typedef scalar_type ST;
typedef Teuchos::ScalarTraits<ST> STS;
if (beta == STS::zero ()) {
if (alpha == STS::zero ()) {
Y.putScalar (STS::zero ()); // Y := 0 * Y (ignore contents of Y)
}
else { // alpha != 0
A_crs_->template localSolve<ST, ST> (X, Y, mode);
if (alpha != STS::one ()) {
Y.scale (alpha);
}
}
}
else { // beta != 0
if (alpha == STS::zero ()) {
Y.scale (beta); // Y := beta * Y
}
else { // alpha != 0
MV Y_tmp (Y, Teuchos::Copy);
A_crs_->template localSolve<ST, ST> (X, Y_tmp, mode); // Y_tmp := M * X
Y.update (alpha, Y_tmp, beta); // Y := beta * Y + alpha * Y_tmp
}
}
}
/// \brief Solve AX=B for X with Chebyshev iteration with left
/// diagonal scaling, imitating ML's implementation.
///
/// \pre A must be real-valued and symmetric positive definite.
/// \pre numIters >= 0
/// \pre eigRatio >= 1
/// \pre 0 < lambdaMax
/// \pre All entries of D_inv are positive.
///
/// \param A [in] The matrix A in the linear system to solve.
/// \param B [in] Right-hand side(s) in the linear system to solve.
/// \param X [in] Initial guess(es) for the linear system to solve.
/// \param numIters [in] Number of Chebyshev iterations.
/// \param lambdaMax [in] Estimate of max eigenvalue of D_inv*A.
/// \param lambdaMin [in] Estimate of min eigenvalue of D_inv*A. We
/// only use this to determine if A is the identity matrix.
/// \param eigRatio [in] Estimate of max / min eigenvalue ratio of
/// D_inv*A. We use this along with lambdaMax to compute the
/// Chebyshev coefficients. This need not be the same as
/// lambdaMax/lambdaMin.
/// \param D_inv [in] Vector of diagonal entries of A. It must have
/// the same distribution as b.
void
mlApplyImpl (const MAT& A,
const MV& B,
MV& X,
const int numIters,
const ST lambdaMax,
const ST lambdaMin,
const ST eigRatio,
const V& D_inv)
{
const ST zero = Teuchos::as<ST> (0);
const ST one = Teuchos::as<ST> (1);
const ST two = Teuchos::as<ST> (2);
MV pAux (B.getMap (), B.getNumVectors ()); // Result of A*X
MV dk (B.getMap (), B.getNumVectors ()); // Solution update
MV R (B.getMap (), B.getNumVectors ()); // Not in original ML; need for B - pAux
ST beta = Teuchos::as<ST> (1.1) * lambdaMax;
ST alpha = lambdaMax / eigRatio;
ST delta = (beta - alpha) / two;
ST theta = (beta + alpha) / two;
ST s1 = theta / delta;
ST rhok = one / s1;
// Diagonal: ML replaces entries containing 0 with 1. We
// shouldn't have any entries like that in typical test problems,
// so it's OK not to do that here.
// The (scaled) matrix is the identity: set X = D_inv * B. (If A
// is the identity, then certainly D_inv is too. D_inv comes from
// A, so if D_inv * A is the identity, then we still need to apply
// the "preconditioner" D_inv to B as well, to get X.)
if (lambdaMin == one && lambdaMin == lambdaMax) {
solve (X, D_inv, B);
return;
}
// The next bit of code is a direct translation of code from ML's
// ML_Cheby function, in the "normal point scaling" section, which
// is in lines 7365-7392 of ml_smoother.c.
if (! zeroStartingSolution_) {
// dk = (1/theta) * D_inv * (B - (A*X))
A.apply (X, pAux); // pAux = A * X
R = B;
R.update (-one, pAux, one); // R = B - pAux
dk.elementWiseMultiply (one/theta, D_inv, R, zero); // dk = (1/theta)*D_inv*R
X.update (one, dk, one); // X = X + dk
} else {
dk.elementWiseMultiply (one/theta, D_inv, B, zero); // dk = (1/theta)*D_inv*B
X = dk;
}
ST rhokp1, dtemp1, dtemp2;
for (int k = 0; k < numIters-1; ++k) {
A.apply (X, pAux);
rhokp1 = one / (two*s1 - rhok);
dtemp1 = rhokp1*rhok;
dtemp2 = two*rhokp1/delta;
rhok = rhokp1;
R = B;
R.update (-one, pAux, one); // R = B - pAux
// dk = dtemp1 * dk + dtemp2 * D_inv * (B - pAux)
dk.elementWiseMultiply (dtemp2, D_inv, B, dtemp1);
X.update (one, dk, one); // X = X + dk
}
}
void
Chebyshev<MatrixType>::
applyImpl (const MV& X,
MV& Y,
Teuchos::ETransp mode,
scalar_type alpha,
scalar_type beta) const
{
using Teuchos::ArrayRCP;
using Teuchos::as;
using Teuchos::RCP;
using Teuchos::rcp;
using Teuchos::rcp_const_cast;
using Teuchos::rcpFromRef;
const scalar_type zero = STS::zero();
const scalar_type one = STS::one();
// Y = beta*Y + alpha*M*X.
// If alpha == 0, then we don't need to do Chebyshev at all.
if (alpha == zero) {
if (beta == zero) { // Obey Sparse BLAS rules; avoid 0*NaN.
Y.putScalar (zero);
}
else {
Y.scale (beta);
}
return;
}
// If beta != 0, then we need to keep a copy of the initial value of
// Y, so that we can add beta*it to the Chebyshev result at the end.
// Usually this method is called with beta == 0, so we don't have to
// worry about caching Y_org.
RCP<MV> Y_orig;
if (beta != zero) {
Y_orig = rcp (new MV (Y));
}
// If X and Y point to the same memory location, we need to use a
// copy of X (X_copy) as the input MV. Otherwise, just let X_copy
// point to X.
//
// This is hopefully an uncommon use case, so we don't bother to
// optimize for it by caching X_copy.
RCP<const MV> X_copy;
bool copiedInput = false;
if (X.getLocalMV().getValues() == Y.getLocalMV().getValues()) {
X_copy = rcp (new MV (X));
copiedInput = true;
}
else {
X_copy = rcpFromRef (X);
}
// If alpha != 1, fold alpha into (a copy of) X.
//
// This is an uncommon use case, so we don't bother to optimize for
// it by caching X_copy. However, we do check whether we've already
// copied X above, to avoid a second copy.
if (alpha != one) {
RCP<MV> X_copy_nonConst = rcp_const_cast<MV> (X_copy);
if (! copiedInput) {
X_copy_nonConst = rcp (new MV (X));
copiedInput = true;
}
X_copy_nonConst->scale (alpha);
X_copy = rcp_const_cast<const MV> (X_copy_nonConst);
}
impl_.apply (*X_copy, Y);
if (beta != zero) {
Y.update (beta, *Y_orig, one); // Y = beta * Y_orig + 1 * Y
}
}